Length derivative of the generating series of walks confined in the quarter plane
Thomas Dreyfus Charlotte Hardouin













 
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counterintro
thmintroTheorem
corointro[thmintro]Corollary

*propintroProposition

thmTheorem[section]
cor[thm]Corollary
lemma[thm]Lemma
prop[thm]Proposition
stepStep
conj[thm]Conjecture

definition
defi[thm]Definition 

remark
rmk[thm]Remark
ex[thm]Example 


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    t-derivative transcendence of the generating function of walks]Length derivative of the generating function of  walks confined in the quarter planeInstitut de Recherche Mathématique Avancée, U.M.R. 7501 Université de Strasbourg et C.N.R.S. 7, rue René Descartes 67084 Strasbourg, FRANCEdreyfus@math.unistra.frUniversité Paul Sabatier - Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse.hardouin@math.univ-toulouse.frThis project has been partially founded by ANR De rerum natura project (ANR-19-CE40-0018). The second author would like to thank the ANR-11-LABX-0040-CIMI within
    the program ANR-11-IDEX-0002-0 for its partial support.[2010]05A15,30D05,39A06amsalpha 
    In the present paper, we use difference Galois theory  to study the nature of the generating function  counting walks with small steps in the quarter plane. These series are trivariate formal power series
    Q(x,y,t) that count the number of walks  confined in the first quadrant of the plane with a fixed  set of admissible steps, called the model of the walk. While the variables x and y are associated to the ending point of the path, the variable t encodes its length. In this paper, we prove 
    that in the unweighted case, Q(x,y,t) satisfies an algebraic differential relation with respect to t if and only if it satisfies an algebraic differential relation with respect  x (resp. y). Combined   with <cit.>,   we  are able to characterize the t-differential transcendence of the 79 models of walks  listed by Bousquet-Mélou  and Mishna.
    [
        Charlotte Hardouin
        December 30, 2023
    ======================
    
    
    
    
    
    § INTRODUCTION
    
    
    Classifying   lattice walks in restricted domains is an important problem in enumerative combinatorics. Recently much progress has been 
    made in the study of walks with small steps in the quarter plane. A small steps model in the quarter plane _≥ 0×_≥ 0 is composed by a set of admissible cardinal directions 𝒟⊂{[scale=0.3]
    ;
    [thick,->](0,0)–(-1,0);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(-1,1);
    
     ,
     [scale=0.3]
    ;
    [thick,->](0,0)–(0,1);
    
     ,[scale=0.3]
    ;
    [thick,->](0,0)–(1,1);
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(1,0);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(1,-1);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(0,-1);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(-1,-1);
    }. Given 𝒟, we consider the walks that start at (0,0), with directions in 𝒟, and that stay in the quarter plane, for instance:
    
    [scale=.8, baseline=(current bounding box.center)]
     in 0,1,2,3,4,5,6,7,8,9,10 in 0,1,2,3,4(,) circle[radius=0pt];
     (0,0)–(10,0);
     (0,0)–(0,4);
    [->](0,0)–(1,1);
    [->](1,1)–(1,0);
    [->](1,0)–(0,1);
    [->](0,1)–(1,2);
    [->](1,2)–(2,1);
    [->](2,1)–(2,0);
    [->](2,0)–(3,1);
    [->](3,1)–(3,0);
    [->](3,0)–(4,1);
    [->](4,1)–(3,2);
    [->](3,2)–(2,3);
    [->](2,3)–(2,2);
    [->](2,2)–(3,3);
    [->](3,3)–(4,2);
    [->](4,2)–(4,1);
    [->](4,1)–(5,0);
    [->](5,0)–(6,1);
    [->](6,1)–(6,0);
    [->](6,0)–(7,1);
    [->](7,1)–(8,0);
    [->](8,0)–(9,1);
    [->](9,1)–(9,0);
    [->](9,0)–(10,1);
    [->](10,1)–(9,2);
    [->](9,2)–(8,3);
    [->](8,3)–(8,2);
        𝒟={[scale=.4, baseline=(current bounding box.center)]
     in -1,0,1 in -1,0,1(,) circle[radius=2pt];
    [thick,->](0,0)–(-1,1);
    [thick,->](0,0)–(1,1);
    [thick,->](0,0)–(1,-1);
    [thick,->](0,0)–(0,-1);
    }
    
    
     For a given model, one defines q_i,j,k to be the number of walks confined to the first quadrant of the plane that 
    begin at (0,0) and end at (i,j) in k admissible steps.  The algebraic nature of the  associated complete generating function Q(x,y,t)=∑_i,j,k=0^∞q_i,j,kx^iy^jt^k captures many important combinatorial properties of the model: symmetries,  asymptotic information, and  recursive relations of the coefficients.
    
    Among the 2^8-1=255 models  in the first quadrant of the plane, Bousquet-Mélou  and Mishna proved in <cit.> that, after accounting for symmetries and eliminating the trivial and one dimensional cases, only 79 cases remained.  It is worth mentioning that the generating function is algebraic in all the trivial and one dimensional cases. 
    
    
    For any choice of a variable  ⋆ among x,y,t, one classifies the algebraic nature of the generating series Q(x,y,t) with respect to ⋆
    as follows:
    
      * Algebraic cases: the series Q(x,y,t)  satisfies a nontrivial polynomial relation with coefficients in (x,y,t). 
    
      * Transcendental ⋆-holonomic cases: the series Q(x,y,t) is transcendental and holonomic with respect to ⋆, i.e. there exists n∈_≥ 0, such that  there exist a_0,…,a_n∈ (x,y,t), not 	all zero, such that 
    
        0 =∑_ℓ=0^na_ℓd/d ⋆^ℓQ(x,y,t).
    
      * Nonholonomic d/d⋆-differentially algebraic cases: the series Q(x,y,t) is nonholonomic and d/d ⋆-differentially algebraic, i.e. 
    there exists n∈_≥ 0, such that there exists nonzero multivariate polynomial P_⋆∈(x,y,t)[X_0,…,X_n], such that 
    
        0=P_⋆(Q(x,y,t),…,d/d ⋆^nQ(x,y,t)).
    
       We stress out the fact that in the above definition, it is equivalent to  require that  P_⋆∈[X_0,…,X_n], see Remark <ref>.
    
      * d/d ⋆-differentially transcendental cases: the series is  not d/d ⋆-differentially algebraic.
    
    
    The authors of <cit.>   proved that the  algebraic nature of the generating series was identical for the variables x and y. The classification of the  models of walks regarding the algebraic nature of their series with respect to the variables x and y  is the culmination of ten years of research and the works of many researchers  (see Figure <ref> below). 
    
    
    
    
     §.§ Statement of the main result
    
    In this paper, we  address the question of the classification with respect to the variable t and we prove that this classification coincides with the classification with respect to x and y. There is a priori no relation between the d/dx and d/dt differential algebraic properties of a function in these two variables. For instance,   the function tΓ(x) is holonomic with respect to t but not differentially algebraic with respect to x, thanks to Hlder's result. In that case, the fibration induced by t is “isotrivial”. The main difficulty in our case is to show that such a situation does not happen and that the x and t-algebraic behavior are intrinsically connected.
    
    
    
    
     Our main result is as follows:
    
     [Theorem <ref> and Corollary <ref> below]
    For any of the 79 models of Figure <ref>,  the complete generating function
      is d/dt-differentially algebraic  over  if and only if 
     it is d/dx-differentially algebraic over . 
     
     
    Theorem <ref> is the corollary  of the following proposition proved in the more general setting of 
    weighted walks that are walks whose directions are weighted  (see <ref>). To any such a  walk, one attaches an algebraic curve of genus zero or one called the kernel curve and a group of automorphisms of that curve called the group of the walk (see <ref>). The following holds.
    
    [Theorems <ref> and <ref> below]
    For a  genus zero kernel curve attached to the models (<ref>), the generating series is d/dt-differentially transcendental over . For a genus one kernel curve with infinite group of the walk,  if the generating series is d/dt-differentially algebraic over , then it is d/dx-differentially algebraic over .
    
    
    In <cit.>, the authors proved that for a genus zero kernel curve attached to the models (<ref>), the generating series was d/dx-differentially transcendental over .
    The authors of <cit.> proved that the nine nonholomic d/dx-differentially algebraic models  of Figure <ref> were also d/dt-differentially algebraic over  by giving an   explicit description of the series in terms of  analytic invariants. In <ref>, we will discuss how the construction of <cit.> and  the results of <cit.> should imply that  the second statement of  Theorem <ref> is in fact an equivalence. 
    
    
    
     §.§ Strategy of the proof
    
    
    The classification results of Figure <ref> come from  many approaches: probabilistic methods,  combinatorial classification,  computer algebra and “Guess and Prove”,  analysis and boundary value problems, and more recently  difference Galois theory and algebraic geometry. The 
    analytic approach consists in studying the asymptotic growth of the coefficients of the generating function, or else showing that it has an infinite number of singularities, in order to prove its nonholonomicity. This approach also allows for the study of some important specializations of the complete generating function as for instance Q(1,1,t) the generating function for the number of nearest neighbor walks in the quarter plane (see <cit.>). Though very powerful, these analytic techniques are unable to detect the differentially algebraic generating functions among the nonholonomic ones. For instance, the generating function ∏_k=1^∞1/(1-x^k) counting the number of partitions has an infinite number of singularities, and yet is d/dx-differentially algebraic.  
    
    In order to detect these more subtle kinds of functional dependencies it is necessary to use new arguments that focus on the functional equation satisfied by the complete generating function.
    Indeed, the combinatorial decomposition of a walk into a shorter walk followed by an admissible step translates into a functional equation for the generating function.   Following the ideas of Fayolle, Iasnogorodski and Malyshev  <cit.>, the authors of <cit.> and <cit.> specialized this functional equation  to the so-called kernel curve to find a linear discrete equation: a linear -difference equation in the genus zero case and a shift difference equation in genus one. Difference Galois theory allowed then to characterize the differentially transcendental complete generating function (<cit.>)  whereas the clever use of Tutte invariants produces explicit differential algebraic relations  for the 9 nonholomic but differentially algebraic cases (<cit.>). 
    Unfortunately, all the above methods for proving the differential transcendence are only valid for a fixed value of the parameter t in the field of complex numbers. This allowed 
    the authors to consider the kernel curve as a complex algebraic curve but prevented them to study the variations of the parameter t.
    
    Our work relies on  an nonarchimedean uniformization of the kernel curve, which we consider as an algebraic curve over (t). 
    We use here the formalism of Tate curves over (t) as in   <cit.> to show that for both situations, genus one and zero, the differential algebraic properties of the complete generating functions are encoded by the differential algebraic properties of a solution of a rank one  nonhomogeneous linear -difference equation which  unifies the genus zero and the genus one cases. 
    Then, we generalize some Galoisian criterias for -difference equations of <cit.> to  
     prove Theorem <ref> and Theorem <ref> below.
    
    
    
     §.§ Organization of the paper
    
    The paper is organized as follows. In Section <ref> we present some reminders and notations for  walks in the quarter plane. In Section <ref> we consider  walks with genus zero kernel curve while  Section <ref>  deals with  the genus one case. Since this paper combines many different fields, nonarchimedian uniformization, combinatorics, and Galois theory, we choose to postpone many technical  intermediate results to the appendices. This should allow the reader to understand the articulation of our proofs in Sections <ref> and <ref>  in three steps without being lost in  too many  technicalities. These three steps are the uniformization of the kernel and the  construction  of a linear -difference equation, the Galoisian criteria, and  finally, the resolution of telescoping problems. Appendix <ref> is devoted to the nonarchimedean estimates that we used in the uniformization procedure. Appendix <ref> contains some reminders on  special functions on Tate curves and their normal forms. Appendix <ref> proves  the Galoisian criteria mentioned above.  Finally, Appendix <ref> studies the transcendence properties of  special functions on Tate curves which  will be used for the descent of our telescoping equations.
     
    
    
    
    
    § THE WALKS IN THE QUADRANT
    
    
    
    The goal of this section is to introduce some basic properties of walks in the quarter plane. In <ref>, we introduce  the generating function Q(x,y,t) of a walk confined in the quarter plane. In <ref>, we attach to any  walk a kernel curve, which is an algebraic curve 
    defined over [t]. This curve has been intensively studied as an algebraic curve over  by fixing a morphism from [t] to . For instance,  <cit.> is concerned
    with t=1 whereas the papers <cit.> and <cit.> focus respectively on t ∈ transcendental over  and t ∈ ]0,1[. Unfortunately, 
    specializing t even generically does not allow to study the t-dependencies of the generating function. In this paper, we do not work with a specialization of t. This 
    forces us to  move away from the archimedean framework of the field of complex numbers  and to consider the kernel curve over a suitable valued field extension
    of (t) endowed with the valuation at 0. 
    
    
    
    
    
     §.§ The walks
    
    The cardinal directions  of the plane  {[scale=0.3]
    ;
    [thick,->](0,0)–(-1,0);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(-1,1);
    
     ,
     [scale=0.3]
    ;
    [thick,->](0,0)–(0,1);
    
     ,[scale=0.3]
    ;
    [thick,->](0,0)–(1,1);
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(1,0);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(1,-1);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(0,-1);
    
    ,
    [scale=0.3]
    ;
    [thick,->](0,0)–(-1,-1);
    }   are identified with  pairs of integers  (i,j)∈{0,± 1}^2\{(0,0)}. A walk 𝒲 in the quarter plane _≥ 0^2 is a sequence of points (M_n)_n ∈_≥ 0 such that 
    
      *  it starts at (0,0), that is, M_0=(0,0);
    
      *  for all n ∈_≥ 0, the point M_n belong to the quadrant _≥ 0×_≥ 0;
    
      *   for all n ∈_≥ 0, the vector M_n+1-M_n belongs to a given subset 𝒟 of the  set of cardinal directions.
    
    Fixing  a family of elements (d_i,j)_(i,j)∈{0,± 1}^2 of ∩ [0,1] such that ∑_i,j d_i,j=1, one can choose to weight the model of the walk in order to add a probabilistic flavor to our study. For (i,j)∈{0,± 1}^2\{(0,0)} (resp. (0,0)), the  element   d_i,j can be viewed as  the probability for the walk  to go in the direction (i,j) (resp.  to stay  at the same position). In that case,  the d_i,j are called the weights and the model is called a weighted model.   The set of steps𝒟  of the walk  is  the 
    set of cardinal  directions with nonzero weight, that is, 
    
        𝒟={(i,j) ∈{0,± 1}^2| d_i,j≠ 0 }.
    
    
    
    A  model  is unweighted if d_0,0=0 and if the nonzero d_i,j's all have the same value. 
    
    In what follows we will represent model of walks with arrows. For instance, the family of models  represented by 
    
        [scale=0.6, baseline=(current bounding box.center)]
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(1,-1);
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(0,1);
         or {[scale=.6, baseline=(current bounding box.center)]
        [thick,->](0,0)–(1,1);
        , [scale=.6, baseline=(current bounding box.center)]
        [thick,->](0,0)–(1,-1);
        , [scale=.6, baseline=(current bounding box.center)]
        [thick,->](0,0)–(0,1);
        , [scale=.6, baseline=(current bounding box.center)]
        [thick,->](0,0)–(-1,-1);
        },
    
    correspond to models with d_1,1,d_1,-1,d_0,1,d_-1,-1≠ 0, d_1,0=d_0,-1=d_-1,1=d_-1,0=0, and where nothing is assumed on the value of d_0,0. In the following results, 
     the behavior of the  kernel curve never depends on d_0,0. This is the reason why, to reduce the amount of notations, we have decided not to mention d_0,0 in the graphical representation of the model.
    
    The  weight of the walk is defined to be the product of the weights of its component steps. For any (i,j)∈_≥ 0^2 and any k∈_≥ 0, we let q_i,j,k be the sum of the weights of all walks reaching  the position (i,j) from the initial position (0,0) after k steps.  We introduce the corresponding trivariate generating function
    
        Q(x,y,t):=∑_i,j,k≥ 0q_i,j,kx^iy^jt^k.
    
    
    Note that the generating function is not exactly the same as the one that we defined in the introduction. To recover the latter, we should take  d_i,j∈{0,1} and d_i,j=1 if and only if  the corresponding direction belongs to 𝒟. Fortunately, the assumption ∑_i,j d_i,j=1 can be relaxed by rescaling the t-variable, and the results of the present paper stay valid for the generating function of the introduction since both generating functions have the same nature.
     
    For simplicity, we assume that the weights d_i,j belong to  . However, we would like to mention that any of the arguments and statements  below will hold with arbitrary real weights in [0,1]. One just needs to  replace the field  with the field (d_i,j). 
    
    The  kernel polynomial of a  weighted model  (d_i,j)_i,j∈{0,± 1}^2  is defined by 
    
        K(x,y,t):=xy (1-t S(x,y))
    
    
    
    where 
    
        [                         S(x,y)                              = ∑_(i,j)∈{0,± 1}^2 d_i,jx^i y^j;                                                             =  A_-1(x) 1/y +A_0(x)+ A_1(x) y;                                                             = B_-1(y) 1/x +B_0(y)+ B_1(y) x, ]
    
    
    
    and A_i(x) ∈ x^-1[x], B_i(y) ∈ y^-1[y].
    
    By  <cit.>, see also <cit.>, the generating function Q(x,y,t) satisfies the following functional equation:
    
        K(x,y,t)Q(x,y,t)=xy+F^1(x,t) +F^2(y,t)+td_-1,-1 Q(0,0,t),
    
    
    
    where 
    
        F^1(x,t):= K(x,0,t)Q(x,0,t),   F^2(y,t):= K(0,y,t)Q(0,y,t).
    
    We shall often use the following symmetry argument between x and y. Exchanging  x and y in the kernel polynomial amounts to consider the kernel  polynomial  of a weighted model  𝒟':={ (i,j)   (j,i) ∈𝒟} with  weights d'_i,j:=d_j,i.
    
     §.§ The kernel curve
    
    
    The kernel polynomial may be seen as a bivariate polynomial in x,y with coefficients in (t). The latter is a  valued field  endowed with the valuation at zero. It  is neither  algebraically closed   nor complete. In order to use the theory of Tate curves, one needs to consider a complete algebraically closed field  extension of (t). The field of Puiseux series with coefficients in  is algebraically closed but not complete. We may consider the field C of Hahn series or  Malcev-Neumann series with coefficients in  , and monomials from . We recall that  a  Hahn series f
    is a formal power series ∑_γ∈c_γ t^γ with coefficients c_γ in  and such that the subset {γ | c_γ≠ 0 } is a well ordered subset of . The valuation v_0( f) of f is the smallest element of the subset {γ | c_γ≠ 0 }. The field C is algebraically closed and complete 
    with respect to the valuation at zero, see <cit.>. One can endow 
    C with a derivation ∂_t  as follows
    
        ∂_t(∑_γ∈c_γ t^γ)= ∑_γ∈c_γγ t^γ.
    
    Then, ∂_t extends the derivation td/dt of (t), see <cit.>. 
    
    Let us fix once for all α∈ such that 0< α <1. For any f ∈ C, we define the norm of f as |f|= α^v_0(f). For any  Hahn series f such that |f|<1, we have |∂_t (f)| <1. This is not true when ∂_t is replaced by d/dt.
    
    
     We need to discard some degenerate cases. Following <cit.>, we have the following  definition. 
     
     
    A weighted model is called degenerate if one of the following holds:
    
      * K(x,y,t) is reducible as an element of the polynomial ring C[x,y], 
    
      * K(x,y,t) has x-degree less than or equal to 1,
    
      * K(x,y,t) has y-degree less than or equal to 1.
    
     In <cit.>, the authors specialize the variable t
     as a transcendental complex number. Then, they study the kernel curve as a complex algebraic curve in 1() ×1(). In this work, we shall use any algebraic geometric result
     of <cit.> by appealing to Lefschetz Principle:  every true statement  about  an algebraic variety defined  over    remains true when  is replaced by an algebraically closed field of characteristic zero.
     
