Functional relations for elliptic polylogarithms
Johannes Broedel Andre Kaderli

[1]Title Pagetitle empty


  

  






  





^aInstitut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin

IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany



^bMax-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut

Am Mühlenberg 1, 14476 Potsdam, Germany







jbroedel@physik.hu-berlin.de, kaderlia@physik.hu-berlin.de









Abstract


13.4cm
  Numerous examples of functional relations for multiple polylogarithms are
  known. For elliptic polylogarithms, however, tools for the exploration of
  functional relations are available, but only very few relations are identified. 

  Starting from an approach of Zagier and Gangl, which in turn is based on
  considerations about an elliptic version of the Bloch group, we explore
  functional relations between elliptic polylogarithms and link them to the
  relations which can be derived using the elliptic symbol formalism. The
  elliptic symbol formalism in turn allows for an alternative proof of the validity of
  the elliptic Bloch relation. 

  While the five-term identity is the prime example of a functional identity
  for multiple polylogarithms and implies many dilogarithm
  identities, the situation in the elliptic setup is more involved: there is no
  simple elliptic analogue, but rather a whole class of elliptic identities.















































§ INTRODUCTION



The majority of calculations in quantum field theory, in particular when
considering quantum chromodynamics, is based on the evaluation of integrals
associated to Feynman graphs. Using Feynman parameters one can rewrite
integrations over loop variables into integrations over Feynman parameters in a
formalised manner. Along with several further advantages, the reparametrisation
allows to bring the integrals to an iterated form.

This is a rather general concept: Feynman integrals seem to be expressible in
terms of iterated integrals over a suitably chosen set of differential forms on
Riemann surfaces of various genera. The exploration of classes of these
iterated integrals and the utilisation of their algebraic properties did not
only change the way calculations are performed, but simultaneously leads to
convenient representations: once a proper class of functions is identified, one
can find functional relations and thus reduce to a basis of
integrals.  

It turns out that suitable differential forms defining classes of iterated
integrals can be identified starting from geometrical considerations: taking
the first abelian differential on the simplest genus-zero surface, the Riemann
sphere, leads to the class of multiple polylogarithms
<cit.>
while abelian differentials on a genus-one Riemann surface are the starting
point for the elliptic polylogarithms <cit.> to be
discussed in this article.  



  
Genus zero: Multiple polylogarithms have been a very active field
of research in the last years: since their motivic version constitute a graded
Hopf algebra <cit.>, with
the shuffle product as algebra multiplication and the deconcatenation
coproduct, there are very strong tools available <cit.> allowing in
particular to derive functional relations.  While the Duval algorithm
<cit.> delivers a basis with respect to the shuffle product,
further relations between different arguments of polylogarithms can be explored
using the coproduct, which is usually referred to as the symbol map. A
non-exhaustive list of examples, where such relations are investigated, is
Kummer1840,Wechsung:1970,GONCHAROV1995197,wojtkowiak1996,Gangl2003,Zagier:2007knq,2015arXiv150902869S. Examples involving the evaluation of Feynman integrals include Tarasov:2008hw,Tarasov:2015wcd,Davydychev:2016dfi,Davydychev:2017bbl,Tarasov:2019mqy. We are mainly interested in functional relations of the dilogarithm. Of
particular importance hereby is the so-called five-term identity


    (t)+(s)+(1-t/1-ts)+(1-ts)+(1-s/1-ts)=0 ,


where (t)= (_2(t)-log(|t|)_1(t)) is the
Bloch-Wigner function, the single-valued version of the dilogarithm. The
five-term identity has a beautiful interpretation in terms of a volume
decomposition in hyperbolic space into (hyperbolic) tetrahedra. In addition, it
is known <cit.> to create a large class of
functional equations
for the dilogarithm which are linear combinations of Bloch-Wigner functions
where the arguments are rational functions of one variable and satisfy a
particular condition, to be explained below. Similar statements are conjectured
to hold in more general situations where the arguments are allowed to be
algebraic functions or rational functions of more than one variable
<cit.>.  
On the physics side, the idea of splitting a given volume into several polyhedra has been used to interpret and reformulate the calculation of various Feynman diagrams, see for example refs. <cit.>.
Linear combinations of values of the Bloch-Wigner
function which satisfy the mentioned condition above and which are equal modulo
finitely many applications of functional relations of the Bloch-Wigner function
are identified in the Bloch group <cit.>.
Similarly, higher Bloch groups have been investigated in the context of higher
order polylogarithms. 