    The following proposition gives very simple conditions on 𝒟 to decide whether  a weighted model is degenerate or not.
    
     
    A weighted model is degenerate if and only if at least one of the following holds:
    
      * There exists i∈{- 1,1} such that d_i,-1=d_i,0=d_i,1=0. This corresponds to walks with steps supported in one of the following configurations
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(0,-1);
        [thick,->](0,0)–(1,-1);
        [thick,->](0,0)–(1,0);
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(0,1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(0,-1);
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(-1,0);
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
    
      *  There exists j∈{-1, 1} such that d_-1,j=d_0,j=d_1,j=0. This corresponds to walks with steps supported in one of the following configurations
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,0);
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(0,-1);
        [thick,->](0,0)–(1,-1);
        [thick,->](0,0)–(1,0);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,0);
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(1,0);
    
      *  All the weights are zero except maybe  {d_1,1,d_0,0,d_-1,-1} or  {d_-1,1,d_0,0,d_1,-1}. This corresponds to walks with steps supported in one of the following configurations
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(1,1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(1,-1);
        [thick,->](0,0)–(-1,1);
    
     
     Note that we only discard one dimensional problems as explained in <cit.>. For all the degenerate cases, the generating function Q(x,y,t) is algebraic.From now on, we shall always assume that the weighted model under consideration is nondegenerate.
    
    
    To any weighted model, we attach a curve E, called the kernel curve, that is defined as  the zero set in 1(C)×1(C) of the following homogeneous polynomial
    
        K(x_0,x_1,y_0,y_1,t)= x_0x_1y_0y_1 -t ∑_i,j=0^2 d_i-1,j-1 x_0^i x_1^2-iy_0^j y_1^2-j=x_1^2y_1^2K(x_0/x_1,y_0/y_1,t).
    
     
     
     
     Let us write K(x_0,x_1,y_0,y_1,t)=∑_i,j=0^2 A_i,jx_0^ix_1^2-iy_0^jy_1^2-j where A_i,j= -t d_i-1,j-1 if (i,j) ≠ (1,1) and A_1,1= 1-td_0,0.  The partial discriminants of K(x_0,x_1,y_0,y_1,t) are defined as the discriminants of the second degree homogeneous polynomials y ↦K(x_0,x_1,y,1,t) and  x ↦K(x,1,y_0,y_1,t), respectively, i.e.
    
        Δ_x (x_0,x_1)= (∑_i=0^2 x_0^i x_1^2-i A_i,1)^2-4(∑_i=0^2 x_0^i x_1^2-i A_i,0)×(∑_i=0^2 x_0^i x_1^2-i A_i,2)
    
    and
    
        Δ_y (y_0,y_1)= (∑_j=0^2 y_0^jy_1^2-j A_1,j)^2-4(∑_j=0^2 y_0^j y_1^2-j A_0,j)×(∑_j=0^2 y_0^j y_1^2-j A_2,j).
    
    
    
    Introduce
    
        𝔇(x):=Δ_x(x,1)=∑_j=0^4α_jx^j  and  𝔈(y):=Δ_y(y,1)=∑_j=0^4β_jy^j,
        
    where
    
        [                                                                               α_4                                                                                 =                                                         (d_1,0^2-4d_1,1d_1,-1)t^2;                                                                               α_3                                                                                 =                              2t^2d_1,0d_0,0-2td_1,0-4t^2(d_0,1d_1,-1+d_1,1d_0,-1);                                                                               α_2                                                                                 = 1+t^2d_0,0^2+2t^2d_-1,0d_1,0-4t^2(d_-1,1d_1,-1+d_0,1d_0,-1+d_1,1d_-1,-1) -2td_0,0;                                                                               α_1                                                                                 =                          2t^2d_-1,0d_0,0-2td_-1,0-4t^2(d_-1,1d_0,-1+d_0,1d_-1,-1);                                                                               α_0                                                                                 =                                                    ( d_-1,0^2-4d_-1,1d_-1,-1) t^2;                                                                                  ;                                                                               β_4                                                                                 =                                                         (d_0,1^2-4d_1,1d_-1,1)t^2;                                                                               β_3                                                                                 =                              2t^2d_0,1d_0,0-2td_0,1-4t^2(d_1,0d_-1,1+d_1,1d_-1,0);                                                                               β_2                                                                                 = 1+t^2d_0,0^2+2t^2d_0,-1d_0,1-4t^2(d_1,-1d_-1,1+d_1,0d_-1,0+d_1,1d_-1,-1) -2td_0,0;                                                                               β_1                                                                                 =                          2t^2d_0,-1d_0,0-2td_0,-1-4t^2(d_1,-1d_-1,0+d_1,0d_-1,-1);                                                                               β_0                                                                                 =                                                   ( d_0,-1^2-4d_1,-1d_-1,-1) t^2. ]
    
    
    The discriminants  Δ_x (x_0,x_1), Δ_y (y_0,y_1) are homogeneous polynomials of degree 4.  
     Their Eisenstein invariants can be defined as follows:
     [ 2.3.5 in <cit.>]
     For any homogeneous polynomial of the form 
        f(x_0,x_1)= a_0 x_1^4 +4a_1 x_0x_1^3 +6a_2 x_0^2 x_1^2 +4 a_3 x_0^3x_1 +a_4 x_0^4∈ C[x_0,x_1],
     we define the Eisenstein invariants of f(x_0,x_1) as 
      * D(f)= a_0a_4 +3a_2^2-4a_1a_3
      * E(f)=a_0a_3^2+a_1^2a_4-a_0a_2a_4-2a_1a_2a_3+a_2^3
      * F(f)= 27 E(f)^2-D(f)^3.
    
    
    
    
     Since C is algebraically closed  of characteristic zero, we can apply <cit.> to the kernel curve. The following proposition characterizes the smoothness  of the kernel curve  in terms of the    invariants F(Δ_x), F(Δ_y).
     
     
     	The following statements are equivalent 
     	
      *  The  kernel curve E is smooth, i.e. it has no singular point;
     	
      * F(Δ_x) ≠ 0;
     	
      * F(Δ_y) ≠ 0.
    	
     	 
     Furthermore, if   E is smooth then it is an elliptic curve with  J-invariant given by the element J(E)∈ C such that 
        J(E)=12^3 D(Δ_y)^3/-F(Δ_y).
     
    Otherwise, if E is nondegenerate and singular,  E has a unique singular point and is a genus zero curve.
     
     
    
     
    We define the genus of a weighted model  as the genus of the associated kernel curve E.  We recall the results obtained in <cit.> and <cit.>,  that classify  all the weighted models  attached to a genus zero kernel.
     
      Any nondegenerate  weighted model of genus zero has steps included in one of the following 4 sets of steps:
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(1,0);
        
        
        [thick,->](0,0)–(1,-1);
        
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        
        
        [thick,->](0,0)–(1,1);
        
        [thick,->](0,0)–(1,0);
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(0,-1);
        [thick,->](0,0)–(1,-1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(-1,0);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(-1,-1);
        
        
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        
        
        [thick,->](0,0)–(-1,0);
        
        [thick,->](0,0)–(-1,-1);
        [thick,->](0,0)–(0,-1);
        [thick,->](0,0)–(1,-1);
    
    Otherwise, for any other nondegenerate  weighted model, the kernel curve E is an elliptic curve. 
    
    The walks corresponding to the fourth configuration never enter the quarter-plane.  As described in <cit.>, if we consider walks corresponding to the second and third configurations we are in the situation where one of the quarter plane constraints implies the other. In the last three configurations, the generating function is algebraic. So the only interesting  nondegenerate genus zero weighted models have steps included in
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(1,1);
        [thick,->](0,0)–(1,0);
        
        
        [thick,->](0,0)–(1,-1);
    
    Note that due to Proposition <ref>, the anti-diagonal steps  have nonzero attached weights.
    
    Moreover, by Theorem <ref>, combined with Proposition <ref>, the nondegenerate weighted models of genus one are the walks where there are no three consecutive  cardinal directions with weight zero. Or equivalently, this corresponds to the situation where the  set of steps is not included in any half plane (See (<ref>) below).
     
     
    Thanks to Theorem <ref>, one can  reduce our study to two cases depending on the genus of the kernel curve attached to 
     a nondegenerate weighted model. The following lemma proves that when the kernel curve is of genus one, its  J-invariant  has modulus strictly greater than 1. This property 
     allows us   to use the theory of Tate curves in order to analytically uniformize the kernel curve.
    
    
     
    When E is smooth, the invariant J(E) belongs to  (t)  and is such that |J(E)|>1, where  | | denotes the norm of (C, | |).
     
    At t=0, Δ_y (y_0,y_1) reduces  to y_0^2y_1^2.  This proves that the reduction of D(Δ_y) (resp. E(Δ_y))  at t=0 is 1/12 (resp. 1/6^3). One concludes that  F(Δ_y) vanishes for t=0.
    By Proposition <ref>, J(E) ∈(t) has a strictly negative valuation at t=0. Thus, |J(E)|>1. 
     
    
     §.§ The automorphism of the walk
    
    
    Following <cit.> or <cit.>, we introduce the involutive birational transformations of 1(C)×1(C) given by 
    
        i_1(x,y) =(x, A_-1(x) /A_1(x)y)  and   i_2(x,y)=(B_-1(y)/B_1(y)x,y),
     
    (see <ref> for the significance of the A_i,B_i's).
    
     They induce two involutive automorphisms 
    ι_1, ι_2 : E  E given by 
    
        [                                                                                 ι_1([x_0: x_1],[y_0:y_1])                                                    = ([x_0: x_1], [A_-1(x_0/x_1) A_1(x_0/x_1)y_0/y_1:1]),;                                                  and                            ι_2([x_0: x_1],[y_0:y_1])                                                    =    ([B_-1(y_0/y_1)B_1(y_0/y_1)x_0/x_1:1],[y_0:y_1]). ]
     
    
    
    Note that ι_1 and ι_2 are nothing but the vertical and horizontal  switches of E, see Figure <ref>. That is,  for any P=(x,y) ∈ E, we have 
    
        {P,ι_1(P)} = E ∩ ({x}×1(C))
         and {P,ι_2(P)} = E ∩ (1(C) ×{y}).
    
    
    
    
    The automorphism of the walk σ is defined
     by 
    
        σ=ι_2 ∘ι_1.
     
    
    The following  holds.
    
    
    Let P ∈ E. The following statements are equivalent: 
    
      * P is fixed by σ;
    
      * P is fixed by ι_1 and ι_2;
    
      * P is the only singular point of E, and  E is of genus zero. 
    
    
    
    
    
    § GENERATING FUNCTIONS FOR WALKS, GENUS ZERO CASE
    
     
    
     In this section, we fix a  nondegenerate weighted model of genus zero. Following Remark <ref>, after eliminating duplications of trivial cases and  the interchange of x and y, we should focus  on  walks   arising  from the following 5 sets of steps:
    
    
        [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        
        
        
        
        
        [thick,->](0,0)–(1,-1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        
        [thick,->](0,0)–(1,1);
        
        
        
        
        [thick,->](0,0)–(1,-1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(1,1);
        
        
        
        
        [thick,->](0,0)–(1,-1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        [thick,->](0,0)–(1,1);
        
        [thick,->](0,0)–(1,0);
        
        
        [thick,->](0,0)–(1,-1);
          [scale=.4, baseline=(current bounding box.center)]
         in -1,0,1 in -1,0,1(,) circle[radius=2pt];
        [thick,->](0,0)–(-1,1);
        [thick,->](0,0)–(0,1);
        
        
        [thick,->](0,0)–(1,0);
        
        
        [thick,->](0,0)–(1,-1);
        G0
     
    
    A function f(x,y,t)∈[[x,y,t]] is (d/dx,d/dt)-differentially algebraic over  if there exists  a nonzero polynomial P with coefficients in  such that P(f(x,y,t), d/dx f(x,y,t),d/dt f(x,y,t)   , …)=0. The function  f(x,y,t) is (d/dx,d/dt)-differentially transcendental over  otherwise. Note that if f is d/dt-differentially algebraic over  then it is (d/dx,d/dt)-differentially algebraic over . We define similarly the notion of   (d/dy,d/dt)-differential algebraicity.
    
     In this section, we   prove the following theorem: 
    
    For any   weighted model  listed in  (<ref>), the generating function Q(x,0,t) is (d/dx,d/dt)-differentially transcendental over . 
    For any   weighted model  listed in (<ref>), the generating function Q(0,y,t) is (d/dy,d/dt)-differentially transcendental over .
    
    
     Theorem  <ref> implies the d/dt-differential transcendence of the complete generating function.
    
    For any  weighted model  listed in (<ref>), the generating function Q(x,y,t) is (d/dx,d/dt) and  (d/dy,d/dt)-differentially transcendental over . Therefore, Q(x,y,t) is d/dt-differentially transcendental over .
    
    Suppose to the contrary  that Q(x,y,t) is (d/dx,d/dt)-algebraic over . Let P  be a nonzero polynomial with coefficients in  such that P(Q(x,y,t), d/dx Q(x,y,t),d/dt Q(x,y,t)   , …)=0. Specializing  at y=0 this  relation and noting that d ^i/dx^id^j/dt^j(Q(x,0,t)) is the specialization of d ^i/dx^id^j/dt^j(Q(x,y,t)), one finds  a nontrivial 
    differential algebraic relations for Q(x,0,t) in the derivatives d/dx and d/dt. This contradicts   Theorem <ref>. The proof for the (d/dy,d/dt)-differential transcendence is similar.
    
    
    
    As detailed in the introduction, our proof has three major steps: 
    
      Step 1: we attach to the incomplete generating functions Q(x,0,t) and Q(0,y,t)  some  auxiliary functions which share the same differential behavior than the generating series but
    satisfy simple -difference equations. This is done via the uniformization of the kernel curve (see <ref> and <ref>).
    
      Step 2: we apply difference Galois theory to the -difference equations satisfied by the auxiliary functions in order to relate the differential algebraicity of the incomplete generating functions to the existence of  telescoping relations. These telescoping relations are of the form (<ref>) below.
    
      Step 3: we prove that there is no such telescoping relation. This allows us to conclude that the generating series is  d/dt-differentially transcendental over   (see <ref>).
    
    
    
    
     §.§ Uniformization of the kernel curve
    
    
    With the notation of <ref>,  especially (<ref>),  any  weighted model   listed in (<ref>)  satisfies α_0=α_1=β_0=β_1=0. Moreover, since the  weighted model  is nondegenerate,  one finds that the  product d_1,-1d_-1,1 is nonzero. Furthermore, 
        -1+d_0,0t±√((1-d_0,0t)^2-4d_1,-1d_-1,1t^2)≠ 0.
    
    
    The uniformization of the kernel curve  of a weighted model  listed in (<ref>) is given by the following proposition.
    
    Let us consider a weighted model  listed in (<ref>) and let E be its kernel curve. There exist λ∈ C^* and a parametrization ϕ :1(C) → E with 
        ϕ(s)=(x(s),y(s))=
        (4_2√(_3^2-4_2_4)( s +1/s) -2_3, 
        4_̱2√(_̱3^2-4_̱2_̱4)( s/λ+λ/s) -2_̱3),
     such that
    
      * ϕ: 1(C) ∖{ 0, ∞}→ E ∖{(0,0) } is a bijection and ϕ^-1((0,0))={0,∞};
    
      *  The automorphisms ι_1,ι_2,σ of E induce  automorphisms _1,_2,σ_ of 1 (C) viaϕ that satisfy _1(s)=1/s, _2 (s)= /s, σ_ (s)= s, with λ^2=∈{,^-1} and 
        =-1+d_0,0t-√((1-d_0,0t)^2-4d_1,-1d_-1,1t^2)-1+d_0,0t+√((1-d_0,0t)^2-4d_1,-1d_-1,1t^2)∈ C^*.
     Thus,  we have the commutative diagrams 
     
        E  @->[r]^ι_k    E  
        ℙ^1(C) @->[u]^ϕ@->[r]__k   ℙ^1(C) @->[u]_ϕ and 
            E  @->[r]^σ    E  
        ℙ^1(C) @->[u]^ϕ@->[r]_σ_   ℙ^1(C) @->[u]_ϕ
    
    
    
    
    
    
    
    
    The following estimate on the norm of  holds:
    
    
    We have  || >1.
    
    We consider the expansion as a Puiseux series of . It is then easily seen that its valuation is negative, which gives  ||>1.
    
    
     §.§ Meromorphic continuation of the generating functions
    
    
    In this paragraph,  we combine the functional equation (<ref>) with the uniformization of the kernel curve obtained 
    above to meromorphically continue the generating function.  
    We define the norm of an element b=[b_0:b_1] ∈1(C) as follows: if b_1 ≠ 0, we set |b|=|b_0/b_1| and |[1:0]|=∞ by convention.
    Since  |t|<1, 
     the generating function Q(x,y,t)  as well as F^1(x,t),F^2(y,t) converge for  any  (x,y) ∈1(C) ×1(C) such that 
    |x| and |y| are smaller than or equal to 1. On that domain, they satisfy
    
        K(x,y,t)Q(x,y,t)=xy+F^1(x,t) +F^2(y,t)+td_-1,-1 Q(0,0,t).
    
    
    
    We claim that there exist two  positive real numbers c_0,c_∞ such that ϕ maps the disks U_0={ s ∈1(C) | |s|< c_0} and U_∞={ s ∈1(C) | |s| > c_∞}  into   the domain  defined by  { 
    (x,y) ∈ E  |x| ≤ 1 |y|≤ 1 }. Indeed,
     the _i and _̱i are of norm smaller than or equal to 1 and |_2|=1 (see (<ref>)). Thus, if  |s|< min(1, |√(_3^2-4_2_4)|), then 
    
        |x(s)|=|4_2s√(_3^2-4_2_4)( s^2 +1) -2_3s|=|4α_2 s |/|√(_3^2-4_2_4)|  < 1.
    
    An analogous reasoning  for y(s) shows that when |s| is sufficiently small,  we find  |x(s)|,|y(s)| ≤ 1. Similarly, one can  prove that, when |s| is sufficiently big, one has  |x(s)|,|y(s)| ≤ 1.   This proves our claim.
    
    We set F̆^1(s)=F^1(x(s),t) and F̆^2(s)=F^2(y(s),t). Based on  the above, these functions are  well  defined on U_0 ∪ U_∞. Evaluating (<ref>) for (x,y)=(x(s),y(s)), one finds
    
        0=x(s)y(s)+F̆^1(s) +F̆^2(s)+td_-1,-1 Q(0,0,t).
    
    
    The following lemma shows that one can use the above equation to meromorphically continue
    the functions F̆^i(s)  so that they satisfy a -difference equation.
    
    
    
    
    
    For i=1,2, the restriction of the function F̆^i(s) to U_0 can be continued to a meromorphic function F^i(s) on C such that
    
        F^1( s)-F^1(s)= b_1=(x( s) -x( s))y( s)
    
    and 
    
        F^2( s)-F^2(s)=b_2=(y(  s) -y(s))x(s).
    
    We just give a sketch of a proof  since  the arguments are the exact analogue in our ultrametric context  of those  employed in <cit.>.
    Since _1 (s)=1/s and _2(s)=/s, we can assume  without  loss of generality that _1(U_0) ⊂ U_∞ and _2(U_∞) ⊂ U_0. 
    Then one can evaluate  (<ref>) at any s ∈ U_0. We obtain 
    
        0=x(s)y(s)+F̆^1(s) +F̆^2(s)+td_-1,-1 Q(0,0,t).
    
    Evaluating (<ref>) at _1(s) ∈ U_∞, we find 
    
        0=x(_1( s))y(_1( s))+F̆^1(_1( s)) +F̆^2(_1(s))+td_-1,-1 Q(0,0,t).
    
    Using the invariance of x(s) (resp. y(s)) with respect to _1 (resp. _2), the second equation is 
    
        0=x(s)y( s)+F̆^1(s) +F̆^2( s)+td_-1,-1 Q(0,0,t).
    
    Subtracting this last equation to the first, we find that, for any s ∈ U_0, we have  
    
        F̆^2( s)-F̆^2(s)= (y( s) -y(s))x(s).
     By Lemma <ref>, the norm of 
    is strictly greater than one  and therefore  the norm of || is distinct from 1. This allows us 
    to use (<ref>) to meromorphically continue F̆^2  to  C
    so that it satisfies (<ref>) everywhere. The proof for F̆^1 is similar.
    
    Note that,  for i=1,2,   the function F^i(s)   does not coincide  a priori with F̆^i(s) in the neighborhood of infinity. 
    
    
    
     §.§ Differential transcendence in the genus zero case
    
    
    
    We recall that any holomorphic function f on C^* can be represented as an everywhere convergent Laurent series with coefficients in C, see <cit.>. Moreover any nonzero meromorphic function on C^* can be written as the quotient of two holomorphic functions on C^* with no common zeros.  We denote by er(C^*) the field of meromorphic functions over C^* and  by _ the -difference operator that maps a meromorphic function g(s) onto g( s). Finally, let  C_ be the 
    the field  formed by the  meromorphic functions over C^*  fixed by _. 
    We now define the -logarithm. If ||>1,  the Jacobi Theta function is the meromorphic function defined by 
    θ_(s)=∑_n∈^-n(n+1)/2s^n∈ er(C^*). It satisfies the the -difference equation
    
        θ_( s)=sθ_(s).
    
    Its logarithmic derivative ℓ_(s)=∂_s(θ_)/θ_∈ er(C^*) satisfies
     ℓ_( s)=ℓ_( s)+1. If ||<1 then the meromorphic function  -ℓ_1/ is solution of _ (-ℓ_1/)=-ℓ_1/+1.  Abusing the  notation, we still denote by ℓ_ the  function -ℓ_1/  when ||<1.   
    Since we want to use the -difference equations of Lemma <ref> as a constraint
       for the form of the differential algebraic relations satisfied by the functions  F^i(s), we need to consider
       derivations that are compatible with σ_ in the sense that they commute with σ_. This is not the case for the derivation ∂_t=td/d t. By Lemma <ref>, the derivations ∂_s=s d/ds and Δ_t,=∂_t()ℓ_(s)∂_s +∂_t commute
    with _. The  following lemma relates the  differential transcendence of the incomplete generating functions  Q(x,0,t) and Q(0,y,t) 
    to the differential transcendence	of the auxiliary functions F^i (s). We refer to Definition <ref> for the notion of (∂_s,Δ_t,)-differential algebraicity over a field.
    
    
    If  the generating function  Q(x,0,t) is (d/dx,d/dt)-differentially algebraic over , then F^1(s) is (∂_s,Δ_t,)-differentially algebraic over K=C_(s, ℓ_(s)).  If  the generating function is Q(0,y,t) is (d/dy,d/dt)-differentially algebraic over , then F^2(s) is (∂_s,Δ_t,)-differentially algebraic over K=C_(s, ℓ_(s)).
    
    The statement being symmetrical in x and y, we  prove it only for  Q(x,0,t).
    Assume  that the generating function is Q(x,0,t) is (d/dx,d/dt)-differentially algebraic over .  Since F^1(x,t) is the product of Q(x,0,t) by the polynomial K(x,0,t)∈[x,t], the function F^1(x,t)  is (d/dx,d/dt)-differentially algebraic over . It is therefore (d/dx,∂_t)-differentially algebraic over (t), and finally (d/dx,∂_t)-differentially algebraic over , since t is ∂_t-differentially algebraic over .  Remember that  F^1(s) coincides with  F^1(x(s),t) for s ∈ U_0 where x(s) is defined thanks to  Proposition <ref>. Therefore, we need  to understand the relations between the x and t derivatives of F^1(x,t) and the derivatives of 
    F^1(x(s),t)  with respect to ∂_s and Δ_t,.
    
    
    
     Let us study these relations for an arbitrary  bivariate function  G(x,t) which   converges on |x|,|y| ≤ 1. Denote   by  δ_x  the derivation d/dx and  by G(s)= G(x(s),t).  From the equality (∂_s G(s))=∂_s(x(s))(δ_x G)(x(s),t), we conclude  that 
    
        ∂_t( G(s))=(∂_t G)(x(s),t)+∂_t(x(s))(δ_x G)(x(s),t)=
        (∂_t G)(x(s),t)+  c ∂_s(G(s)),
     where c =∂_t(x(s))/∂_s(x(s)). The element c belongs to K because x(s) ∈K and K is stable by ∂_s, Δ_t, and  thereby by ∂_t= Δ_t,- ∂_t()ℓ_(s)∂_s, see  Lemma <ref>. An easy induction shows that 
    
        (∂_t^n G)(x(s),t)= ∂_t^n(G(s)) + ∑_i ≤ n ,j <n b_i,j∂_t^j∂_s^i(G(s)),
    
    where the b_i,j's belong to  K. By Lemma <ref>, we have ∂_s Δ_t,-Δ_t,∂_s=f∂_s,  where  f=∂_t() ∂_s(ℓ_)∈K.
    Combining (<ref>) with  ∂_t= Δ_t,- ∂_t()ℓ_(s)∂_s, we find that 
    
        (∂_t^n G)(x(s),t)= Δ_t,^n(G(s)) + ∑_i ≤ 2n,j <n d_i,jΔ_t,^j∂_s^i(G(s)),
    
    for some  d_i,j's  in K.
    Moreover, an easy induction shows that, for any m ∈^*, we have  
    
        (δ^m_x G)(x(s),t)= 1/∂_s(x(s))^m∂_s^m(G(s)) + ∑_i=1^m-1 a_i ∂_s^i(G(s)),
    
    where a_i ∈K.
    Applying (<ref>) with G replaced by δ^m_x G, we find that  for every m,n∈,
    
        (∂_t^n δ^m_x G)(x(s),t)=Δ_t,^n((δ^m_x G) (x(s),t)  ) + ∑_i ≤ 2n,j <n d_i,jΔ_t,^j∂_s^i((δ^m_x G )(x(s),t) ).
    
    Combining this equation with (<ref>), we conclude that
    
        (∂_t^n δ^m_x G)(x(s),t)=
         1/∂_s(x(s))^mΔ_t,^n ∂_s^m (G(s)) + ∑_i ≤ 2n +m,j <n r_i,jΔ_t,^j∂_s^i(G(s)),
    
    where the r_i,j's are elements of  K.
    
     Applying the computations above to G =F^1(x,t), we find that  any nontrivial polynomial equation in the derivatives δ^m_x∂_t^n F^1(x,t) over  yields to a nontrivial polynomial equation over K between the derivatives  Δ_t,^j∂_s^i (F^1(s)). 
    
    Thus, we have reduced the proof of Theorem <ref> to the following proposition:
    
    
    
    The functions F^1(s)  and F^2(s) are (∂_s,Δ_t,)-differentially transcendental over K.
    
    
    Suppose to the contrary that  F^1(s) is (∂_s,Δ_t,)-differentially algebraic over K. By Lemma <ref>, the meromorphic function F^1(s) satisfies
    F^1( s)-F^1(s)= b_1=(x(  s) -x(s))y( s) with b_1 ∈ C(s)⊂ C_(s). We now apply difference Galois theory to this -difference equation. More precisely, 
     by   Proposition <ref>  and Corollary <ref> with K=C_(s),  there exist  m ∈, d_0,…,d_m ∈ C_ not all zero and h ∈ C_(s) such that 
    
        d_0 b_1+d_1∂_s(b_1)+… +d_m ∂_s^m(b_1)=_(h)-h.
    
    Let   (e_β)_β∈ B be  a C-basis of C(s). Then,  (e_β)_β∈ B is   a C_-basis of C_(s) by <cit.>.  Now, decompose  the d_k's and h over (e_β)_β∈ B. Since b_1∈ C(s), it is easily seen that (<ref>) amounts into a collection of polynomial equations  with coefficients in C that should satisfy the coefficients of the d_k's and h with respect to the basis (e_β)_β∈ B.
    Since this collection  of polynomial equations has a nonzero solution in C_, we can conclude   that it has a nonzero solution in C because  C is algebraically closed. 
    Therefore,
    there exists c_k ∈ C not all zero and g ∈ C(s) such that 
    
        ∑_k c_k∂_s^k(b_1) = _(g)-g.
     
    By <cit.>  there exist f ∈ C(s) and c∈ C, such that 
        F^1( s)-F^1(s)=b_1=_ (f)-f +c.
     Since F^1 is meromorphic at s=0, we conclude that  c must be equal to zero. Finally, we have shown that there exist f ∈ C(s) such that
    
        b_1=_(f)-f.
    
    
    By duality, the morphism ϕ :1 → E gives rise to a field isomorphism ϕ^* from  the field C(E)=C(x,y)[Here  x and y denote the coordinate functions on the curve E. ] of rational functions on  E and the field C(s)  of rational functions  on 1.
    Moreover, one has σ_ϕ^*= ϕ^* σ^*, where σ^* is the action induced by the automorphism of the walk on C(E). Then, it is easily seen that the equation (<ref>) is equivalent to 
    
        (σ(x)-x)σ(y) = σ(f̃)-f̃,
    
    where f̃∈ C(x,y) is the rational function corresponding to f via ϕ^*.
    The coefficients of f̃ as a rational function over E belong to a finitely generated extension F of (t). 
    
    There exists a -embedding ψ of F into   that maps t onto a transcendental complex number. 
    Since σ and E are defined over (t), we   apply ψ to  (<ref>) and  we find 
    
        (σ(x)-x)σ(y)= σ (f)-f,
    
    where f belongs to  (E) the field of rational functions on the complex algebraic curve E defined by the kernel polynomial K(x,y,ψ(t)) and  where  σ is the automorphism of (E) induced by the automorphism of the walk corresponding to E.  In <cit.>, the authors  proved that there is no such equation. This concludes the proof by contradiction.
    
    
    
    § GENERATING FUNCTIONS OF WALKS, GENUS ONE CASE
    
    
    
    
    In this section we consider the situation where the kernel curve  E is an elliptic curve.  By Remark <ref>, this corresponds to the case where the set of steps is not included in an  half plane.  Unlike  the genus zero cases of (<ref>), the group of the walk might be finite for genus one walks. For unweighted walks of genus one  with  finite group, it was proved in <cit.> that the series was  holonomic with respect to the three variables. More recently, the authors of  <cit.>  studied weighted walks of genus one  with  finite group. They proved that the uniformization of the generating series was a product of zeta functions and elliptic functions over curve isogeneous  to the kernel curve. This allowed them to conclude that the generating series was holonomic with respect to the variables x and y. Their description should also allow to conclude to the d/dt-differential algebraicity of the series but in that case the question of the holonomy with respect to the variable t is still open.  
    
    In this section, we shall focus on the weighted walks of genus one with infinite group and we will prove analogously to the genus zero case that the (d/dx,d/dt)-differential algebraicity of the series implies its d/dx-differential algebraicity. This result combined to <cit.> shows that, for unweighted walks of genus one with infinite group, the series is d/dx-differentiallly algebraic if and only if it is d/dt-differentially algebraic (see Corollary <ref> below).
    
    Our strategy  follows the basic lines of   the one employed in the genus zero situation. However, the  uniformization procedure  in  the genus one case is more delicate and  differs from previous works such as <cit.> which relied on the uniformization of elliptic curves over  by a fundamental parallelogram of periods. Over a  nonarchimedean field C, there might be a lack 
    of nontrivial lattices. One has to consider  multiplicative analogues, that is, discrete subgroups of C^* of the form  q^. Then, rigid analytic geometry gives a geometric meaning to the quotient C^*/q^. This geometric quotient is    called  a Tate curve (see <cit.> for more details). For simplicity of exposition, we will not give here many details on this nonarchimedean geometry
    The multiplicative uniformization of the kernel curve allows us as in <ref> to   attach to the incomplete generating functions Q(x,0,t) and Q(0,y,t) some meromorphic functions F^i(s) satisfying 
    
        F^i( s) -F^i(s)=b_i(s),
    
    for some ∈ C^*  and  b_i(s) ∈ C_q, the field of q-periodic meromorphic functions over C^*. This 
    process detailed in <ref>, <ref> and <ref> has many advantages. Though technical, it is much more simple than the uniformization
    by a fundamental parallelogram of periods since we only have to deal with one generator
    of the fundamental group of the elliptic curve, precisely the loop around the origin in C^*. Moreover, it gives a unified framework to study the genus zero and one case, namely, the Galois theory of -difference equations. This is the content of <ref> 
    where we apply the Galoisian  criteria of Appendix <ref>
    to translate the differential algebraicity of the generating function in terms of the existence of  a telescoper. 
    
    
    
     §.§ Uniformization  of the kernel curve
    
    
    Let us fix a  weighted model of genus one. By Lemma <ref>, the norm  of the J-invariant J(E) of  the kernel curve is such that |J(E)|>1. By Proposition <ref>, there exists q ∈ C such that 0<|q|<1 and J(E)=J(E_q)=1/|q|, where E_q is the elliptic curve attached to the Tate curve C^*/ q^ (see Proposition <ref>, Lemmas <ref>, and <ref>). The curve E_q can be analytically uniformized by C^*  thanks to special functions, which have their origins in the theory of Jacobi q-theta functions (see Proposition <ref> below). Finally, since E and E_q have the same J-invariant, there exists an algebraic isomorphism between these two elliptic curves.  In order to describe the uniformization 
    of the kernel curve E, one needs to explicit this algebraic isomorphism. This is not completely obvious since E_q is given by its Tate normal form in P^2, i.e. by an equation of the form
    
        Y^2+ XY =X^3 +BX +C.
    
    
    Therefore, many  intermediate technical  results are  postponed    to the appendix <ref>. The following proposition describes the multiplicative uniformization of an elliptic curve given by a Tate normal form.
    
    Following <cit.>, we   set  s_k =∑_n >0n^k q^n/1-q^n∈ C  for k ≥ 1.
    
     The  series
    
      * X(s)=∑_n ∈q^n s/(1-q^n s)^2 -2s_1; 
    
      * Y(s)=∑_n ∈(q^ns)^2/(1-q^n s)^3 +s_1;
    
    are q-periodic meromorphic functions over C^*. Furthermore X(s)=X(1/s), and X(s) has  a pole of order 2 at any element of the form q^. Moreover, the analytic map  
    
        [            π:           C^*             →       P^2(C),;                           s             ↦ [X(s):Y(s):1] ]
    
    is onto and his image is E_q, the elliptic curve defined by the following Tate normal form
    
        Y^2+XY= X^3 +BX +C,
    
    where B=-5s_3 and C =-1/12(5s_3 +7s_5). Moreover, π(s_1)=π(s_2) if and only if s_1 ∈ s_2 q^.
    
    This is <cit.>.
    
    In the notation of Section <ref>,  set 𝔇(x):=Δ_x(x,1).  Let us write the kernel polynomial 
        K(x,y,t)=A_0(x)+A_1(x)y+A_2(x)y^2=B_0(y)+B_1(y)x+B_2(y)x^2
     with A_i(x)∈ C[x] and B_i(y)∈ C[y]. For i≥ 1, let 𝔇^(i)(x) denote the i-th derivative 
     with respect to x of 𝔇(x).
    The analytic uniformization of the kernel curve is given by the following proposition.   
     
    There exists a root  a of 𝔇(x) in C  such that |a|, |𝔇^(2)(a)-2|,|𝔇^(i)(a)| <1 for i=3,4,  |q|^1/2<|𝔇^(1)(a)|<1. For any such a, there exists   u∈ C^* with |u|=1 such that the  map ϕ given by 
    
        [           ϕ:          C^*            →           E,;                         s            ↦ (x(s),y(s)), ]
    
    is surjective where
     
        x(s)   =   a +𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6
        y(s)   =   𝔇^(1)(a)( 2u^3Y(s) +u^3X(s))/2(u^2 X(s)+u^2/12-𝔇^(1)(a)/6)^2- A_1( a +𝔇^(1)(a)/u^2 X(s)+u^2/12-𝔇^(2)(a)/6)    /2 A_2(a +𝔇^(1)(a)/u^2 X(s)+u^2/12-𝔇^(2)(a)/6).
    
    Lemma <ref> and Lemma <ref> guaranty the existence of a.
    The element a allows us to write down the isomorphism between the kernel curve E and one of its Weierstrass normal form E_1. More precisely, by Proposition <ref>,  the application  w_E
        [                             E_1               → E ⊂1 (C) ×1 (C);                     [x_1:y_1:1]               ↦           (x,y) ]
    
    where 
        x=a +𝔇^(1)(a)/x_1-𝔇^(2)(a)/6 and y=𝔇^(1)(a) y_1/2(x_1-𝔇^(1)(a)/6)^2- A_1( a +𝔇^(1)(a)/x_1 -𝔇^(2)(a)/6)    / 2A_2(a +𝔇^(1)(a)/x_1-𝔇^(2)(a)/6),
     is an isomorphism  between the elliptic curve E_1 ⊂P^2(C) given by the  equation y_1^2=4x_1^3 -g_2x_1-g_3 and the kernel curve E. Now, it remains to explicit the isomorphism between E_q and one of its Weierstrass normal form E_1. By Lemma <ref>,  the application 
    [              w_T:               E_q                 →             E_1 ,;                             [X:Y:1]                 ↦ [X +1/2: 2Y +X:1] ]
    induces  an isomorphism between  E_q and the curve E_1 given by y^2=4x^3-h_2x-h_3. Since E and E_q have the same J-invariants and are therefore isomorphic, the same holds for their Weierstrass normal forms.  Thus,   there exists u ∈ C^*
    such that [              ψ:             E_1               →            E_1,;                         [x:y:1]               ↦ [u^2 x:u^3 y:1] ] induces an isomorphism of elliptic curves (see Lemma <ref>). To conclude, we set ϕ= w_E ∘ψ∘ w_T ∘π where  π is  the uniformization of E_q by C^* given in  Proposition  <ref>. The norm estimate on u is Lemma <ref>.
      *  Note that by construction ϕ(s_1)=ϕ(s_2) if and only if if s_1 ∈ s_2 q^ (see Proposition <ref>).
    
      *  Via ϕ, the field of rational functions over E can be identified with  field of q-periodic meromorphic functions over C^*.
    
      *  The conditions on a are crucial to guaranty the meromorphic continuation of the generating function (see the proof of Lemma <ref>).
    
      *  The symmetry  arguments between x and y of Remark <ref> can be pushed further and one can   construct another  uniformization of E as follows. Denoting by 𝔈(y) the polynomial Δ_y(y,1). One can prove that there exist a root  b ∈ C^* of 𝔈 such that |b|, |𝔈^(2)(b)-2|,|𝔈^(i)(b)| <1 for i=3,4 and |q|^1/2<|𝔈^(1)(b)|<1 and 
     v∈ C^* with |v|=1 such that the analytic  map ψ  given by 
    
        [           ψ:          C^*            →           E,;                         s            ↦ (x(s),y(s)), ]
    
    is surjective with 
    y(s)=b +𝔈^(1)(b)/ v^2X(s)+ v^2/12  -𝔈^(2)(b)/6 (see <cit.> for similar arguments).
    
    
    
    
     §.§ The group of the walk
    
    
    
    
    
    The following proposition gives an explicit form for the  automorphisms of C^*  induced via ϕ by  the automorphisms σ,ι_1,ι_2 of E.   
     
    There exists   in C^* such that the automorphism of C^* defined by  _ : s↦ s induces via ϕ the automorphism σ, that is σ∘ϕ= ϕ∘σ_.
    Similarly, the involutions _1,_2  of C^*,  that are defined by _1(s)=1/s and _2(s)= /s, induce via ϕ the automorphisms  ι_1,ι_2.
    