  
Genus one: While elliptic polylogarithms have been explored for a
long time <cit.>, it is only recently that
they have been facilitated in the calculation of scattering amplitudes in
physics <cit.>. However, as became apparent,
many of the structures inherent in multiple polylogarithms can be taken to
genus one easily: iterated integrals on genus one allow for a natural shuffle
multiplication and an associated coaction or symbol map <cit.>. 

Given the existence of the symbol map for elliptic iterated integrals, it is a
natural problem to investigate functional relations for elliptic
polylogarithms. In particular, an elliptic analogue of the Bloch group has been
considered in ZagierGangl, which is based on a class of
functional relations for an elliptic generalisation of the Bloch-Wigner
function, the elliptic Bloch-Wigner function , and given by relations of
the form


    (η_F)=0


where the object η_F is parametrised by (some of the zeros and
singularities of) any non-constant elliptic function F <cit.>.
However, a similar functional relation for the elliptic Bloch-Wigner function
and the construction of an elliptic analogue of the Bloch group has already
been discussed in 1995alg.geom..8008G.  In contrast to the genus-zero
case, where the five-term identity suffices to represent a large class of
functional identities of the dilogarithm, a whole class of functional
identities given by eqn:ellBlochRel needs to be investigated in the
genus-one case <cit.>. The considerations therein, however, remain
on the level of a few particular examples, e.g. an implicitly defined elliptic
analogue of the five-term identity. As will be described in detail below, the
answer to the question of an explicit elliptic five-term identity and the
explicit description of the other elliptic functional identities generated by
eqn:ellBlochRel requires substantially more technical effort than for
classical polylogarithms.



In this article, we are going to put Zagier's and Gangl's method to work in
order to find several examples of functional identities between simple elliptic
polylogarithms.  The resulting relations are going to be contrasted with
relations derived using the elliptic symbol map. In order to compare the two
types of relations, one has to translate between different formulations of the
elliptic curve, and thus different types of (iterated) integrals, which is a
source of the complexity of the problem. Despite those difficulties we find
several relations connecting elliptic polylogarithms of rather complicated
arguments. In some cases, the relations found can be trivially accounted to
known symmetry relations for the elliptic Bloch-Wigner function. 

The translation of the elliptic Bloch-Wigner function to the torus, represented
as the complex plane modulo a two-dimensional lattice /Λ, allows a
new perspective on the elliptic Bloch relation: the condition encoded in
eqn:ellBlochRel above translates into rather simple relations between
iterated elliptic integrals on the torus, whose correctness is not difficult to
show.  Thus the translation combined with the elliptic symbol calculus provides
an alternative proof of the elliptic Bloch relation. 

As an aside, we are going to translate Ramakrishnan's generalisations of the
elliptic Bloch-Wigner dilogarithm <cit.> as well as
Zagier's generalised single-valued elliptic polylogarithms <cit.> to the
torus formulation of the elliptic curve. These representations will be serving
as a starting point for the investigation of relations between higher
elliptic functions in a forthcoming project.   

Given the general structure of the elliptic curve, it was not to be expected
that functional relations are at the same level of simplicity as their
genus-zero cousins. On the one hand, the calculation of zeros and poles of
elliptic functions is more complicated than in the case of rational functions
on the Riemann sphere. On the other hand, the translation from the projective
formulation of the elliptic curve, where the mentioned zeros and poles may be
described in terms of rational functions, to the torus given by Abel's map is
not algebraic and highly non-trivial.