    By <cit.>, the automorphism  corresponds to the addition by a prescribed point Ω of E. Let  π: C^* → E_q be the  surjective map  defined in Proposition <ref>.  By <cit.>, the map π is  a group isomorphism between the multiplicative group (C^*,*)  and the Mordell-Weil group of E_q[ This is the group whose underlying set is the set of points of E_q and whose group law is given by the addition on the elliptic curve E.]. Moreover, since E_q and E are elliptic curves, any isomorphism between	E_q and E is a group morphism between their respective Mordell-Weil groups.  This proves that ϕ is a group morphism.  Then, there exists ∈ C^* 
     such that σ∘ϕ= ϕ∘σ_. Since ϕ is q-invariant, the element  is determined modulo q^ (see Remark <ref>). This proves the first statement.
    
    Let us denote by   _1,_2 some automorphisms of C^*, obtained by   pulling back to C^* via ϕ the automorphisms ι_1,ι_2 of E. The automorphisms    _1,_2 are uniquely determined up to multiplication by some power of q. The automorphisms of C^* are of the form s↦ l s^± 1 with l ∈ C^*. Note that x(q^)=a, and (a, -B(a)/2A(a)) ∈ E is fixed by ι_1. Indeed, by construction 𝔇(a)=0. This proves that  _1(1) belongs to q^. Since ι_1 is not the identity, we can modify _1 by a suitable power of q to get 
     _1 (s)=1/s. The expression of _2 follows with σ=ι_2 ∘ι_1.
    
      *  The choice of the element  is unique up to multiplication by q^.  Since |q| ≠ 1, we can choose  such that  |q|^1/2≤ ||<|q|^-1/2. 
    
    
      *  Pursuing the symmetry arguments of  Remark <ref>, we easily note that  Proposition <ref> has a straightforward analogue when one replaces ϕ by ψ and one exchanges _1 and _2.
    
    
    The proof of the following lemma is straightforward.
    The automorphism σ has infinite order if and only if 
     and q  are multiplicatively independent[Note that multiplicatively independent is sometimes  replaced in the literature  by noncommensurable  (see  <cit.>).], that is,  there is  no (r,l) ∈^2 ∖ (0,0) such that q^r=^l.
    
    
     §.§ Meromorphic continuation
    
    
    In this section, we prove that the functions 
        F^1(x,t):= K(x,0,t)Q(x,0,t),  and   F^2(y,t):= K(0,y,t)Q(0,y,t),
     can be meromorphically  continued  to C^*. We follow some of the ideas initiated in <cit.>. We note that, since |t|<1, the series F^1(x,t) and F^2(y,t) converge on the affinoid subset  U={(x,y) ∈ E ⊂1 (C)×1 (C)||x|≤ 1, |y|≤ 1 } of E. With Lemma <ref>, U is not empty. For (x,y) ∈ U, we have 
    
        0=xy+F^1(x,t) +F^2(y,t)+td_-1,-1 Q(0,0,t).
    
    Set U_x={(x,y) ∈ E ⊂1 (C)×1 (C)||x|≤ 1}.  Note that F^1(x,t) is analytic on U_x. We continue F^2(y,t)
     on U_x by setting 
      
        F^2(y,t)= -xy -F^1(x,t) -td_-1,-1 Q(0,0,t).
    
    
    
    Composing  F^i(x,t) with the  surjective map 
        [           ϕ:          C^*            →            E;                         s            ↦ (x(s),y(s)), ]
     we   define the functions  F̆^1(s)=F^1(x(s),t) and F̆^2(s)=F^2(y(s),t) for any s in the set 
        𝒰_x:= ϕ^-1(U_x)∩{s∈ C^*| |s|∈ [|q|^1/2,|q|^-1/2[}.
    
    
    The goal of the following lemma is to prove that 𝒰_x is an annulus 
     whose size is large enough in order to continue the functions F̆^1, F̆^2, to the whole C^* (see Figure <ref>).
    
    
    Let |s|∈ [|q|^1/2,|q|^-1/2[. The following statements  hold:
    
      *  if |s|∈ ]|𝔇^(1)(a)|,|𝔇^(1)(a)|^-1[, then |x(s)|< 1;
    
      *  if |s|=|𝔇^(1)(a)|^± 1, then |x(s)|= 1;
    
      *  otherwise |x(s)|> 1.
    
    In conclusion,  𝒰_x =[|𝔇^(1)(a)|,|𝔇^(1)(a)|^-1].
    
    From the definition of X(s), we have X(s)=X(1/s) so that x(s)=x(1/s). Using this symmetry, we just have to  prove Lemma <ref> for |s|∈ [|q|^1/2,1].
    We have 
     
        |x(s)| =|a +𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6|≤max(|a|,|𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6|),
     
    
    with equality if |a| ≠|𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6|.
    Remember  that  |u|=1, |a|<1, and  |q|^1/2<|𝔇^(1)(a)|<1, see Theorem <ref>.
    Let us first assume that |s|∈ [|𝔇^(1)(a)|,1[. By Lemma <ref>, |u^2X(s)|=|s|  and by Lemma <ref>,
      |u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|. Therefore   
     
        |𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6|=|𝔇^(1)(a)/s|.
    
    Combining this equality with (<ref>) and |a|<1, we find that |x(s)| <1 if |s|∈ ]|𝔇^(1)(a)|,1[,  and |x(s)| =1 if |s|=|𝔇^(1)(a)|.   
     Assume now that |s|=1. By construction,  |x(1)|=|a|<1. So let us assume that s≠ 1. Since |u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|<1  and |u^2X(s)| ≥ 1 by Lemma <ref>, we find 
    
        | 𝔇^(1)(a)/ u^2X(s)+ u^2/12  -𝔇^(2)(a)/6| =|𝔇^(1)(a)/u^2X(s)| ≤ |𝔇^(1)(a)|<1
        .
     
    
    This concludes the proof of the first two points.  
    Assume that |s|∈ ]|q|^1/2,|𝔇^(1)(a)|[. By Lemma <ref>, |u^2X(s)|=|X(s)|=|s|. Since 
        |u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|<1,
     we find that |u^2X(s)+u^2/12  -𝔇^(2)/6|<|𝔇^(1)(a)| and therefore, |x(s)|> 1. If we have |s|=|q|^1/2<|𝔇^(1)(a) | then Lemma <ref> implies  that |u^2X(s)|=|X(s)|≤ |s|<|𝔇^(1)(a)|. Since  |u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|, we deduce that |u^2X(s)+u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|
     and therefore, |x(s)|> 1. This concludes the proof.
    
    By symmetry between x and y, one could have define U_y={(x,y) ∈ E ⊂1 (C)×1 (C)||y|≤ 1} and continue F^1(x,t)
     on U_y by setting 
      
        F^1(x,t)= -xy -F^2(y,t) -td_-1,-1 Q(0,0,t).
      Then, the composition of the  F^i with the surjective map ψ defined in Remark <ref> yields to functions F̆^i that are defined on
    𝒰_y:= ψ^-1(U_y)∩{s∈ C^*| |s|∈ [|q|^1/2,|q|^-1/2[}. The analogue of Lemma <ref> is as follows. For   |s|∈ [|q|^1/2,|q|^-1/2[, the following statements  hold:
    
      *    if |s|∈ ]|𝔈^(1)(b)|,|𝔈^(1)(b)|^-1[, then |y(s)|< 1;
    
      *  if |s|=|𝔈^(1)(b)|^± 1 then |y(s)|= 1;
    
      *  otherwise |y(s)|> 1.
    
    
    
    By Proposition <ref>, the automorphism of the walk corresponds to 
    the -dilatation on C^*. The following lemma shows that one can cover C^* either  with the -orbit of the  set 𝒰_x or with  the -orbit of 𝒰_y.
    
    
    The following statement  hold:
    
      * ||≠ 1;
    
      *  moreover, up to replace  by some convenient q^-multiple, the following hold: 
      *  if  either d_-1,1= 0 or d_1,-1≠ 0, then, 
    
        ⋃_ℓ∈_^ℓ (𝒰_x)=C^*;
    
      *  if either d_-1,1≠ 0 or d_1,-1= 0 then, 
    
        ⋃_ℓ∈_^ℓ (𝒰_y)=C^*.
    
    Let us first prove that  ||≠ 1. By Remark <ref>, one can choose  so that we have
    |q|^1/2≤ || < |q|^-1/2.
     By construction, x(1)=a. Let b∈1 (C) such that (a,b)∈ E. Since ι_1(a,b)=(a,b) we have ι_2(a,b)≠ (a,b) by  Lemma <ref>. So let a' ∈1 (C) distinct from a such that σ (a,b)=(a',b). Then, x()=a'. By Lemma <ref>, |x(s)|<1 for |s|=1. Thus,  it suffices to prove that  |x()|=|a'|≥  1 to conclude that || ≠ 1. 
    Remember that  K(x,y,t)=A_-1(x)+A_0(x)y+A_1(x)y^2=B_-1(y)+B_0(y)x+B_1(y)x^2 with A_i(x)∈ C[x] and B_i(y)∈ C[y].
    With ι_1(a,b)= (a,b) and the formulas in <ref>, one finds that 
    
        b^2 = A_-1(a) /A_1(a)=A_-1(a) /A_1(a).
     
    
    Let ν be the valuation at  X=0   of A_-1 (X)/A_1(X). Lemma <ref> with |a|<1 gives |b|^2=|a|^ν. Note that A_1 and A_-1 are polynomial of degree at most two  in X, so the integer  ν belongs to {-2,-1,0,1,2}. We have
    
        a' = B_-1(b) /B_1(b)a.
    
    We will prove that |a'|≥1 with a case by case study of the values of ν.
    
    Remember that 
    
        [                      A_-1                         = d_-1,-1+d_0,-1x+d_1,-1x^2;                       A_1                         =    d_-1,1+d_0,1x+d_1,1x^2;                      B_-1                         = d_-1,-1+d_-1,0y+d_-1,1y^2;                       B_1                         =   d_1,-1+d_1,0y+d_1,1y^2. ]
    Case ν≥ 1. Then, |b|=|a|^ν/2 <1. Combining (<ref>) and Lemma <ref>, we find
    |a||a'|=|b|^l where l is the valuation at X=0 of B_-1 (X)/B_1(X). This gives |a'|=|a|^lν/2 -1. Since l belongs to { -2,…,2} and ν is in {1,2}, we get  -3 ≤ l ν/2 -1 ≤ 1. If lν/2 -1 equals 1 then ν must be equal to 2  and by (<ref>), we must have d_-1,-1=d_0,-1=0 and d_-1,1≠ 0. By Remark <ref>, we must have d_-1,0d_1,-1≠ 0 so that l=1 and lν/2-1=0. A contradiction. Then, lν/2 -1 ≤ 0 and 
    |a'| ≥ 1.
    
    Case ν=0. Then, |b|=1. With Lemma <ref> and |a|<1, we obtain |a'|>1. Case ν≤ -1. Then |b |= |a|^ν/2>1. Combining (<ref>) and Lemma <ref>, we find
    |a'|=|a|^lν/2 -1 where l ∈{-2,…,2} is  the degree in X of B_-1 (X)/B_1(X).  Since l belongs to { -2,…,2} and ν is in {-1,-2}, we get  1 ≥ lν/2 -1 ≥ -3. If l ν/2 -1=1 then ν =-2   and by (<ref>), we must have d_-1,1=d_0,1=0 and d_-1,-1≠ 0. By Remark <ref>, we must have d_-1,0d_1,1≠ 0 so that l=-1 and lν/2-1=0. A contradiction. Then, lν/2 -1 ≤ 0 and 
    |a'| ≥ 1.
     
    
    
    
    Assume that either d_-1,1= 0 or d_1,-1≠ 0 and let us prove that
    
        ⋃_ℓ∈_^ℓ (𝒰_x)=C^*.
    
    By Lemma <ref>, there exists  (a_0,b_0) ∈ E such that |a_0|=1 and (a_0,b_0)=(a_1,b_1) with |a_1| ≤ 1.
    By Lemma <ref>, there exists  s_0∈ C^* with  |s_0|=|𝔇^(1)(a)|^± 1  such that 
    x(s_0)=a_0. Since |q|^1/2≤ ||<|q|^-1/2 and |q|^1/2<|𝔇^(1)(a)|<1, we find that   |q|<| s_0| < |q|^-1/2. Since |x( s_0)|=|a_1| ≤ 1, we conclude using  Lemma <ref> that 
    
      *  either | s_0 | ∈_x. This proves that  
        _x ∩σ_(_x)=[|𝔇^(1)(a)|, |𝔇^(1)(a)|^-1]∩σ_([|𝔇^(1)(a)|, |𝔇^(1)(a)|^-1])≠∅ .
     Since || ≠ 1, we deduce that
    
        ⋃_ℓ∈_^ℓ (𝒰_x)=C^*.
    
      *  or | s_0 | ∈ [|q||𝔇^(1)(a)|, |q||𝔇^(1)(a)|^-1]. Replacing  by /q allows to conclude.
    
      *  or | s_0 | ∈ [|q|^-1|𝔇^(1)(a)|, |q|^-1|𝔇^(1)(a)|^-1]. Replacing  by q allows to conclude.
    
    The proof for  𝒰_y is obtained by a symmetry argument using   Lemma <ref> and  Remark <ref>.
    
    
    According to Lemma <ref>, we define some auxiliary functions    as follows 
    
      *   if  d_-1,1= 0, we define, for i=1,2,  the function  F^i(s) on 𝒰_x as  F^i(ϕ(s),t); 
    
      *  if  d_-1,1≠ 0,   the function  F^i(s) is defined on  𝒰_y  as  F^i(ψ(s),t).
    
    A priori the auxiliary functions F^1(s), F^2(s) are defined on _x if  d_-1,1= 0 and on _y otherwise.   Theorem <ref> below shows that one can meromorphically 
    continue the functions F^i(s) on C^* so that they satisfy 
    some  nonhomogeneous rank 1 linear -difference equations.
    
    
     The auxiliary functions F^1(s), F^2(s) can be continued meromorphically  on C^* so that they satisfy 
    
        F^1( s)-F^1(s)= b_1
    
    and 
        F^2( s)-F^2(s)=b_2,
    
    where  b_1=(x(  s) -x( s))y( s) and b_2 = (y( s) -y( s))x(s)  are two q-periodic meromorphic functions over C^*. 
    
    
    The proof is completely similar to the proof of Lemma <ref> and relies on the fact that either the -orbit of 𝒰_x or the -orbit of  𝒰_y covers C^*.
    
    Note that by Remark <ref>, the coefficients b_1,b_2 of the -difference can be identified with rational functions on the algebraic curve E.
    
    
    
     §.§ Differential transcendence
    
    
    The strategy to study the differential transcendence of generating functions  of nondegenerate weighted models of   genus one with infinite group is similar to the one employed in <ref>.  One first relate the differential behavior  of the incomplete generating functions to the differential algebraic properties of their associated auxiliary functions. Then,    one applies to these auxiliary functions the Galois theory of -difference equations. 
    However, since the coefficients of the -difference equations satisfied by the auxiliary functions  are no longer rational but elliptic, the Galoisian criteria as well as the descent method to obtain some “simple  telescopers” are quite technical and postponed to Appendix  <ref>. Theorem <ref> below gives a  first  criteria to guaranty the differential transcendence of the incomplete generating function.
    
    
    
    Assume that the weighted model is nondegenerate, of  genus one, and that the group of the walk is infinite. If 
    Q(x,0,t) is (d/dx,d/dt)-differentially algebraic over  then there exist c_0, …,c_n ∈ C not all zero and h ∈ C_q such that 
    
        c_0 b_1+c_1 ∂_s(b_1)+ … +c_n ∂_s^n (b_1) =_(h)-h.
    
    A symmetrical result holds for Q(0,y,t) replacing b_1 by b_2.
    
    Since the group of the walk is of infinite order, the 	automorphism σ is of infinite order. Therefore
    by Lemma <ref> the elements  and q defined in  Proposition <ref> are multiplicatively independent. 
    Assume that  Q(x,0,t) is (d/dx,d/dt)-differentially algebraic over . Let F^1(s)  be the auxiliary  function defined above.
    
     We  denote by C_.C_q  the compositum of  the  fields C_q and C_ inside the field of meromorphic functions over C^*.
     We claim that  F^1(s)
    is (∂_s,Δ_t,)-differentially algebraic over C_.C_q(ℓ_,ℓ_q).  Let us prove this claim when d_-1,1=0,  the proof when d_-1,1≠ 0 being similar.  Reasoning as in Lemma <ref>, one can show that, for n,m ∈, one has 
    
        (∂_t^n ∂_x^m F^1)(x(s),t)=
         1/∂_s(x(s))^mΔ_t,q^n ∂_s^m (F^1(s)) + ∑_i ≤ 2n +m,j <n r_i,jΔ_t,q^j∂_s^i(F^1(s)),
    
    where r_i,j∈ C_(ℓ_)(x(s),∂_s^l∂_t^k(x(s)),…).  By construction, x(s) is in C_q so that  Lemma <ref> implies  that ∂_s^l∂_t^k(x(s)) ∈ C_q(ℓ_q) for any positive integers  k,l. Then,  the field  C_(ℓ_)(x(s),∂_s^l∂_t^k(x(s)),…) generated by x
    and its derivatives with respect to ∂_s and ∂_t is contained in C_.C_q(ℓ_,ℓ_q). Thus,
    any nontrivial polynomial relation between the x-t-derivatives of Q(x,0,t) yields to a 
    nontrivial polynomial relation between the derivatives of F^1(s) with respect to ∂_s and Δ_t,
    over C_.C_q(ℓ_,ℓ_q). This proves the claim. 
    
    By Theorem <ref>, the function F^1(s) satisfies F^1( s)- F^1(s) =b_1(s) with b_1(s) ∈ C_q ⊂ C_.C_q(ℓ_,ℓ_q). Since F^1(s) is (∂_s,Δ_t,)-differentially algebraic over C_.C_q(ℓ_,ℓ_q), Proposition <ref> and 
    Corollary <ref> imply that 
    there exist m ∈ and d_0,…,d_m ∈ C_ not all zero and g ∈ C_.C_q(ℓ_q) such that 
    
        d_0 b_1+d_1∂_s(b_1)+… +d_m ∂_s^m(b_1)=_(g)-g .
    
    Since b_1 is in C_q, Lemma <ref> allows to perform a descent on the coefficients of the telescoping relation above. Thus, there exist c_0, …,c_n ∈ C not all zero and h ∈ C_q such that 
    
        c_0 b_1 +c_1∂_s( b_1)+ … +c_n ∂_s^n (b_1) =_(h)-h .
    
    This  concludes the proof. The symmetry argument between x and y gives the proof for Q(0,y,t).
    
    Theorem <ref> has an easy corollary concerning  the differential transcendence of the complete generating function for  weighted models of genus one with infinite group.
    
    
    For any nondegenerate  weighted model of genus one with  infinite group, the following statements are equivalent:
    
      *  the series Q(x,0,t) is (d/dx,d/d t)-differentially algebraic over ;
    
    
      *  the series Q(x,0,t) is d/dx-differentially algebraic over .
    
    An analogous result holds for Q(0,y,t) replacing the derivation d/dx by d/dy.
    
    
    Since the group is infinite, the 	automorphism σ is of infinite order. Therefore
    by Lemma <ref> the elements  and q defined in  Proposition <ref> are multiplicatively independent.  
     Assume that (1) holds. By Theorem  <ref>, there exist c_0, …,c_n ∈ C not all zero and h ∈ C_q such that 
    
        c_0b_1+c_1 ∂_s (b_1)+ … +c_n ∂_s^n (b_1) =_(h)-h.
    
    Combining (<ref>) with the functional equation satisfied by F^1(s) and using the commutativity of _ and ∂_s, one finds that 
    
        _[ c_0 F^1(s) +… + c_n∂_s^n( F^1(s))-h ]= c_0 F^1(s) +… + c_n∂_s^n( F^1(s))-h.
    