This article is structured in the following way: in sec:review we
present some of the well-known results for functional relations of the
Bloch-Wigner function and in particular the construction of the Bloch group and
the Bloch relation. In sec:toolbox we review several known concepts:
we set the notation for different formulations of elliptic curves as well as
elliptic functions and review known results about the Bloch group in the
genus-one situation, which are mostly formulated on the Tate curve describing
the corresponding elliptic curve. *sec:connection is devoted to the
translation of the above and further concepts to the torus and the projective
elliptic curve. In particular, notions of (conjecturally) single-valued
elliptic generalisations of polylogarithms defined on the Tate curve are
related to the elliptic multiple polylogarithms as holomorphic iterated
integrals on the torus, which further allows to formulate (and prove) the
elliptic Bloch relation (<ref>) on the torus and the projective
elliptic curve, respectively. 



§ BLOCH GROUPS FOR POLYLOGARITHMS

 The description of functional relations of polylogarithms
and in particular of the single-valued dilogarithm – the Bloch-Wigner function
– can be formalised using the concept of (higher) Bloch groups. These are
certain (abelian) groups _m which capture functional relations satisfied
by single-valued polylogarithms of order m.

In sec:review:BG we are going to review the geometric construction
and interpretation of _2 in terms of hyperbolic three-manifolds.
Afterwards, in ssec:dilogrel we introduce the Bloch relation of the
Bloch-Wigner function, which generates functional identities such as the
five-term identity. In the subsequent section this Bloch relation will be
generalised to the elliptic curve and will be used to define the elliptic
analogue of _2, the elliptic Bloch group, which is discussed in
sec:toolbox:ellBloch. 



 §.§ The Bloch group

The functional relations of the dilogarithm _2 often take a very simple
form when expressed in terms of the Bloch-Wigner function 


    (t)   = (_2(t)-log(|t|)_1(t)) ,


which is the single-valued version of the dilogarithm
(see Zagier:2007knq for an extensive review of the Bloch-Wigner
function). The Bloch-Wigner function is continuous on the Riemann sphere and
real analytic except at the points 0, 1 and ∞, where it is defined
to vanish. 

The Bloch-Wigner function and its functional relations admit a broad variety of
mathematical interpretations and applications, ranging from  periodicities of a
cluster algebra
<cit.>, volumes in
hyperbolic space <cit.> and the symbol calculus
<cit.> to functional identities
generated by rational functions on the Riemann sphere <cit.>, the
latter is the main focus of our considerations.

The Bloch-Wigner function satisfies the symmetry relations 


    (t)=(1-1/t)=(1/1-t)=-(1/t)=-(1-t)=-(-t/1-t)


and the duplication relation


    (t^2)=2(t)+2(-t) ,


which can be easily proven using the properties of the logarithm and _2.
In addition, there is the famous five-term identity already mentioned in the
introduction, which can be described as a consequence of the periodicity of the
A_2 cluster algebra <cit.>. It reads 


    (t)+(s)+(1-t/1-ts)+(1-ts)+(1-s/1-ts)   =0 .


In order for the above equation to yield a valid new relation, t and s are numbers
chosen such that neither of the arguments yields 0,1 or ∞, i.e. s,t≠ 0,1 and st≠ 1. In those
special cases, however, sec:invitation:BlochGroup:fiveterm degenerates to
the symmetry relations in
sec:invitation:BlochGroup:symD above. 

Alternatively, one can interpret the five-term identity as a relation between
volumes of hyperbolic three-simplices in the so-called Poincaré half-space
model <cit.>. As it is this volume interpretation of
the Bloch-Wigner function which leads to an illustrative geometric construction
of the Bloch group [The Bloch group ℬ_2 has originally
been introduced in Bloch78 and has been extended in refs.<cit.> to higher orders.], let us describe this
construction in a little more detail following the lines of refs.<cit.>.  The volume of a complete, finite,
hyperbolic three-manifold M can be triangulated and, thus, expressed as the
sum over the volumes of a finite number of three-simplices  


    (M)   =∑_i (t_i) ,


each of which can be labelled by a cross ratio t_i∈  such that its
volume is given by (t_i).  Considering the geometric properties of such a
triangulation, one can show that the associated coordinates t_i in
eqn:VolM have to satisfy the following algebraic constraint
<cit.>:


    ∑_i t_i∧ (1-t_i)=0∈^∗∧^∗ .