    Since F^1 and h are meromorphic over C^*,   there exists g ∈ C_ such that 
    
        c_0 F^1(s) +… + c_n∂_s^n( F^1(s))-h =g.
    
    Therefore, 
    F^1(s) is ∂_s-differentially algebraic over C_q. Reasoning as in Lemma <ref>, one finds a nontrivial algebraic relation with coefficients in C_q between the  first n-th derivatives of  F^1 with respect to ∂_x evaluated in (x(s),t). Any  element of C_q=C(x(s),y(s)) is algebraic over C(x(s)). Therefore, the 
    first n-th derivatives of  F^1 with respect to ∂_x evaluated in (x(s),t) are still algebraically dependent over  C(x(s)). We conclude that   F^1(x,t)=K(x,0,t)Q(x,0,t) is d/dx-differentially algebraic over C(x) and therefore over  by Remark <ref>.  This proves that  (1) ⇒(2). Statement  (2) implies obviously  (1). 
    
    
    
    A corollary of Theorem <ref> is that the d/dt-differential algebraicity of the series implies the d/dx-algebraicity of the series of the series. One of the major breakthrough of <cit.> is to show that for unweighted walks, the series was d/dx-differentially algebraic over  if and only if the models was decoupled, that is, there exist f,g ∈(t)(X) such that 
    
        xy=f(x)+g(y)  K(x,y,t).
     
    The authors  of <cit.> used   boundary value problems and the notion of  analytic invariants to deduce from (<ref>)  a closed form of the generating series allowing them to conclude that the series was also d/dt-algebraic (see <cit.>). Combining our result to <cit.>, one finds the following corollary
    
    
    If the walk is unweighted of genus one with infinite group, the following statement are equivalent
    
      *  the generating series is d/dx-differentially algebraic over ;
    
      *  the generating series is d/dt-differentially algebraic over ;
    
    
    In a recent publication  <cit.>, M.F.Singer and the second author generalized the results of <cit.> and proved that a weighted model of genus one with infinite group was decoupled if and only if the series was d/dx-differentially algebraic. There is no doubt that 
    the arguments of <cit.> proving that if a model is decoupled then the series is d/dt-algebraic over  will hold in a weighted situation. Combined to Theorem <ref>, this  will prove that  Corollary <ref> is also true for weighted walks. 
    
    
    
    
    
    § NONARCHIMEDEAN ESTIMATES
    
    
    In this section, we give some nonarchimedean estimates, which will be crucial to uniformize the kernel curve.
    
    
     §.§ Discriminants of the kernel equation
    
    Lemma <ref> relates the genus of the kernel curve to the simplicity of the roots of the discriminant 
    of the kernel polynomial. It also ensures the existence of a root with convenient norm estimates. Let us remind, see (<ref>), that we have defined 𝔇(x):=Δ_x(x,1), where  Δ_x(x_0,x_1) is the discriminants of the second degree homogeneous polynomials y ↦K(x_0,x_1,y,1,t).
         
    
    For any nondegenerate weighted model  of genus one, the following holds:
     
      *   all the roots of Δ_x(x_0,x_1) in 1(C) are simple;
     
      *  the discriminant 𝔇(x):=Δ_x (x,1) has a root a ∈ C such that |a|<1, |𝔇^(2)(a) -2|<1, and |𝔇^(1)(a)|,|𝔇^(3)(a)|,|𝔇^(4)(a)|<1 where 𝔇^(i) denote the i-th derivative 
     with respect to x of 𝔇(x).
     
     A symmetric statement holds for Δ_y(y_0,y_1) by replacing 𝔇 by 𝔈.
     
    The first assertion is <cit.>. First, let us prove the existence of a root a ∈ C of 𝔇(x) such that |a|<1.  Suppose to the contrary that all the roots of 𝔇(x) have a norm greater than or equal to 1.  If α_0 is zero  then  zero is a root: a contradiction. Thus, we can assume that α_0 is nonzero.
     
     Let us first assume that α_4 ≠ 0.  The product of  the roots of 𝔇(x) equals 
        α_0/α_4=t^2(d_-1,0^2 -4d_-1,-1d_-1,1)/t^2(d_1,0^2 -4d_1,-1d_1,1).
    
    Then  we conclude that |α_0/α_4|=1  so that each of the roots must have norm  1. Then, considering the symmetric functions of the roots of 𝔇(x), we conclude that, for any i=0,…, 3, the element α_i/α_4 should have  norm smaller than or equal to 1.  Since 
        α_2/α_4=-4d_-1, -1d_1, 1t^2-4d_0, -1d_0, 1t^2-4d_1, -1d_-1, 1t^2+2d_-1, 0d_1, 0t^2+d_0, 0^2t^2-2td_0, 0+1/t^2(d_1,0^2 -4d_1,-1d_1,1),
    
     has norm strictly greater than 1, we find  a  contradiction. 
     Assume now that α_4 = 0. Since the roots of Δ_x(x_0,x_1) in 1(C) are simple, the coefficient  α_3 is nonzero.  The product of the roots of 𝔇(x) equals 
        -α_0/α_3=-t^2(d_-1,0^2 -4d_-1,-1d_-1,1)/2t^2d_1,0d_0,0-2td_1,0-4t^2(d_0,1d_1,-1+d_1,1d_0,-1).
    
    Then, it is clear that  |α_0/α_3|≤ 1  and that each of the roots has norm 1.  Thus, the  symmetric function α_2/α_3
    should also  have norm   smaller than  or equal to 1.
    But
    
        -α_2/α_3=-4d_-1, -1d_1, 1t^2-4d_0, -1d_0, 1t^2-4d_1, -1d_-1, 1t^2+2d_-1, 0d_1, 0t^2+d_0, 0^2t^2-2td_0, 0+1/2t^2d_1,0d_0,0-2td_1,0-4t^2(d_0,1d_1,-1+d_1,1d_0,-1),
     has  norm strictly bigger than 1. We find a contradiction again.  
     
    Let  a be a root of 𝔇(x) in C with |a|<1.  Since a,α_1,α_3,α_4 have norm smaller than 1, |α_2 -1| <1, and 
    
      * 𝔇^(1)(a)=α_1 +2α_2a +3 α_3 a^2 +4 α_4a^3;
    
      * 𝔇^(2)(a)=2 α_2+ 6 α_3 a +12 α_4 a^2;
    
      * 𝔇^(3)(a)=6 α_3 +24 α_4 a;
    
      * 𝔇^(4)(a)=24α_4,
    
     we have |𝔇^(2)(a)-2|<1, and |𝔇^(1)(a)|,|𝔇^(3)(a)|,|𝔇^(4)(a)|<1.
    The statement for Δ_y(y_0,y_1) is symmetrical and we omit its proof.
     
    
     §.§ Automorphisms of the  walk on the domain of convergence
    
    In this section, we study the action of the group of the walk on the product of the unit disks in P^1(C) ×P^1(C). This product is the  fundamental  domain of convergence of the generating function.
    
    
    We need a preliminary lemma that  explains how one can compute the norm of 
    the values of a rational function.
    
    Let f ∈ C(X) be a nonzero rational function and  let  a ∈1(C). Let ν (resp. d) be the valuation at X=0 (resp. ∞) of f with the convention that ν=+∞, d=-∞ if f=0. The following statements  hold:
    
      *  if |a| <1, then |f(a)| =|a|^ν;
    
      *  if |a|>1, then |f(a)|=|a|^d.
    
    Let us prove the first case, the second being completely symmetrical.  Let us write f(X) as  ∑_i=ν_1^r_1 c_i X^i/∑_j=ν_2^r_2 d_j X^j with c_ν_1d_ν_2≠0. If k>l, we note that  |a^k| <|a^l|.  Then 
        |f(a)|=|∑_i=ν_1^r_1 c_i a^i|/|∑_j=ν_2^r_2 d_j a^j|=|a|^ν_1-ν_2=|a|^ν.
    
    
    
    
    
    
    The following lemma explains how  the fundamental involutions permute the interior and the exterior of the fundamental domain  of convergence. 
    
    
    For any nondegenerate weighted model, the  following statements  hold: 
    
      *   for any a ∈ C with |a|=1, there exist b_±∈1(C) with |b_-|<1, and |b_+|>1, such that  K(a,b_±,t)=0;
    
      *  for any b ∈ C with |b|=1, there exist a_±∈1(C) with |a_-|<1, and |a_+|>1, such that  K(a_±,b,t)=0.
    
    See <cit.> for a similar result in the situation where C is replaced by .
    
    The statements are symmetrical, so we only prove  the first one. Since C is algebraically closed and   the model  is nondegenerate, Proposition <ref> implies that K(x,y,t) is of degree 2 in y. Then,  for any a ∈ C,  there are two elements b_±∈1(C) such that 
    K(a,b_±,t)=0. let a ∈ C with  |a|=1. We write 
        K(a,y,t) =t α + β y + tγ y^2
    
    
     where 
      * α=-∑_i=-1^1 d_i,-1a^i+1;
    
      * β =a - t∑ _i=-1^1 d_i,0a^i+1;
    
      * γ=- ∑_i=-1^1 d_i,1 a^i+1.
    
    
    Since |a|=1, we find |β|=1, |α |,|γ| ≤ 1. First let us prove that there is no point (a_0,b_0) ∈ E such that |a_0|=|b_0|=1. Indeed, suppose to the contrary that |a_0|=|b_0|=1 and K(a_0,b_0,t)=0. Then, |β|=|a_0|=1 and  |γ|,|α | ≤ 1 so that the equality 
    |β b_0|= |t(α + γ b_0^2)| implies
    |b_0| <1. We find a contradiction. From the equation K(a,b,t)=0, we deduce that
    
        if |b |<1,   then |t α| =|β b + t γ b^2|=|β b| which gives |b|=|t α|;
    
        if |b| >1,   then |1/b| <1  and we find |t γ|=|tα/b^2+β/b|=|β/b|=|1/b|.
    
    
    Using K(a,b_±,t)=0, we find 
    
        b_-b_+=α/γ,
    
    with the convention that b_+  is [1:0] if γ=0. If γ=0 then b_-=-t α/β has  norm smaller than 1, which  concludes the proof in that case. Assume now  that γ≠ 0.
     Since |b_+| and |b_-| cannot have norm 1, we just need to discard the cases “|b_+| <1 and |b_-| <1” or  “|b_+| >1 and |b_-| >1”. If α =0, then one of the root  is zero, say  b_-=0, and |b_+|=|β|/|t γ| >1, which concludes the proof in that case. If  α≠ 0 then one can  suppose to the contrary that |b_+| <1 and |b_-| <1.  From  (<ref>), we obtain  |b_+|=|b_-|=|t α|, which gives
     
        |b_+b_-|=|t α|^2= |α |/|γ|.
    
    Then,
    |t^2 α | =1/|γ | ≥ 1, which contradicts  |t^2 α |<1.   Suppose to the contrary that |b_+| >1 and |b_-| >1. By (<ref>), |b_+|=|b_-|= 1/|t γ| which gives
     
        |b_+b_-|= 1/|t γ|^2= |α |/|γ|.
    
    Thus, |t^2 α | =1/|γ | ≥ 1,  and once again, we find a  contradiction.
    
     Lemma <ref> explains how the  the intersection of the  fundamental domain of convergence of the generating function and its image by σ  is nonempty. This result is  therefore crucial in order  to continue the generating function to the whole  C^*.
    
    For any nondegenerate weighted model, the  following statements hold:
    
      *  if d_-1,1=0 or d_1,-1≠ 0 there exists (a,b) ∈ E with |a|=1 such that σ(a,b)=(a',b') with |a'| ≤ 1;
    
      *   if d_-1,1≠ 0 or  d_1,-1 =0 there exists (a,b) ∈ E with |b|=1 such that σ(a,b)=(a',b') with |b'| ≤ 1.
    
    Using  the symmetry between x and y mentioned in Remark <ref>, we  only  prove the first statement of Lemma <ref>. 
    
    Let a ∈1(C) such that |a|=1. By Lemma <ref>, there exist b_+∈1(C) with |b_+|>1 and b_-∈ C with |b_-|<1 such that (a,b_±) ∈ E. Let B_i as in (<ref>) and note that by Proposition <ref>, B_1 is not identically zero.
    Let ν (resp. d) be the valuation at 0 (resp. ∞) of the rational fraction B_-1(y)/B_1(y)=∑_j=-1^1 d_-1,jy^j /∑_j=-1^1 d_1,jy^j ∈ C(y). We claim that either ν≥ 0 or d ≤ 0.  If d_1,-1≠ 0 then ν≥ 0. If d_-1,1 =0 then either d ≤ 0 or d =1. In the latter situation, we must have d_1,1=d_1,0=0 and d_-1,0≠ 0. Since the model is nondegenerate, we must have d_1,-1≠ 0 by Proposition <ref>. In that case, ν≥ 0. This proves the claim.
    
    
    Let a_+,a_-∈1(C) such that ι_2 (a,b_+)= (a_+, b_+) and  ι_2 (a,b_-)= (a_-, b_-). This gives 
    
        a_+  = B_-1(b_+)/B_1(b_+)a    a_-  =B_-1(b_-)/B_1(b_-)a.
    
    
    Since σ(a,b_-)= (a_+,b_+) (resp. σ(a,b_+)= (a_-,b_-)),    it is enough to prove that either a_+ or a_- has norm smaller or equal  to 1.  If d ≤ 0, we combine (<ref>), Lemma <ref> and |b+|>1 to find  |a a_+|=|a_+|=|b_+|^d ≤ 1. If ν≥ 0, we combine (<ref>), Lemma <ref> and  |b_-|<1 to find  |a a_-|=|a_-|=|b_-|^ν≤ 1. This ends the proof.
    
    
    
    
    § TATE CURVES AND THEIR NORMAL FORMS
    
    
    Let (C,| |) be a complete nonarchimedean algebraically closed valued field of zero characteristic and let q ∈ C such that 0<|q| <1.  In this section, we recall 
    some of the basic properties of elliptic curves over nonarchimedean fields. The period  lattice is here replaced by a discrete  multiplicative group of the form q^. Then, the quotient of  by a  period lattice is replace by the so called Tate curve, which  corresponds to the naive quotient of the multiplicative group C^* by q^. 
    However, in the nonarchimedean context, only elliptic curves with J-invariant of norm greater than equal to one can be analytically uniformized by Tate curves (see Proposition <ref>). The analytic geometry behind is the rigid analytic geometry as developed in <cit.>. We will not introduce this theory here   but  we just recall briefly the algebraic geometrical and special functions aspects of Tate curves.  
    
    
    
    
     §.§ Special functions on a Tate curve
    
     
     We recall that any holomorphic function f on C^* can be represented by an everywhere convergent Laurent series ∑_n ∈ a_n s^n with a_n ∈ C. Moreover any nonzero meromorphic function on C^* can be written as g/h such that the holomorphic functions g and h have no common zeros.  We shall denote by er(C^*) the field of meromorphic functions over C^*. 
    
    
    
     If k is a complete nonarchimedian sub-valued field of C and q belongs to  k, every result quoted above still holds over k. 
    
    The analytification of the  elliptic curve E_q  is isomorphic to the Tate curve, which  is the rigid analytic space corresponding to the naive quotient of C^*/q^. The curve E_q is therefore a“canonical” elliptic curve. A natural question is "Given an elliptic curve E  defined over C, is there a q such that E is isomorphic to E_q?" The answer is positive under certain assumption on the J-invariant J(E) of E. 
    
    
    Let E be an elliptic curve  over C  such that |J(E)|>1. Then, there exists q ∈ C such that 0<|q|<1 and  E is isomorphic to the elliptic curve  E_q.
    
    Remind that we  have defined  s_k =∑_n >0n^k q^n/1-q^n∈ C  for k ≥ 1, and 
    
        X(s)=∑_n ∈q^n s/(1-q^n s)^2 -2s_1,   Y(s)=∑_n ∈(q^ns)^2/(1-q^n s)^3 +s_1.
    
     They are  q-periodic  meromorphic  functions over C^*.
    By Proposition <ref>, the field C_q  of   q-periodic  meromorphic functions over C^*   coincides with the field generated over C by X(s) and Y(s).  
     Since we need to understand what is the pullback of the fundamental domain of convergence of the generating function via this uniformization, we prove some basic properties on the norm of  X(s). Remind that X(s)=X(1/s) and X(qs)=X(s). Thus it suffices to study |X(s)| for |q|^1/2≤ |s|≤ 1.   The following study follows the arguments of <cit.>.
    
    
    Let  s∈ C^*. The following holds:
    
      *  If |q|^1/2<|s|<1, then |X(s)|= |s|;
    
      *  If |s|=1,  then |X(s)|≥ 1;
    
      *  If |s|=|q|^1/2, then |X(s)|≤  |s|.
    
    Since X(s) has a pole in s=1 we may further assume that s≠ 1. Let us rewrite  X(s):
    
        X(s)=s/(1-s)^2+∑_n>0q^n s/(1-q^n s)^2+q^n s^-1/(1-q^n s^-1)^2-2 q^n /1-q^n.
    
    This means that we have 
    
        |X(s)| ≤max( |s/(1-s)^2|, | ∑_n>0q^n s/(1-q^n s)^2+q^n s^-1/(1-q^n s^-1)^2-2 q^n /1-q^n|),
    
    with equality when |s/(1-s)^2| ≠  | ∑_n>0q^n s/(1-q^n s)^2+q^n s^-1/(1-q^n s^-1)^2-2 q^n /1-q^n|. 
    Let us consider s∈ C^*∖{1} with |q|^1/2≤ |s|≤ 1.
    Using |q|<1 we find that 
    |q^n s|≤ |qs|<1  for every n≥ 1. This shows that the norm of q^n s is strictly smaller than 1. Then, |q^n s/(1-q^n s)^2|=|q^n s|<|s|. On the other hand, 
    |q^n |≤ |q|<|s| and 
    |q^n /1-q^n|<|s|.
    Finally, when |q|^1/2< |s|, we have  |q^n s^-1|≤ |qs^-1|<|qq^-1/2|<|s| and therefore |q^n s^-1/(1-q^n s^-1)^2|=|q^n s^-1|<|s|. 
    This proves  that,  for any s ∈1(C) such that |q|^1/2< |s|≤ 1, we have
    
    
        |∑_n>0q^n s/(1-q^n s)^2+q^n s^-1/(1-q^n s^-1)^2-2 q^n /1-q^n|<|s|.
    
    When, |q|^1/2=|s| and n≥ 2, we have  |q^n s^-1|≤ |q^2s^-1|=|q^2q^-1/2|<|s|, and therefore  |q^n s^-1/(1-q^n s^-1)^2|=|q^n s^-1|<|s|.  Moreover, if   |q|^1/2= |s| then |q s^-1| =|qq^-1/2|=|s|. Therefore |q s^-1/(1-q s^-1)^2|=|q s^-1|=|s|.  We conclude that 
    
        |∑_n>0q^n s/(1-q^n s)^2+q^n s^-1/(1-q^n s^-1)^2-2 q^n /1-q^n|=|s|.
    
    It remains to  consider  the term s/(1-s)^2.  If  |s| <1 then   we have |s/(1-s)^2|= |s|. Combining with (<ref>), (<ref>) and (<ref>) respectively, we obtain the result when |q|^1/2<|s|<1 and |q|^1/2=|s|<1 respectively.
    If  |s|=1 and s ≠ 1 then |1-s|≤ 1. Thus, |s/(1-s)^2|≥ |s|=1, which, combined with (<ref>) and (<ref>) concludes the proof.
    
     §.§ Tate and Weierstrass normal forms
    
    In <cit.>, the authors generalize the results  of <cit.> and attach a Weierstrass normal form to the kernel curve. The following proposition proves that, with some care, their result passes to a nonarchimedean framework.
    