Correspondingly, one can in general express the volume of M as  


    (M)   =∑_i (t_i)=(ξ) ,


for an element ξ∈_2(), where


    _2()   ={∑_i=1 ^n n_i(t_i) | t_i∈^∗∖{1},n∈, n_i∈,  ∑_i=1^n n_i(t_i∧ (1-t_i))=0}⊂_ ,


_ is the free abelian group[The free abelian group generated
	by a set S is the group of formal finite sums ∑_s∈ S n_s (s)
	with n_s∈, all but finitely many equal to zero. The group
	operation is defined by ∑_s∈ S n_s (s)+∑_s∈ S m_s
	(s)=∑_s∈ S (n_s+m_s) (s) and the identity element is the empty
sum. Note that in contrast to the usual notation where square brackets are used
to denote an element of the free abelian group, we use parentheses in agreement
with the notation of divisors introduced in ssec:dilogrel.]
generated by  and the Bloch-Wigner function is extended by linearity to
_, i.e.


    (∑_i n_i(t_i))=∑_i n_i (t_i) .


Let us briefly discuss the definition (<ref>) of _2(). The
condition t_i∉{0,1} corresponds to the definition of
(0)=0=(1). The fact that now, we allow in ∑_i=1^n
n_i(t_i∧ (1-t_i))=0 the coefficients n_i to be any integer
and not only to equal 1, as in eqn:TriangulationCondition, is required
to turn _2() into a subgroup of _ and to
(uniquely) shorten the sum (<ref>) in the case of t_i=t_j for
i≠ j. 

The geometric interpretation of the five-term identity corresponds to a change
of triangulation: it describes two distinct triangulations of a volume defined
by five vertices and the edges being geodesics, which can either be described
by a disjoint union of two hyperbolic three-simplices or of three such
simplices. The five-term identity expresses the equality of the sum of the
volumes of the former two and the latter three simplices. The change of
triangulation and the associated applications of the five-term relation
motivate the definition of the following subgroup of 𝒜_2(),
which can be thought of as constituting the group of relations of the
Bloch-Wigner function with the generator[For T⊂ S, the
subgroup ⟨ t|t∈ T⟩ of _ generated by T is the group
of formal finite sums ∑_t∈ T n_t (t) with n_t∈, all but
finitely many equal to zero.] being the arguments occurring in the five-term
identity,


    _2()   =⟨(t)+(s)+(1-t/1-ts)+(1-ts)+(1-s/1-ts)| s,t∈^∗∖{1}  , st≠ 1⟩  .


Thus, the volume of M can be expressed as the value


    (M)   =(ξ_M)


for a canonical ξ_M∈_2() associated to M with 


    _2()=_2()/_2()


being the Bloch group[Higher Bloch groups _m() for m>2 can
	be constructed recursively <cit.>. In analogy to the case
	_2() considered above, the subgroup 𝒞_m() of the
	group of "allowable" points 𝒜_m() (where allowable can
be defined recursively and corresponds to the condition ∑_i=1^n
n_i(t_i∧ (1-t_i))=0 in the case m=2) is constructed to be
the span of functional relations among polylogarithms of order m.]. 