    
    Let us consider a nondegenerate weighted  model of genus one and let us write its kernel polynomial as follows: K(x,y,t)=A_0(x)+A_1(x)y+A_2(x)y^2=B_0(y)+B_1(y)x+B_2(y)x^2 with A_i(x)∈ C[x] and B_i(y)∈ C[y]. The following proposition gives a Weierstrass normal form for the kernel curve.
    
     Let a ∈ C be  as in Lemma <ref>. Let E_1 be the elliptic curve defined by the Weierstrass equation
     
        y_1^2=4x_1^3 -g_2x_1-g_3,
    
     with 
        g_2   =   𝔇^(2)(a)^2/3-2𝔇^(1)(a)𝔇^(3)(a)/3
        
         g_3   =   -𝔇^(2)(a)^3/27+𝔇^(1)(a)𝔇^(2)(a)𝔇^(3)(a)/9 -𝔇^(1)(a)^2𝔇^(4)(a)/6.
      
    Then, the rational map 
    
        [                             E_1               → E ⊂1 (C) ×1 (C);                     [x_1:y_1:1]               ↦           (x,y) ]
    
    where 
        x=a +𝔇^(1)(a)/x_1-𝔇^(2)(a)/6 and y=𝔇^(1)(a) y_1/2(x_1-𝔇^(1)(a)/6)^2- A_1( a +𝔇^(1)(a)/x_1 -𝔇^(2)(a)/6)    / 2A_2(a +𝔇^(1)(a)/x_1-𝔇^(2)(a)/6),
     is an isomorphism of elliptic curves that sends the point 𝒪=[1:0:0]  in E_1 to the point (a, -A_1(a)2A_2(a)) ∈ E.
     
    This is the same proof as in <cit.>. Note that there is only one configuration here since we have chosen a root of the discriminant |a|<1 which can not be infinity.
     
     
    We recall that the J-invariant  J(E_1) of the elliptic curve E_1  given in a  Weierstrass form y_1^2=4x_1^3-g_2 x_1 -g_3 equals to J(E_1)=12^3g_2^3/g_2^3-27g_3^2.  For a weighted model of genus one, the J-invariant J(E) of the  kernel curve  has modulus strictly greater than 1 by  Lemma <ref>.  Since J(E)=J(E_1),
     Proposition <ref>  shows that  there exists q∈ C^*  such that 0<|q|<1 and E_1 is isomorphic to E_q. In order to explicit this isomorphism, we 
    need  to understand how one passes  from  to a Tate normal form to a Weierstrass normal form. This is the content of the following lemmas.
    
    [ 6, Page 29 in <cit.>] 
    In the notation of Proposition <ref>, the  change of variable X=x- 1/12 and Y=1/2(y-x + 1/12) maps the Tate equation
    
        Y^2 +XY =X^3 +BX +C
    
    onto the Weierstrass equation 
    
        y^2= 4x^3 -h_2 x -h_3,
    
    where h_2= 1/12+20s_3 and h_3=-1/6^3+ 7/3s_5. 
    
    
     As detailed above, the elliptic curves E_1 and E_q are isomorphic. The following lemma gives the form of an explicit isomorphism between theses two curves. 
    
    Let y^2= 4x^3 -h_2 x -h_3 be the Weierstrass normal form   (resp. Y^2 +XY =X^3 +BX +C  its Tate normal form )    of E_q  as  in   Lemma <ref> and let 
    y_1^2= 4x_1^3 -g_2 x_1-g_3 be the Weierstrass normal form of E_1 as in  Proposition <ref>.
    
    
    There exists u ∈ C^* such that the following map
    
        [                       E_q                         →                      E_1,;                     (X,Y)                         ↦ (u^2(X+1/12), u^3(2Y +X)) ]
     is an isomorphism of elliptic curves.   Moreover, the following holds
     
      * h_2=g_2/u^4 and h_3=g_3/u^6;
     
      * Δ_q=Δ_1/u^12 where Δ_1 and Δ_q denote the discriminants 
     of the Weierstrass equations of E_1 and E_q respectively.
     
     From <cit.>, we deduce that  any isomorphism between the  elliptic curves E_ 1 and E_q   is given by 
    x_1=u^2x+α and y_1=u^3 y+β u^2x+γ with u ∈ C^*, α,β,γ∈ C. Since both equations are in Weierstrass normal form, we necessarily have α=β=γ=0. This  proves the first point. From y_1^2= 4x_1^3 -g_2 x_1-g_3, we substitute x_1,y_1 by x,y to find
    
        u^6  y^2= 4  u^6 x^3 -g_2 u^2 x-g_3.
    
    Dividing the both sides by u^6 we find h_2=g_2/u^4 and h_3=g_3/u^6. The assertion on the discriminants follows from  Δ_q=
    h_2^3-27h_3^2 and Δ_1=
    g_2^3-27g_3^2. 
    
     
     
     
    
    
    The  lemma below gives some  precise estimate  for the norms of  Δ_q=h_2^3-27h_3^2 and Δ_1 =g_2^3-27g_3^2,  the discriminants of the elliptic curves E_q ,E_1, and the element  u defined in Lemma <ref>.
    
     
     The following statement  hold:
     
      * |Δ_q|=|q|,  with |h_2 -1/12|=|q| and |h_3 -(-1/6^3)| =|q|;
     
      * |Δ_1|=|q|  with  |g_2 -4/3|<1,  |g_3-(-8/27)|<1;
     
      * |u|=1;
     
      * |𝔇^(1)(a)|∈ ]  |q|^1/2,1[.
     
     Following <cit.>, we  find |Δ_q|=|q|, |s_3|=|q|=|s_5|.  Combining  the latter norm estimates with  Lemma <ref>, we find |h_2 -1/12|=|q| and |h_3 -(-1/6^3)|=|q|. 
    
    Let us prove the second point.  It follows from (<ref>) that |1-_2|<1 and |_i|<1 for i=0,1,3,4. By Lemma <ref>, |𝔇^(1)(a)|,|𝔇^(3)(a)|,|𝔇^(4)(a)|<1, |𝔇^(2)(a) -2|<1. Combining these norm estimates with  (<ref>), we find  |g_2 -4/3|<1,  |g_3-(-8/27)|<1. Since |J(E_1)|=|J(E_q)|=|12^3g_2/Δ_1|= |12^3h_2/Δ_q| and |g_2|=|h_2|=1, we find |Δ_q|=|Δ_1|=|q|. By Lemma  <ref>,  Δ_q=Δ_1/u^12, and then |u|=1.
    
    
    Let us prove the last point. Let us expand Δ_1 =g_2^3-27g_3^2 with the expression of g_2,g_3 given in (<ref>):
    
        [                                                                                                       Δ_1                                                                                                         = (𝔇^(2)(a)^2/3-2𝔇^(1)(a)𝔇^(3)(a)/3)^3-27(-𝔇^(2)(a)^3/27+𝔇^(1)(a)𝔇^(2)(a)𝔇^(3)(a)/9-𝔇^(1)(a)^2𝔇^(4)(a)/6)^2;                                                                                                                                                                                                                   =    𝔇^(2)(a)^6/27-2𝔇^(1)(a)𝔇^(2)(a)^4𝔇^(3)(a)/9+4𝔇^(1)(a)^2𝔇^(2)(a)^2𝔇^(3)(a)^2/9-8𝔇^(1)(a)^3𝔇^(3)(a)^3/27;                                                                                                               -𝔇^(2)(a)^6/27-𝔇^(1)(a)^2𝔇^(2)(a)^2𝔇^(3)(a)^2/3-3𝔇^(1)(a)^4𝔇^(4)(a)^2/4+2𝔇^(1)(a)𝔇^(2)(a)^4𝔇^(3)(a)/9;                                                                                                                                                  -𝔇^(1)(a)^2𝔇^(2)(a)^3𝔇^(4)(a)/3+𝔇^(1)(a)^3𝔇^(2)(a)𝔇^(3)(a)𝔇^(4)(a);                                                                                                                                                                                                                   =                         𝔇^(1)(a)^2𝔇^(2)(a)^2𝔇^(3)(a)^2/9-8𝔇^(1)(a)^3𝔇^(3)(a)^3/27-3𝔇^(1)(a)^4𝔇^(4)(a)^2/4;                                                                                                                                                 -𝔇^(1)(a)^2𝔇^(2)(a)^3𝔇^(4)(a)/3+𝔇^(1)(a)^3𝔇^(2)(a)𝔇^(3)(a)𝔇^(4)(a). ]
    
    Since |𝔇^(1)(a)|,|𝔇^(3)(a)|,|𝔇^(4)(a)|<1, |𝔇^(2) -2|<1 , the previous expression is a sum of terms that are all strictly smaller in norm than |𝔇^(1)(a)|^2. This proves that  |Δ_1| =|q| < |𝔇^(1)(a)|^2.
    
    
    The following estimate will be required to uniformize the generating function.
    
    In the notation of Theorem <ref>, we have  |u^2/12  -𝔇^(2)(a)/6|<|𝔇^(1)(a)|.
    
    Using (<ref>) and the norm estimate on the 𝔇^(i)(a)'s, we get
    
        g_2 =𝔇^(2)(a)^2/3 + 𝔇^(1)(a) ω,    g_3 =-𝔇^(2)(a)^3/27 + 𝔇^(1)(a) ω',
    
    where |ω|, |ω'|<1.  This proves that 
    
        g_3/g_2 =-𝔇^(2)(a)/9 +𝔇^(1)(a) ω”
    
    with |ω”|<1. Then, we find
    
        | u^2/12 - 𝔇^(2)(a)/6|= | u^2/12 + 3g_3/2g_2 -3g_3/2g_2 -  𝔇^(2)(a)/6| ≤max(|u^2/12+3g_3/2g_2|, |3/2𝔇^(1)(a) ω”|).
    
    Finally, with  the norm estimate of Lemma <ref>, it is sufficient to show that |u^2/12+3g_3/2g_2|≤ |q|.
    By Lemma <ref>, we have u^2/12=g_3h_2/12g_2h_3. By Lemma <ref>, |h_2 -1/12|=|q| and |h_3 -(-1/6^3)|=|q|. Then, by Lemma <ref> again, we find 
    
        |u^2/12+3g_3/2g_2|=|g_3h_2/12g_2h_3+3g_3/2g_2|=|g_3/g_2| |h_2/12 h_3+3/2|=|h_2+18h_3/12 h_3|
        
        =|h_2+18h_3 |=|(h_2-1/12)+18(h_3 -(-1/6^3)) |≤max(|h_2-1/12|,|h_3 -(-1/6^3)|)≤ |q| .
    
    
    
    
    § DIFFERENCE GALOIS THEORY
    
    
    In this section, we establish some criteria to guaranty the transcendence of functions satisfying a difference  equation of order 1. This criteria is based on the Galois theory of difference fields as developed in <cit.> but generalizes some of the existing results in the literature, for instance the assumption  that the field of constants is   algebraically closed
     (see for instance Theorem <ref>). 
    
    
    The  algebraic framework of this section is difference algebra and more precisely the notion of difference fields.
    A difference field is a pair (K,) where K is a field and  is  an automorphism of K. The  field
    -constants K^ of (K,) is formed by the elements f ∈ K such that (f)=f. An extension (K,_K) ⊂ (L,_L) of difference fields is a field 
    extension K⊂ L such that _L coincides with _K on K. If there is no confusion, we shall denote by  the automorphism _K and _L. For a complete introduction on difference algebra, we shall refer to <cit.>. 
    
     §.§ Rank one difference equations
    
    
    In this section, we focus  on  rank one difference equations.
    
    Let (K,) ⊂ (L,)  be an extension of difference fields such that L^=K^. Let x∈ L. The following statements are equivalent
    
      * x is algebraic over K^;
    
      *  there exists r ∈^* such that ^r(x)=x.
    
    Assume that x is algebraic over K^. Then,  induces a permutation on the set of roots of the minimal polynomial of x over K^. Thus, there exists r ∈^* such that ^r(x)=x. Conversely, if there exists r ∈^* such that ^r(x)=x, the polynomial P(X)=∏_i=0^r-1 (X-^i(x)) ∈ L[X] is fixed by  and thereby P(X) ∈ L^[X]=K^[X ]. Since P(x)=0, we have proved that x is algebraic over K^.
    
    Let (K,) ⊂ (L,)  be an extension of difference fields such that L^=K^. Let f ∈ L and 0≠ c ∈ K, such that 
    (f)=f +c. The following statements are equivalent
    
      * f ∈ K;
    
      * f is algebraic over K;
    
      *  There exists  α∈ K such that (α)=α +c.
    
    Moreover, let  K be the algebraic closure of K  endowed  with a structure of -field extension of K.  For all α∈K, i ∈ we denote by α_i the element of K such that σ^i( f-α)=f-α_i.  If f is transcendental over K then for i,j ∈ such that  i ≠ j, the elements α_j and α_i are distinct.
    
    Let us prove the first part of the proposition. The first statement implies trivially the second one. Assume that f is algebraic over K and let P(X)=X^n +a_n-1X^n-1+… a_0 ∈ K[X] be its minimal polynomial over K. Note that n ≠ 0. Using (f)-f=c and P(f)=0, we find that 
    (P(f))-P(f)=0=( nc +(a_n-1)-a_n-1) f^n-1 +b_n-2f^n-2+…+b_0 with b_i ∈ K for i=0,…,n-2. By minimality of P(X), we find that (a_n-1)-a_n-1=-nc with a_n-1∈ K. Then, (α)- α=c with α=a_n-1/-n∈ K. We have shown that the second statement implies the third. Finally, assume that there exists α∈ K such that (α)=α +c. With (f)-f=c, we find that (α-f)=α -f. This gives that α -f ∈ L^=K^ and the element f belongs to K. 
    
    Now, let us assume that f is transcendental over K. Suppose to the contrary that there exist α∈K and i >j ∈ such that  
        α_i=(^i(α) -c - σ(c) -…-σ^i-1(c))= α_j=(^j(α) -c - σ(c) -…-σ^j-1(c)).
     
    The latter equality gives ^r(β)- β= γ where r=i-j >0, β =^j(α) and γ =^i-1(c) +… +^j(c). Since α is algebraic over K, the same holds for β. Let P(X)=X^n +a_n-1X^n-1+… +a_0 ∈ K[X] ∖ K be the  minimal polynomial of β over K. Using the fact that ^r(β)-β=γ and the minimality of P, we conclude, as above, that ^r(a_n-1)-a_n-1=-nγ, that is ^r(β̃)-β̃=γ where β̃=a_n-1/-n∈ K. Combining this equality with 
    ^r(^j(f))-^j(f)=γ, we find that β̃- ^j(f) ∈ L is fixed by ^r. By Lemma <ref>, this means that β̃- ^j(f) is algebraic over K^, which yields to f algebraic over K. We find a contradiction. 
    
    Let (K,) ⊂ (L,)  be an extension of difference fields such that L^=K^. Let f ∈ L and 0≠ c ∈ K, such that 
    (f)=f +c.  Assume that f is transcendental over K. If there exists g ∈ K(f) such that
     (g)-g ∈ K[f], then  g ∈ K[f].
    
    Let K be an algebraic closure of K, endowed with a structure of -field extension of K.  Since  f is transcendental over K, we can  write  a partial fraction decomposition of g∈K(f). Let  R be  the  largest  integer such that there exists α∈K  so that the element 1/(f -α)^R appears in the partial fraction decomposition of g.  Suppose to the contrary that R >0 and let  α∈K such that 1/(f -α)^R appears in the partial fraction decomposition of g. We deduce from  Lemma <ref>  applied to K and f, that the elements  {α_i, i∈}  are all distinct. Then, there exists N, the largest  integer such that ^N(1/(f -α)^R) appears in the partial fraction decomposition of g. The element ^N+1(1/(f -α)^R) appears in the partial fraction decomposition of (g). This proves that   ^N+1(1/(f -α)^R) appears in the partial fraction decomposition of (g)-g. A contradiction with  (g)-g ∈ K[f]. This proves that g ∈ K[f].
    
    
    
    
     §.§ Differential transcendence criteria
    
    
    
    
    In this section, a (,∂,Δ)-field K is a difference field (K,) endowed with two   derivations ∂,Δ commuting with  such that ∂Δ-Δ∂ =c_K∂ with c_K ∈ K^. We assume that ∂ is nontrivial on K, that is, it is not the zero derivation. The element c_K has to be considered as part of the data of the notion of (,∂,Δ)-field. An extension of (,∂,Δ)-fields is an  inclusion of two (,∂,Δ)-fields  (K,_K,∂_K,Δ_K ) ⊂ (L,_L,∂_L,Δ_L ) such that
    
      * K⊂ L is a field extension;
    
      * _K,∂_K,Δ_K are the restrictions of _L,∂_L,Δ_L to K;
    
      * c_K=c_L. 
      
    If there is no confusion, we shall omit the subscripts _K, _L. If  is the identity, we  shall speak of  (∂,Δ)-fields, (∂,Δ)-fields extension for short.
    
    
    
    As proved in <ref>, the following fields  are (σ,∂,Δ)-fields, that correspond respectively to the framework of the genus zero and genus one kernel curve. Remind that _ denote the automorphism of er(C^*) defined by f(s)↦ f( s) and  C_ denote the field of meromorphic functions fixed by _.
     In the two examples, we have Δ_,t=∂_t()ℓ_(s)∂_s +∂_t where ℓ_ is the so called -logarithm. That is,    an element of er(C^*)  satisfying _ (ℓ_)=ℓ_+1, and c_K=∂_t() ∂_s(ℓ_) ∈ C_.
     
      *  Let ∈ C^* with ||≠ 1. Then, the inclusion 
     
        (C_(s,ℓ_),σ_,∂_s,Δ_t, ) ⊂ ( er(C^*),σ_,∂_s,Δ_t, )
    
     is an extension of (σ,∂,Δ)-fields.
    
      *   Let  and q two elements of C^* such that |q|,||≠ 1, that are multiplicatively independent, that is,  there are no r,l ∈^2∖ (0,0) such that q^r=^l. Since C_⊂ℳer(C^*) and  C_q⊂ℳer(C^*), we 
    consider C_.C_q⊂ℳer(C^*),   the field  compositum  of  C_ and C_q inside ℳer(C^*). Then, the inclusion
    
        (C_.C_q(ℓ_,ℓ_q),σ_,∂_s,Δ_t, ) ⊂ ( er(C^*),σ_,∂_s,Δ_t, )
    
     is an extension of (σ,∂,Δ)-fields.
    
    Let (K,∂,Δ) ⊂ (L,∂,Δ). An element f ∈ L is said to be (∂,Δ)-differentially algebraic over K if there exists N∈, such that the elements 
    
      * ∂^i (f) for i ≤ N are algebraically dependent over K if  Δ is a K-multiple of ∂; 
    
      * ∂^iΔ^j(f) for i,j≤ N are algebraically dependent over K otherwise.
     Otherwise, we will say that f is  (∂,Δ)-transcendental over K.
    
    Note that since ∂Δ-Δ∂=c∂ with c ∈ K^σ⊂ K, the (∂, Δ)-field extension of K generated by some element f ∈ L coincides with the field extension of K generated by the set  
    {∂^iΔ^j(f),  i,j ∈}.
    
    
    Let us make a remark concerning the field of definition of the coefficients of the differential polynomials. 
    
    
    Let (K,∂,Δ) ⊂ (K',∂,Δ)⊂ (L,∂,Δ) and assume that K' is a field generated over K by elements that are (∂,Δ)-differentially algebraic  over K.
    By <cit.>, f∈ L  is (∂,Δ)-differentially transcendental over K if and only if it is  (∂,Δ)-differentially transcendental over K'.
    
    
    The following lemma will be crucial in many arguments: 
    
    If K ⊂ M is a -field extension  such that M^σ=K and K ⊂ L is a -field extension with L^=L. Then M and L are linearly disjoint over K.
    
    Let c_1,…,c_r ∈ L be K-linearly independent elements,  that become dependent 
    over M. Up to a permutation of the c_i's,  a minimal linear relation among these elements over M has the following form
    
        c_1 +∑_i=2^r λ_i c_i =0,
      
    with λ_i ∈ M for i=2,…,r. Computing ((<ref>))-(<ref>), we find
    
        ∑_i=2^r ((λ_i)-λ_i) c_i =0.
     