Besides this geometric construction, the Bloch group _2() is an
elementary algebraic structure for the description of dilogarithmic functional
relations, i.e. identities of finite sums such as ∑_i n_i
(t_i(s_j))=c, for rational or algebraic functions t_i of one or more
variables s_j and some constant c∈. In the case of only one variable
s and rational functions t_i(s)∈(s), the element ξ =∑_i n_i
(t_i(s)) evaluates under  to a constant if and only if ∑_i n_i
(t_i(s))∧(1-t_i(s)) is independent of s
<cit.>. For the particular condition ∑_i n_i
(t_i(s))∧(1-t_i(s)) =0, the element ξ belongs
to the Bloch group ℬ_2((s)) of the field of rational functions
(s). As proven in wojtkowiak1996, all such elements are equal to
zero in ℬ_2((s)). Thus, in this case the functional equation
∑_i n_i (t_i(s))=0 is indeed obtained by a finite number of
applications of the five-term identity. Similar statements are not known in the
case of algebraic functions or rational functions in more than one variable,
but they are expected to exist, see e.g. <cit.>.



 §.§ Bloch's dilogarithm relations


Bloch describes a concept to formalise the generation of functional
identities for the Bloch-Wigner function and of its generalisation to elliptic
curves <cit.>. In this subsection we state his results in the
classical situation and generalise it to the elliptic case in
sec:toolbox below.


In the following, we are going to make use of the concept of a
divisor: for any meromorphic function g defined on a compact Riemann
surface X, the divisor of g is defined as 


    (g)=∑_p∈ X_p(g) (p)  ,


where _p(g) is the order of the pole (a negative integer) or the order of
the zero (a positive integer), respectively, of g at p. If p is neither a
pole nor a zero of g, _p(f)=0, which renders the number of terms in the
above sum finite. In the definition above, divisors are elements of the free
abelian group generated by the Riemann surface X.  

Let f:  P^1→ P^1 be a non-trivial rational function on the
Riemann sphere satisfying 


    f(0)=f(∞)=1 ,


which can be realized by representing f as a finite product 


    f(t)=∏_i (t-a_i)^d_i ,    ∑_i d_i=0 ,    ∏_i a_i^d_i=1 ,


where a_i∈ and d_i∈. Furthermore, let us write 


    1-f(t)   =b∏_j (t-b_j)^e_j ,


where b, b_j∈ and e_j∈. The divisor of the function f defined in 
sec:invitation:Bloch:fandCoefs reads 


    (f)=∑_i _a_i(f)(a_i)=∑_i d_i(a_i) ,    (1-f)=∑_j e_j(b_j) .


In BlochRelation Bloch proves that for any
rational function f as defined above, the Bloch-Wigner function satisfies


    ∑_i,jd_i e_j(a_i/b_j)   =0 ,


abbreviated in terms of the element 


    η_f   =∑_i,jd_i e_j(a_i/b_j)


of the free abelian group _ and the Bloch-Wigner function extended by
linearity as in eqn:DlinExample, this so-called classical Bloch
relation reads


    (η_f )   =0 .


Letting the zeros a_i of f vary subject to the conditions in
sec:invitation:Bloch:fandCoefs, this dilogarithm relation of Bloch
becomes a functional relation. Choosing different rational functions satisfying
sec:invitation:Bloch:condf in the first place,
sec:invitation:Bloch:abbreviatedBlochRel yields a whole class of
functional relations for the Bloch-Wigner function parametrized by rational
functions f on the Riemann sphere, which is however not independent. In fact,
it is conjecturally generated by the single example of the five-term identity
(see the discussion at the end of sec:review:BG), which is
discussed in the following paragraph.

As the most fundamental example and an application of the Bloch relation, let
us discuss how to recover the five-term identity
sec:invitation:BlochGroup:fiveterm from
sec:invitation:Bloch:abbreviatedBlochRel following the lines of
ZagierGangl: let a,b∈, a'=1-a, b'=1-b and consider the
rational function 


    f(t)   =(t-a)(t-a')(t-bb')/(t-b)(t-b')(t-aa') .


It satisfies f(0)=f(∞)=1 and 


    1-f(t)   =(bb'-aa')t^2/(t-b)(t-b')(t-aa') ,


such that Bloch's relation can be applied, which yields the identity


    (a/b)+(a'/b')+(a/b')+(bb'/aa')+(a'/b)   =0 ,
 

where we have used that (0)=(1)=(∞)=0 and the symmetry relations
(<ref>). Changing variables to t=a/b,
s=a'/b' finally leads to the five-term identity in the usual
form (<ref>).