    By minimality, (λ_i)=λ_i and λ_i ∈ M^=K. By K-linear independence of the c_i, we find that λ_i=0 for i=2,…,r and then c_1=0. A contradiction.
    
    
    The following statement, whose proof is due to Michael Singer, is a  version of an old theorem of Ostrowski <cit.> and  its proof follows the lines of   the proof of <cit.>. In this last paper,  it was assumed that K^ is algebraically closed, which is not the case in  this article.  One could use the powerful scheme-theoretic tools  developed in  <cit.> to prove the result in our more general setting.  Instead we will argue in a more elementary way to reduce  Theorem <ref> to the case where K^ is algebraically closed. 
     Let (K,,∂, Δ) be a  (,∂, Δ)-field such that 
    K^ is relatively algebraically closed in K, that is there are no proper algebraic extension of K^ inside K. Let (L,,∂, Δ) be a (,∂, Δ)-ring extension of (K,,∂, Δ).  Let f ∈ L and b ∈ K such that (f)=f+b. If f is (∂,Δ)-differentially algebraic over K then there exist ℓ_1,ℓ_2∈,  c_i,j∈ K^ not all zero and g ∈ K such that 
    
        ∑_
        0≤ i≤ℓ_1,
         0≤ j≤ℓ_2c_i,j∂^iΔ^j(b)= (g)-g.
      
    Furthermore, we may take ℓ_2=0 in the case where ∂ and Δ are K-linearly dependent.  We call (<ref>) a telescoping relation for b.
    
    
    The proof of this result depends on results from the Galois theory of linear difference equations and we will refer to <cit.> and the references given there for relevant facts from this theory. Let (K, σ) be a difference field and consider the system of difference equations 
    
        σ(y_0) - y_0 = b_0,  … , σ(y_n) - y_n = b_n ,  b_0, … , b_n  ∈ K.
    
    Let us see (<ref>) as a system σ (Y)=AY, where A∈_2(n+1)(K) is a diagonal bloc matrix A=Diag(A_0,…,A_n) with A_i=[   1 b_i;   0   1 ] which correspond to the equation σ(y_i) - y_i = b_i. A Picard-Vessiot extension for σ (Y)=AY is a difference ring extension (R,σ) of (K,σ) such that: 
    
      *  there exists U∈_2(n+1)(R) such that σ (U)=AU;
    
      * R is generated as a K-algebra by the entries of U and (U)^-1;
    
      * R is a simple difference ring, that is, the σ-ideals of R are {0} and R.
    
     We will need the following result.
     
      Assume that (K,) is a difference field with K^ algebraically closed. Let R be a Picard-Vessiot extension  for the system (<ref>) and z_0, … , z_n ∈ R be solutions of this system.  If z_0, … , z_n are algebraically dependent over K, then there exist c_i ∈ K^, not all zero,  and g ∈ K such that 
    
        c_0 b_0 + … + c_n b_n = (g) - g.
    
    
    Before proving Theorem <ref>, we give a slight generalization of Lemma <ref>.
    
    Let (K,) be  a difference field with K^ relatively  algebraically closed in K and let b_0,…, b_n be some elements in K. Let (L,) be a -ring extension of (K,). Let  z_0, … , z_n ∈ L be solutions of  σ(z_i)-z_i=b_i.  If z_0, … , z_n are algebraically dependent over K, then there exist c_i ∈ K^, not all zero,  and g ∈ K such that 
    
        c_0 b_0 + … + c_n b_n = (g) - g.
    
     Let 𝐤 be the algebraic closure of K^. We extend  to be the identity on 𝐤[On the other hand, there is no unique procedure to extend a field automorphism of K^ to  the algebraic closure 𝐤. Indeed, these extensions are controlled by the Galois group of the field 𝐤 over K^.].  Under the assumption  that K^ is relatively algebraically closed, the ring K = K⊗_K^𝐤 is an integral domain and in fact is a field.   We have K^σ = 𝐤. Let L= L⊗_K^𝐤. We then have a natural inclusion of K⊂L. Let S=K[z_0,…,z_n] ⊂L. It is easily seen that S is a -ring extension of K. Let I be a maximal difference ideal in S and let R = S/I.  For each r = 0, … , n, let u_r be the image of z_r in R. Since K^σ = 𝐤 is algebraically closed and  R is a simple difference ring, we have that R is a Picard-Vessiot ring for the system associated to  (y_r) -y_r= b_r, r= 0 , …, n, over K. The elements u_0,…,u_n are algebraically dependent over K and solutions of (
    y_r) -y_r= b_r, r= 0 , …, n. Lemma <ref>  proves that  there exist c_i∈𝐤, not all zero, and g ∈K such that 
    
        ∑_0≤ i≤ n c_i b_i= (g)-g.
    
     Let {d_r}⊂𝐤 be a K^-basis of 𝐤. By Lemma <ref>, it is also  a K-basis of K.  We may write each c_i and g as
    
        c_i = ∑_r c_i,rd_r     and     g = ∑_r g_rd_r
    
    for some c_i,r∈ K^ and g_r ∈ K. Since not all the c_i are zero, there exists r such that c_i,r are not all zero. For this r, we have
    
        ∑_i ≤ n c_i,r b_i = σ(g_r) - g_r.
    
    This yields the conclusion of the proof.
     
    
    
     Assuming that f is (∂,Δ)-differentially algebraic over K, there is some finite set {∂^i_0Δ^j_0(f), … , ∂^i_nΔ^j_n(f)}⊂ L of elements that are algebraically dependent over K. Note that  j_k=0 for all k if Δ is K-linearly dependent from ∂. Since  commutes with Δ and ∂, we have for all r=0,…,n,
    
        σ(∂^i_rΔ^j_r(f)) - ∂^i_rΔ^j_r(f) = ∂^i_rΔ^j_r(b).
    
    To conclude it remains to apply Lemma <ref> with z_r= ∂^i_rΔ^j_r(f) and b_r =∂^i_rΔ^j_r(b) for r=0,…,n.
    
    
    
    
    § MEROMORPHIC FUNCTIONS ON A TATE CURVE AND THEIR DERIVATIONS
    
    
    
    In this section we translate the galoisian criteria of Theorem <ref> in the context of elliptic functions field. We start by defining the derivations.
     Studying the transcendence properties of the -logarithm, we then perform a descent on the field of coefficients and on the number of derivations involved in the telescoping relation.
      
    
    
     §.§ Derivation on nonarchimedean elliptic functions field
    
    
    Let ∈ C^* such that ||≠ 1 and let   _ denote the automorphism of er(C^*) defined  by _(f(s))=f( s). We denote 
    by C_ the field of meromorphic functions fixed by _. By Proposition <ref>, it is the 
    field of rational  functions on the Tate curve E_ or E_1/, depending whether ||< 1 or ||>1.  In this section, we construct, as in   <cit.> a derivation of these functions that encode their t-depencies and commute with _. 
    
    The fact that ∂_s=sd/ds acts on er(C^*), and its  commutation with σ_ is straightforward. Unfortunately,  the t-derivative of  may be nontrivial, implying a more complicated commutation rule between  ∂_t=td/dt and σ_. 
    More precisely, we have
    
        [               ∂_s∘_=_∘∂_s;; ∂_t∘_= ∂_t()_∘∂_s +_∘∂_t . ]
    
    
    The following statement holds.
    
    
    The ∂_s-constants er(C^*)^∂_s={ f ∈ er(C^*) | ∂_s(f)=0} of er(C^*) are precisely the constant functions C. 
     
    
     Next Lemma introduces a twisted t-derivation that commutes with _. Remind that the q-logarithm ℓ_ has been defined in <ref>.
    
    
    The following derivations of er(C^*)
        {[                      ∂_s; Δ_t,= ∂_t()ℓ_(s)∂_s+∂_t, ].
    
    commute with _. Moreover,  we have 
    
        ∂_s Δ_t,q - Δ_t,q∂_s=∂_t() ∂_s(ℓ_) ∂_s,
    
    where ∂_t() ∂_s(ℓ_) ∈ C_.
    
    Note that since ∂_s,Δ_t, commute with _, we can derive the equation _(ℓ_)=ℓ_+1 to find  _ (∂_s(ℓ_))=∂_s(ℓ_) and _(Δ_t,(ℓ_))=Δ_t,(ℓ_). We then conclude that   ∂_s(ℓ_),Δ_t,(ℓ_) belong to C_.
    
    
     
    The link with the iterates of Δ_t, and the derivatives ∂_s,∂_t is now made in the following lemma.
    
    For any i ∈, there exist c_j,k,l∈ C_ such that 
    
        Δ_t,^i= (∂_t() ℓ_)^i  ∂_s^i +∑_k=0^i-1∑_j=0^k∑_l=0^i c_j,k,lℓ_^j∂_s^k ∂_t^l.
    
    Let us prove the result by induction on i. For i=1, this comes from the fact that Δ_t,= ∂_t()ℓ_∂_s +∂_t. Let us fix i∈ and assume that the result  holds for i.  We find
    
        Δ_t,^i+1= ( ∂_t()ℓ_∂_s +∂_t)((∂_t() ℓ_)^i  ∂_s^i +∑_k=0^i-1∑_j=0^k∑_l=0^i c_j,k,lℓ_^j∂_s^k ∂_t^l ),
    
    that is
    
        Δ_t,^i+1  =  (∂_t() ℓ_)^i+1∂_s^i+1 +Δ_t,((∂_t() ℓ_)^i)  ∂_s^i +(∂_t() ℓ_)^i  ∂_t ∂_s^i +  
        ∑_k=0^i-1∑_j=0^k∑_l=0^i Δ_t,(c_j,k,lℓ_^j) ∂_s^k ∂_t^l +∑_k=0^i-1∑_j=0^k∑_l=0^i c_j,k,l∂_t() ℓ_^j+1∂_s^k+1∂_t^l+ ∑_k=0^i-1∑_j=0^k∑_l=0^i c_j,k,lℓ_^j∂_s^k ∂_t^l+1  .
     
    Note that the commutation of σ_ with Δ_t, implies that  C_ is stabilized by Δ_t,.
    Since by Remark <ref>,  Δ_t,(ℓ_) belongs to C_, we get that,  for any integer j, any c̃∈ C_, we have Δ_t,(c̃(ℓ_)^j)=Δ_t,(c̃)(ℓ_)^j+ c̃c(ℓ_)^j-1 where c =j Δ_t,(ℓ_) ∈ C_. Therefore, with Δ_t,(c̃)∈  C_, we find that Δ_t,(c̃(ℓ_)^j)∈ C_[ℓ_] is of degree at most j in ℓ_. With
    ∂_t(),c_j,k,l∈ C_, this ends the proof.
    
    
    From now on, let us fix q∈ C^* with |q|≠ 1, that is multiplicatively independent to ,
     that is there are no r,l∈^2∖(0,0)  such that q^r=^l.
     Remind that C_.C_q⊂ℳer(C^*) is the compositum of fields and ℓ_∈ℳer(C^*) is a solution of _ (ℓ_)=ℓ_+1. We now give examples of difference differential fields for σ_,∂_s and Δ_t,.
     
      The following statement	 hold.
    
      *  The field C_(s,ℓ_) is stabilized by _, ∂_s and Δ_t,. The field C_(s) is stabilized by _, and ∂_s. The field C(s) is stabilized by ∂_s, ∂_t.
    
      *  The field C_.C_q(ℓ_, ℓ_q) is stabilized by _, ∂_s and Δ_t,. The field C_.C_q( ℓ_q) is stabilized by _, and ∂_s. The field C_q(ℓ_q) is stabilized by ∂_s, ∂_t.
    
      *  (1) Since _ (ℓ_ )=ℓ_ +1, we easily see that  C_(s,ℓ_),C_(s) are stabilized by _. Since _ commutes with ∂_s and Δ_t,, the field C_ is stabilized by ∂_s and Δ_t,. It is now clear that C_(s) is stabilized by ∂_s and Δ_t,(C_(s))⊂ C_(s,ℓ_).
    By Remark <ref>, Δ_t,(ℓ_),∂_s(ℓ_)∈ C_. Combining the lasts assertions, we obtain the result for C_(s,ℓ_).  Finally, the field C(s) is stable by ∂_s, ∂_t, since C is stable by ∂_s,∂_t, and ∂_s(s)=s, ∂_t(s)=0.
     
    
      *  (2) Let us prove that C_q(ℓ_q) is stabilized by _. Using _q(ℓ_q)=ℓ_q+1 and the commutation between _ and _q, we find that 
    _ (ℓ_q)-ℓ_q∈ C_q. Similarly,  _ (C_q)⊂ C_q, proving that C_q(ℓ_q) is stabilized by _. Using ∂_s(C_)⊂ C_ and ∂_s (ℓ_q) ∈ C_q, we find that the field C_.C_q( ℓ_q) is stabilized by _ and ∂_s.  
    Let us now consider the field C_.C_q(ℓ_, ℓ_q). The field C_(ℓ_)  is clearly stable by _. From what preceede, C_q(ℓ_q) is stable by _, and therefore, C_.C_q(ℓ_, ℓ_q) is stable by _. The same arguments than those used in (1), prove that  Δ_t, (C_(ℓ_)) ⊂ C_.C_q(ℓ_)  and ∂_s(C_(ℓ_)) ⊂ C_(ℓ_).
    It remains to prove that Δ_t, (C_q(ℓ_q)) ⊂ C_.C_q(ℓ_,ℓ_q). We note that  ∂_t()ℓ_∂_s+∂_t =Δ_t,=Δ_t,q +(∂_t() ℓ_ - ∂_t(q) ℓ_q)∂_s. Since C_q is stabilized by Δ_t,q and ∂_s, we find that Δ_t,(C_q)⊂ C_.C_q(ℓ_,ℓ_q). Moreover, since ∂_s(ℓ_q),Δ_t,q(ℓ_q)
     belong to C_q, see Remark <ref>, we find that Δ_t,(ℓ_q) ∈ C_.C_q(ℓ_,ℓ_q). We have shown the inclusion Δ_t, (C_q(ℓ_q)) ⊂ C_.C_q(ℓ_,ℓ_q). This concludes the proof for C_.C_q(ℓ_, ℓ_q). 
      Let us now consider C_q(ℓ_q). By Remark <ref> and ∂_t = Δ_t,q- ∂_t(q)ℓ_q ∂_s, we find that the inclusion holds ∂_s(ℓ_q), ∂_t(ℓ_q)∈ C_q(ℓ_q). Since ∂_s,Δ_t,q commute with σ_q, C_q is stable by   ∂_s,Δ_t,q. With ∂_t = Δ_t,q- ∂_t(q)ℓ_q ∂_s, it follows that ∂_t (C_q)⊂ C_q(ℓ_q). Finally, we obtain that the field C_q(ℓ_q) is stable by ∂_s, ∂_t.
     
    
     §.§ Difference Galois theory  for elliptic function fields
    
    In this section, we  apply the results of <ref> to the specific cases of elliptic function fields  introduced in Lemma <ref>. We recall that  the following fields extensions   are (σ,∂,Δ)-fields extensions. 
      *  Let ∈ C^* with ||≠ 1. Then, let us consider 
     
        (C_(s,ℓ_),σ_,∂_s,Δ_t, ) ⊂ ( er(C^*),σ_,∂_s,Δ_t, ).
    
      *   Let  and q two elements of C^* such that |q|,||≠ 1, that are multiplicatively independent. Let us consider
    
        (C_.C_q(ℓ_,ℓ_q),σ_,∂_s,Δ_t, ) ⊂ ( er(C^*),σ_,∂_s,Δ_t, ).
    
    In that framework,  the  criteria obtained in <ref> to guaranty  the (∂_s,Δ_t,)-differential transcendence of a solution  of a rank one  -difference equation can be simplified and some descent arguments prove that the existence of a telescoping relation involving the two derivatives implies the existence of a telescoping relations involving only the derivation ∂_s. More precisely, we find the following proposition:
    
    
    
    
    Let K⊂ℳer(C^*) be a (σ_,∂_s)-field and let us assume that 
    
      * (H1)  L=K(ℓ_) is a (_,∂_s,Δ_t,)-field; 
    
      * (H2)  K^_=L^_ =C_ is relatively algebraically closed in L;
    
      * (H3)  ℓ_ is transcendental over K.
    
    Let f∈ℳer(C^*),  that satisfies _(f)=f+b, for some b that belongs to a subfield of K stable by ∂_s,∂_t.
    
    
    If f is (∂_s,Δ_t,)-differentially algebraic over L then, there exist m ∈, d_0,…,d_m ∈ C_ not all zero, and h ∈ K such that 
    
        d_0 b_1+d_1∂_s(b)+… +d_m ∂_s^m(b)=_(h)-h.
    
    Since f is (∂_s,Δ_t,)-differentially algebraic over L and K^σ_ is relatively algebraically closed, Theorem <ref> yields that there exist  M ∈, c_i,j∈ L^_ not all zero, and g ∈ L such that
    
        ∑_i,j ≤ M c_i,j∂_s^iΔ_t,^j (b)= _(g)-g.
    
    
    
    By Lemma <ref>, for all i ∈, there exist c_j,k,l∈ C_ such that 
    
        Δ_t,^i= (∂_t() ℓ_)^i  ∂_s^i +∑_k=0^i-1∑_j=0^k∑_l=0^i c_j,k,lℓ_^j∂_s^k ∂_t^l.
     
    
    The left hand side of (<ref>) is a polynomial in ℓ_ with coefficients in K. By Lemma <ref> with (H2) and (H3),  we find that  g∈ K[ℓ_] as well.
    Thus, let us write  g=∑_k=0^R α_k ℓ_^k with α_k ∈ K and α_R≠ 0.
    Let 
        N=max{j∈ |∃ i  such that  c_i,j≠ 0}.
     
    By  (<ref>), the coefficient of highest degree in ℓ_ of the left hand side of (<ref>) is  
    
        (∑_i ≤ M c_i,N (∂_t())^N ∂_s^N+i(b))  ℓ_^N.
    
     
    On the other hand, we have 
    
        _(g)-g =ℓ_^R( _(α_R)-α_R)) + ℓ_^R-1(_(α_R-1) -α_R-1 +R_(α_R))  + P(ℓ_),
     
    where P(X) ∈ K[X] is a polynomial of degree strictly smaller than R-1. Then, comparing (<ref>) and (<ref>),  we find that  
     
      *  either R< N so that 
    
        ∑_i ≤ M c_i,N (∂_t())^N ∂_s^N+i(b) = 0,
    
      *   either R=N so that 
        ∑_i ≤ M c_i,N (∂_t())^N ∂_s^N+i(b) = _(α_N)-α_N,
    
      *  or R > N so that R>0,  0≠α_R ∈ L^_. We claim that R=N-1. Indeed, R>N-1 implies _(α_R)=_(α_R), _(α_R-1) -α_R-1 +R α_R=0 and then _(α_R-1/α_R) - α_R-1/α_R +R =0  with α_R-1/α_R∈ K in contradiction  with  Lemma <ref>  applied to  f=ℓ_. Thus, we get R=N-1 and
    
        ∑_i ≤ Mc_i,N/α_R (∂_t())^N ∂_s^N+i(b)
         = _(α_R-1/α_R)-α_R-1/α_R +R.
     
    
    For all these cases, note that there exists i_0 such that c_i_0,N≠ 0 by definition of N.  Since ∂_s commutes with _, we can derive  (<ref>) with respect to   ∂_s and obtain that in any case, there exists d_k ∈ L^_= C_ not all zero and h ∈ K such that 
     
        ∑_k≤ M+1 d_k∂_s^k(b) = _(h)-h.
    