§ ELLIPTIC CURVES, THE DIVISOR FUNCTION AND BLOCH'S RELATION



The aim of this section is twofold: after reviewing mathematical tools for the
description of elliptic curves in various formulations and a particular type of
elliptic iterated integrals in subsections <ref>
and <ref> we are going to discuss and exemplify the generalisation of
the concepts of the divisor function and the Bloch relation from the previous
section to the genus-one Riemann surfaces / elliptic curves in subsections
<ref> and <ref>. In
particular, sec:toolbox:ellBloch contains three examples of
functional relations on the elliptic curve parametrised by various rational
functions.



 §.§ Elliptic curves and functions


This subsection begins with the introduction of the torus description of elliptic
curves: being a Riemann surface of genus one, the torus is the natural geometry
underlying an elliptic curve due to its two periodicities. Along with the
discussion of the torus formulation, several properties of elliptic functions are
reviewed. 
Afterwards, two isomorphisms are discussed, where the first one relates the
torus to the projective (elliptic) curve and the second one maps the torus to
the so-called Tate curve given by the exponential map. 
These are well-known mathematical concepts, but 
in particular the map from the torus to the Tate curve is rarely mentioned in the physics
literature.  A thorough introduction which relates to the common physics
language can e.g. be found in Broedel:2017kkb, which is the basis for
the discussion in this subsection.


A torus can be described as the quotient /Λ of the complex plane
and a lattice 


    Λ=ω_1 +ω_2


where the periods ω_1 and ω_2 are complex numbers and taken to be linearly
independent over the real numbers.
The domain P_Λ={aω_1+bω_2 | 0≤ a,b<1} is called the
fundamental parallelogram of /Λ which defines the torus upon
identifying the opposite sides of its closure. Due to this immediate relation,
we will simply refer to /Λ as the torus itself. The torus is often scaled
such that τ=ω_2/ω_1 and 1 are its periods and without loss of
generality τ is assumed to be an element of the upper half plane,
(τ)>0, in this case the fundamental parallelogram can be depicted as in figure <ref>.



A function is called elliptic on  if it is Λ-periodic,
i.e. a function defined on /Λ, and meromorphic. However, in the
case of generalisations of multiple polylogarithms to the elliptic curve, we
sometimes also refer to multi-valued functions on the torus /Λ
(i.e. not necessarily Λ-periodic functions), as elliptic functions if
they are meromorphic. This is in particular the case for the elliptic multiple
polylogarithms introduced in ssec:EMP. 

Two explicit examples of elliptic functions are the even
Weierstrass ℘ function


    ℘(z)=℘(z;ω_1,ω_2)=1/z^2+∑_(m,n)≠ (0,0)(1/(z+mω_1+nω_2)^2-1/(mω_1+nω_2)^2)


and its odd derivative ℘'(z). Note that ℘ has a double pole at any
lattice point, whereas ℘' has a triple pole at the lattice points. Closed
expressions of zeros of ℘ are generally complicated, while the zeros of
℘' are exactly the half periods ω_i/2, for i=1,2,3 and
ω_3=ω_2-ω_1.  Moreover, these elliptic functions satisfy the
differential equation 


    ℘'(z)^2=4(℘(z)-e_1)(℘(z)-e_2)(℘(z)-e_3)=4℘(z)^3-g_2℘(z)-g_3


where the three roots e_i are defined by 


    e_i=℘(ω_i/2)


and sum to zero. The Weierstrass invariants g_2 and g_3 in the above
equation can be expressed in terms of Eisenstein series


    g_2=60∑_(m,n)≠(0,0)1/(mω_1 +nω_2)^4 ,     g_3=140∑_(m,n)≠(0,0)1/(mω_1 +nω_2)^6


and are related to the roots by 


    e_1+e_2+e_3=0 ,     e_1e_2+e_2e_3+e_3e_1=-1/4g_2 ,     e_1e_2e_3=1/4g_3 .