    
     §.§ Transcendence properties
    
    
    
    The goal of this subsection is to prove  some transcendence properties of the -logarithm in order to perform some descent procedure on telescopers. More precisely, we need to prove that the assumptions (H1) to (H3) of Proposition <ref> are satisfied for the fields C_(s) and C_.C_q(ℓ_,ℓ_q) for   and q  two multiplicatively independent elements of C^* with |q|≠ 1, ||≠ 1.  We  recall  that q and  are multiplicatively independent if  there are no (r,l) ∈^2∖(0,0)  such that q^r=^l. Remind that C_.C_q⊂ℳer(C^*) is the compositum of fields and ℓ_∈ℳer(C^*) is a solution of _ (y)=y+1. With Lemma <ref>,
     (H1) of Proposition <ref> is satisfied for K=C_(s) and K=C_.C_q(ℓ_q).
     
    
    Any element  in a _-extension of   C_q[We recall that since  _ and _q commute, the field C_q is a _-field.] that is algebraic over C_q and 
    invariant by _  is in C.    Any element  in a _q-extension of   C_ that is algebraic over C_ and 
    invariant by _q  is in C. 
    The two statements are symmetrical, so let us only prove the first one. First let us prove that C_q ∩ C_=C. Let f be an element of C_q that is _-invariant.
    Suppose to the contrary that f is nonconstant. Then f has a nonzero pole c.
     Since _(f)=f, the multiplication by  induces a permutation of the poles of f modulo q. Since the set of poles modulo q is a finite set,  there exists m ∈   such that ^m c =q^dc for some d ∈. A contradiction with the fact that q and  are multiplicatively independent. 
     Now, let f be  in a _-extension of   C_q  algebraic over C_q and 
    invariant by _. Let μ(X) ∈ C_q[X] be the monic minimal polynomial of f above C_q. Since _(f)=f, we easily see that the coefficients of μ must be fixed by _. Then, 
    these coefficients belong to C_q∩ C_, which is equal to C. Then, f is algebraic over C. The latter field being algebraically closed, we conclude that f ∈ C.
     
    The following statements hold:
    
      *   the fields C_ and C_q are linearly disjoint over C;
    
      *  for all  α∈ C_.C_q, _q(α) ≠α +1 and 
    _(α) ≠α +1;
    
      *  for all  α∈ C_(s),  
    _(α) ≠α +1.
    
      *  (1) This is Lemmas   <ref> and <ref>  with K=C, M=C_q and 
    L=C_, σ=σ_.
    
    
    
      *  (2) Suppose to the contrary  that there exists α∈ C_.C_q, such that _q(α) = α +1. Since 
    C_q is by Proposition <ref>, the field of meromorphic functions over a Tate curve, there exist x,y  ∈ C_q such that x is transcendental over C, y algebraic of degree 2 over C(x) and C_q=C(x,y). Since C_ is linearly disjoint from C_q over C, the field  C_.C_q equals C_(x,y) and there are P(X),Q(X) ∈ C_(X) such that α =P(x)y+Q(x). Since x,y are fixed by _q and y is of degree 2 over C_(x), we deduce from _q(α)=α +1 that
    P^_q(x)=P(x) and
     Q^_q(x)-Q(x)=1 where P^_q(X) (resp.  Q^_q(X)) denotes the fraction obtained from P(X) (resp. Q(X)) by applying _q to the coefficients.
    Let  C_  be some algebraic closure of C_. We endow C_ with a structure of _q-field extension of C_.  Let us  write  Q(X)=c_r/X^r+…+c_1/X +R(X) with R ∈C_(X) with no pole at X=0. Then, since x is   transcendental over C_ and fixed by _q
        Q^_q(x)-Q(x)=1=_q(c_r)-c_r/x^r+…+_q(c_1)-c_1/x +R^_q(x)-R(x).
    
    Using the transcendence of x over C_, we find that 1=_q(β̃)-β̃ for β̃=R(0) ∈C_.  There exists a unique derivation extending  ∂_s to C_ and this derivation commutes with _q. Denoting this derivation by ∂_s and  
     deriving  1=_q(β̃)-β̃, we conclude  that
     ∂_s (β̃) ∈ C_q ∩ C_^r. Note that q and ^r are multiplicatively independent. By Lemma <ref>, we  find that  ∂_s(β̃) ∈ C which leads to β̃=cs+d for some c,d ∈ C. A contradiction with 1=_q(β̃)-β̃.
    The proof for  is similar. 
    
    
      *  (3)  Let  α∈ C_(s). Using the partial fraction decomposition of α in C_(s), the fact that _(s)= s and  the transcendence of s over C_, one can easily see that _(α)-α≠ 1.
     
    The following statements hold:
    
      *  the function ℓ_ (resp. ℓ_q) is transcendental over C_.C_q;
    
      *   the function ℓ_
    is transcendental over C_(s). In particular, (H3) of Proposition <ref> is satisfied for K=C_(s).
    
      *  (1)
    Since _(ℓ_) =ℓ_ +1 and C_⊂(C_.C_q)^_⊂ er(C^*)^_= C_, we can apply Lemma <ref> and find that ℓ_ is algebraic over C_.C_q if and only if there exists α∈ C_.C_q such that
    _(α) =α +1. We conclude by Lemma <ref>. The proof for ℓ_q is symmetrical. 
    
      *  (2)
    Since _(ℓ_) =ℓ_ +1 and C_⊂ (C_(s))^_⊂ er(C^*)^_=C_, we can apply Lemma <ref> and find that ℓ_ is algebraic over C_(s) if and only if there exists α∈ C_(s) such that
    _(α)=α +1. We again conclude by Lemma <ref>.
    
     The following statement  hold:
     
      *   let f ∈ C_q. If  there exists α∈ C_.C_q  satisfying _(α)-α= f, then there exists β∈ C_q such that 
    _(β)-β =f; 
     
      *   let f ∈ C_.C_q. If there exists α∈ C_.C_q(ℓ_q)  satisfying _(α)-α= f, then, there exist ã∈ C_, b̃∈ C_.C_q such that 
    _(ãℓ_q +b̃)- (ãℓ_q +b̃) =f.
    
    
      *  (1)
     Analogously to the proof of Lemma <ref>, let us write α =P(x)y+Q(x) for P(X),Q(X) ∈ C_q(X) and C_=C(x,y).  Reasoning as in the proof of Lemma <ref>,  we find that Q^_(x)-Q(x)=f. Since x is transcendental over C_q, we conclude as in Lemma <ref> that  there is β̃∈C_q, for some C_q algebraic closure of C_q such that   _(β̃)-β̃ =f. Since by Lemma <ref>, C_q^σ_= C_q^σ_=C, Lemma <ref> implies that there exists  β∈ C_q such that 
    _(β)-β=f.
    
      *  (2) First of all, let us note that since _ and _q commute, there exists d ∈ C_q such that 
    
        _(ℓ_q)=ℓ_q+d.
    
    By Lemma <ref>, the function ℓ_q is transcendental over C_.C_q. This implies that ℓ_q∉ C_ and then d≠ 0.  Since C_.C_q(ℓ_q)^_=C_ =ℳer (C^*)^_=C_.C_q^_=C_,  Lemma <ref>, applied to _(ℓ_q)=ℓ_q +d, implies that there exists P ∈ C_.C_q[X] such that 
    
        f= _( P(ℓ_q)) -P(ℓ_q).
    
    Now, let us write P(X)=∑_k=0^N a_k X^k with a_k ∈ C_.C_q,  and N minimal. We find 
    
        f =(_(a_N)-a_N) ℓ_q^N+ (_(a_N-1)-a_N-1 +Nd _(a_N))ℓ_q^N-1+ 
         N-1.
    
    
     We conclude in view of (<ref>) that if N =0 we are done by setting ã=0 and b̃=a_N. Let us now assume that N>0. Then, by minimality of N, _(a_N)=a_N. We claim that _(a_N-1)-a_N-1 +Nd _(a_N) =_(a_N-1)-a_N-1 +Nd a_N ≠ 0. To the contrary, _(a_N-1)=a_N-1 -Nd a_N implies _( a_N-1/a_N +N ℓ_q)=a_N-1/a_N +N ℓ_q and a_N-1/a_N +N ℓ_q∈ C_, contradicting the transcendence of ℓ_q over C_.C_q, see Lemma <ref>. This proves the claim. If N>1, then (<ref>) with _(a_N)=a_N and
     _(a_N-1)-a_N-1 +Nd a_N ≠ 0, would give an equation of order N-1 which would contradicts the transcendence of ℓ_q over C_.C_q. This proves that N=1 and f = _(a_1 ℓ_q +a_0) - (a_1 ℓ_q +a_0) for some a_1 ∈ C_,a_0∈ C_.C_q.
    
    The function ℓ_ is transcendental over C_.C_q(ℓ_q). In particular, the assumption (H3) of Proposition <ref> holds for  K=C_.C_q(ℓ_q).
    
    By Lemma <ref>, the function ℓ_ is algebraic over C_.C_q(ℓ_q) if and only if we have ℓ_∈ C_.C_q(ℓ_q).  Suppose to the contrary that ℓ_∈ C_.C_q(ℓ_q). Since 1 =_(ℓ_)-ℓ_, we conclude by Lemma <ref> that there exist ã∈ C_, b̃∈ C_.C_q such that 1 =_(ãℓ_q +b̃)- (ãℓ_q +b̃). Combining this equation with _(ℓ_)-ℓ_ =1, we find that _(ℓ_)-ℓ_=_(ãℓ_q +b̃)- (ãℓ_q +b̃), proving that _(ãℓ_q +b̃-ℓ_)=ãℓ_q +b̃-ℓ_∈ C_. Then, there exists b_1∈ C_.C_q such that 
    
        ℓ_ =aℓ_q +b_1.
    
    Deriving (<ref>) with respect to ∂_s, we find
    
        ∂_s(ℓ_) =∂_s(a ) ℓ_q+ a∂_s(ℓ_q) +∂_s(b_1 ).
    
    By Remark <ref>, ∂_s(ℓ_), ∂_s(ℓ_q)∈ C_.C_q. In virtue of the commutation between ∂_s and _,_q, the fields C_q,C_ are stabilized by ∂_s, which implies ∂_s(a ),  ∂_s(b_1 ) ∈ C_.C_q. By Lemma <ref>, the function  ℓ_q is transcendental over the latter field, we conclude that ∂_s(a)=0 and therefore a∈ C. In particular it belongs to C_ and C_q. Using 1 =_(ãℓ_q +b̃)- (ãℓ_q +b̃), we find 
    
        1-a d =_(b)-b,
    
    where d =_(ℓ_q)-ℓ_q ∈ C_q, see (<ref>).
    Since 1-a d∈ C_q, we conclude by Lemma <ref>, that there exists b_2∈ C_q such that 1-a d =_(b_2)-b_2. Replacing the left hand side gives 
        _(ℓ_)-ℓ_-
        _(aℓ_q)+aℓ_q=_(b_2)-b_2.
    
    
    This shows that ℓ_-
    aℓ_q-b_2∈ C_ and then, there exists c ∈ C_ such that ℓ_ + c= aℓ_q +b_2. Deriving this equation with respect to ∂_s, we find (we use  
    ∂_s(a)=0)
    
        ∂_s(ℓ_) +∂_s (c) = a∂_s(ℓ_q)+ ∂_s(b_2).
     
    By Remark <ref>, the left hand side  of the equation belongs to C_ whereas the right hand side is in C_q. By Lemma <ref>, we conclude that ∂_s(ℓ_ +c) ∈ C. This means that  there exist a_0, b_0 ∈ C such that ℓ_ =a_0 s+ b_0 -c in contradiction with ℓ_ transcendental over C_(s), see Lemma <ref>.
    
    
    We can now prove  that our fields satisfy the assumption (H2) of Proposition <ref>.
    
    
    The following holds:
    
      * C_ is relatively algebraically closed in C_  (s, ℓ_);
    
      * C_ is relatively algebraically closed in C_.C_q ( ℓ_,ℓ_q).
    
    In particular, (H2) of Proposition <ref> holds for K=C_(s) and K=C_.C_q(ℓ_q).
      *  (1)
    The first point is a consequence of transcendence of s over C_, and the transcendence of ℓ_
     over C_(s), see Lemma <ref>.
    
    
      *  (2) Let us prove the second point.  Let us start by proving that C_ is relatively  algebraically closed in C_.C_q. As in the proof of Lemma <ref>, we have C_= C(x,y) and C_.C_q = C_q(x,y)  where  y is of degree 2 over both C(x) and C_q(x). Let f ∈ C_q(x,y). Then f = P(x) y + Q(x) with P(x), Q(x) ∈ C_q(x). 
    If f is algebraic over C_ then Lemma <ref> implies that _^r(f) = f for some r ∈^* and therefore _^r(P(x)) = P(x) and _^r(Q(x)) = Q(x). We claim that P(x) and Q(x) are in C(x), and therefore that f ∈ C_. Let P(x) = P_1(x)/P_2(x) where P_1(x), P_2(x) ∈ C_q[x] are relatively prime and P_1(x) is monic.  We then have that _^r(P_1(x))P_2(x)  =_^r(P_2(x)) P_1(x) and consequently P_1(x) divides _^r(P_1(x)) (resp. _^r(P_1(x)) divides P_1(x)). Since P_1(x) is monic,  P_1(x) = _^r(P_1(x)) and P_2(x) = _^r(P_2(x)). This implies that the coefficients of P_1(x) and P_2(x) are left fixed by _^r . Note that by assumption, q and ^r are multiplicatively independent. Therefore, by Lemma <ref>, applied with  replaced by ^r, P_1, P_2 ∈ C[X]. The proof for Q is similar. This proves our claim and show that f∈ C_. Then C_ is relatively  algebraically closed in C_.C_q. 
    
    Note that Lemma <ref> implies that ℓ_ is transcendental over C_.C_q  and Lemma <ref> implies that ℓ_q is transcendental over C_.C_q(ℓ_).  Therefore C_ is relatively algebraically closed in C_.C_q (ℓ_, ℓ_q).
    
    
    Finally, we prove a lemma that will allows us to descend some telescoping relations on smaller base fields.
    
    
    Let b ∈ C_q such that there exist N∈, c_i∈ C_ with c_N ≠ 0, and g ∈ C_.C_q (ℓ_,ℓ_q) that satisfy
    
        ∑_i=0^N c_i∂_s^i(b)=_(g)-g.
    
    Then, there exist m ∈, d_0,…,d_m ∈ C not all zero and h ∈ C_q such that 
    
        d_0 b_2+d_1∂_s(b_2)+… +d_m ∂_s^m(b_2)=_(h)-h.
    
     First of all note that the left hand side of (<ref>) belongs to C_.C_q.
    By Lemma <ref>, the function ℓ_ is transcendental over 
    C_.C_q(ℓ_q). 
    By Lemma <ref>,  g ∈ C_.C_q(ℓ_q)[ℓ_]. So let us write   g=∑_k=0^R α_k ℓ_^k with α_k ∈ C_.C_q(ℓ_q), α_R≠ 0.
    
    Claim. There exist m ∈, c'_k ∈ C_, c'_m ≠ 0, and α∈ C_.C_q(ℓ_q) such that 
     
        ∑_k=0^m c'_k∂_s^k(b) = _(α)-α.
    
    If R=0 the claim is proved. Assume that R>0.
    Then, we have 
    
        _(g)-g =ℓ_^R( _(α_R)-α_R)) + ℓ_^R-1(_(α_R-1) -α_R-1 +R α_R)  + P(ℓ_),
     
    where P(X) ∈ C_.C_q(ℓ_q)[X] is a polynomial of degree smaller than R-1.
    Then, comparing (<ref>) and (<ref>),  we find, by transcendence of ℓ_ over C_.C_q(ℓ_q), see Lemma  <ref>, that _(α_R)=α_R. Let us prove that _(α_R-1) -α_R-1 +R α_R≠ 0. Indeed if _(α_R-1) -α_R-1 +R α_R=0 then _(α_R-1/α_R) - α_R-1/α_R +R =0  with α_R-1/α_R∈ C_.C_q in contradiction  with  Lemma  <ref> and Lemma <ref>. We then obtain that R=1 since otherwise we would deduce from (<ref>) an algebraic relation for ℓ_ over C_.C_q(ℓ_q), contradicting Lemma <ref>. Thus, 
    
        ∑_i=0^Nc_i/α_1∂_s^i(b)  = _(α_0/α_1)-α_0/α_1 +1.
    
    Remind that α_1 ∈ C_ and the latter field is stable by ∂_s due to the commutation between ∂_s and _.  By Lemma <ref>, the field C_.C_q(ℓ_q) is stabilized by ∂_s. We can  derive (<ref>) with respect to  ∂_s and using the commutation between _ and ∂_s, we obtain our claim.
    
    Claim. There exist 
    M ∈, d_k ∈ C_, d_M ≠ 0 and β∈ C_.C_q such that 
    
        ∑_k=0^M d_k∂_s^k(b) = _(β)-β.
    
    Indeed, by Lemma <ref>, we can find a ∈ C_, b ∈ C_.C_q such that 
    
        ∑_k=0^m c'_k∂_s^k(b) = _(aℓ_q +b)-(a ℓ_q +b).
    
    Either a =0 and ∑_k c'_k∂_s^k(b) = _(b)-(b) for some b ∈ C_.C_q. Or a ≠ 0 and dividing (<ref>) by a and deriving with respect to ∂_s, we find
    
        ∑_k=0^m+1 d_k∂_s^k(b) = _(∂_s(ℓ_q)  +∂_s(b/a))-(∂_s( ℓ_q) +∂_s(b/a)),
    
    where the d_k are in C_, d_m+1=c'_m/a≠ 0. Furthermore, by Remark <ref> and the fact that C_, C_q, are stable by ∂_s, we find
     ∂_s(ℓ_q)+∂_s(b/a) ∈ C_.C_q. This proves the claim.
     
    Now, let us consider an equation of the form
    
        ∑_k=0^M d_k∂_s^k(b) = _(β)-β,
    
     with β∈ C_.C_q, d_k ∈ C_ and d_M ≠ 0, minimal with respect to the maximal  order of derivation M of b.  We can write this minimal equation as follows
    
        d_M∂_s^M(b)+ ∑_k=0^M-1 d_k∂_s^k(b) = _(β)-β,
    
    with d_M ∈ C_^*. Then dividing by d_M, we find
    
        ∂_s^M(b)+ ∑_k=0^M-1d_k/d_M∂_s^k(b) = _(β/d_M)-β/d_M.
    
    Therefore, we can without loss of assumption assume that d_M=1. Now, if we compute the element _q(_(β)-β))-(_(β)-β)) and use the fact that b ∈ C_q, we find
    
        ∑_k=0^M-1(_q( d_k)- d_k)  ∂_s^k(b) = _(  _q( β)-β)-( _q( β)-β).
    
    By minimality, we find that, for all k, the element d_k ∈ C_ is fixed by _q. This means that d_k ∈ C by Lemma <ref>.
    
    Since ∂_s^M(b)+ ∑_k=0^M-1 d_k∂_s^k(b) ∈ C_q and 
    ∂_s^M(b)+ ∑_k=0^M-1 d_k∂_s^k(b)=_(β)-β with β∈ C_.C_q, Lemma   <ref>  shows that we have the existence of h ∈ C_q such that 
     
        ∂_s^M(b)+ ∑_k=0^M-1 d_k∂_s^k(b)=_(h)-h.
    
    
    
    The   results of Appendix <ref>  are summarized in  the following crucial corollary.
    
    
    
    The assumptions  of Proposition <ref> are satisfied for 
    
      *  Genus zero case: K=C_(s) and b ∈ C(s) with ∈ C^* such that ||≠ 1;
    
      *  Genus one case: K=C_.C_q(ℓ_q) and b ∈ C_q(ℓ_q) with ,q∈ C^* such that ||,|q|≠ 1 and  and q are multiplicatively independent.
    
    The fact that the field K and b satisfy the assumptions (Hi)  is  Lemmas <ref>,  <ref>, <ref>, and <ref>.
    alpha