It turns out that the notion of ellipticity is quite restrictive: for example,
the zeros and poles of an elliptic function F are subject to the
conditions[See e.g. the lecture notes <cit.> for the
  derivation of equation
  (<ref>) and the
following statements about Weierstrass functions.]


    ∑_z∈ P_Λ_z(F)=0 ,     ∑_z∈ P_Λ_z(F) z∈Λ ,


where the order _z(F) of F at z is the usual order of zeros and poles
of meromorphic functions, in analogy to the definition of the  order in the
context of rational functions on the Riemann sphere used in
eqn:defdivisor. In particular, points which are neither zeros nor
poles are of order zero. Thus, the sums over the fundamental parallelogram in
equation
(<ref>) are finite and
include the non-vanishing terms at zeros and poles of F only. 

Moreover, an elliptic function can not have a single simple pole: using
Cauchy's residue theorem and integration along the fundamental parallelogram,
where the (reversed) parallel paths cancel pairwise due to the
Λ-periodicity, the sum of the residues has to vanish, which can not be
satisfied by a single simple pole alone.  The conditions in
sec:toolbox_ellipticfunctions:conditionsEllipticFctTorus follow from the
same cancellation of the integration along the fundamental parallelogram in the
(generalised) argument principle. 

Furthermore, any elliptic function is determined up to scaling by its zeros and
poles: the quotient of two elliptic functions with the same zeros and poles,
counting multiplicities, is bounded on the fundamental parallelogram
P_Λ and hence, by Λ-periodicity, is a bounded entire
function, such that Liouville's theorem implies that these two elliptic
functions are proportional to each other. This fact, in turn, implies that any
elliptic function on /Λ is a rational function in ℘ and ℘':
those rational functions are elliptic by construction and can be combined to
have the same zeros and poles as any given elliptic function.  

Alternatively, any elliptic function can be expressed in terms of the
Weierstrass σ function


    σ(z)   =s_C exp(∫_z_0^zdz' ζ(z')) ,


where the scaling factor s_C and the base point z_0 are
chosen[Both, s_C and z_0, can be chosen canonically by adjusting
the integration constant ξ(ω_i) in
sec:toolbox_ellipticfunctions:trafoZeta.] such that σ'(0)=1.  The
logarithmic derivative (and thus the integrand in
sec:toolbox_ellipticfunctions:sigma) of the Weierstrass σ function
is the Weierstrass ζ function


    ζ(z)=1/z+∑_(m,n)≠ (0,0)(1/z-mω_1-nω_2+1/mω_1+nω_2+z/(mω_1+nω_2)^2) ,


which itself is the negative odd primitive of ℘. 

The Weierstrass σ function has no poles and one simple zero at the
lattice points, hence, it can not be elliptic. In fact, neither ζ nor
σ is Λ-periodic. For the Weierstrass ζ function and a
lattice period ω_i, integrating the equation ℘(z+ω_i)=℘(z)
implies that ζ changes by a z-independent integration constant


    ζ(z+ω_i)   =ζ(z)+2η(ω_i)


with the quasi-period η(ω_i)=ζ(ω_i/2), which follows from
the evaluation of sec:toolbox_ellipticfunctions:quasiperiod at
. In a similar manner one can determine the transformation
behaviour of the Weierstrass σ function, which reads 


    σ(z+ω_i)   =exp(2η(ω_i)z+ξ(ω_i))σ(z) ,


where ξ(ω_i) is yet another integration constant (see
e.g. deShalit). This shows explicitly that σ is indeed not
elliptic. The transformation (<ref>) of
σ and the fact that it has one simple zero at any lattice point and no
poles at all leads to the alternative representation of a given elliptic
function F mentioned above: one can always choose particular representatives
A_i of the zeros and poles of F in /Λ (not necessarily in the
fundamental domain) such that


    ∑_i d_i=0 ,    ∑_i d_i A_i=0 ,


where d_i=_A_i(F). It is then the set of
conditions (<ref>), satisfying the
natural
constraints (<ref>)
for the zeros and poles of an elliptic function, which ensures that the
combination[Note that compared to the definition of the Weierstrass
  σ function (<ref>), the factors of
  s_C from the product on the left-hand side of
  sec:toolbox_ellipticfunctions:prodSigma multiply to one and the base
point z_0 of the integrals in the exponential can be shifted to zero due to
the condition ∑_i d_i=0 in
sec:toolbox_ellipticfunctions:ZeroPolesCond.]


    ∏_iσ(z-A_i)^d_i   =exp(∑_i d_i ∫_0^z-A_idz'ζ(z'))


is elliptic. Indeed, under a lattice displacement the exponential
proportionality factor in sec:toolbox_ellipticfunctions:trafoZeta from
the transformation of the individual factors σ(z-A_i)^d_i in
sec:toolbox_ellipticfunctions:prodSigma form a product with an exponent
which is a linear combination of the left-hand sides of the two conditions
(<ref>), such that the overall
proportionality constant evaluates to one. Since σ has only one simple
zero at the lattice points and no pole, the above product has exactly the same
zeros and poles including multiplicities as the function F. Correspondingly,
any elliptic function F can be written as


    F(z)=s_A∏_iσ(z-A_i)^d_i=s_Aexp(∑_i d_i ∫_0^z-A_idz'ζ(z'))


for some scaling factor s_A∈. The behaviour of the zeros and poles of an
elliptic function can be conveniently captured in terms of divisors, which are
introduced in sec:toolbox:divisorFunction.


The fact that all elliptic functions can be expressed as rational functions of
℘ and ℘' facilitates their description in terms of rational functions
on a complex projective algebraic curve. The Weierstrass ℘ function induces an
isomorphism between /Λ and the complex projective algebraic curve


    E()   ={xy| y^2=4x^3-g_2(Λ) x-g_3(Λ)}∪{[0:1:0]} ,


where [0:1:0] is denoted by infinity ∞. Note that the
cubic equation in x and y of the curve in definition
(<ref>) is of the same form as
the differential equation (<ref>) for
℘: this representation of the constraint equation on the projective
formulation of the elliptic curve is called the Weierstrass form or
Weierstrass equation. Furthermore, the projective algebraic curve E() is
often called the projective formulation of the elliptic curve or the projective elliptic curve. 

The isomorphism of Riemann surfaces is given by


    ξ_Λ,E: /Λ→ E() ,      0≠ z↦℘(z)℘(z)' ,     0↦[0:1:0]=∞ ,


see e.g. Broedel:2017kkb for more details. The addition on
E() is provided by the so-called chord-tangent construction with the
additive unity being ∞. It has a nice geometric interpretation, which is
described in app:chord-tangent. 

The inverse of the isomorphism ξ_Λ,E is called Abel's map and can
be determined from the differential
equation (<ref>). Given a point
P=x_Py_P with y_P≠ 0, one finds


    z   =±∫_∞^x_Pdx/yΛ ,


where the correct sign is determined by the requirement that ℘'(z)=y_P, and
ξ^-1_Λ,E(e_i)=ω_i/2 for P=e_i0. Upon identifying


    x=℘(z) ,     y=℘'(z)

as well as using the fact that these two functions generate any elliptic
function on the torus in terms of rational functions, it follows that the
elliptic functions can be described as the rational functions in x and y
on the projective elliptic curve E().

The above choice of signs in Abel's map
(<ref>) is not the only issue that needs
some care if a translation from a given projective elliptic curve E() with
elliptic invariants g_2 and g_3 to the torus has to be implemented
explicitly. 

A first ambiguity has to be addressed by making a choice for the periods
ω_1 and ω_2 associated to the elliptic curve with Weierstrass
equation y^2=4x^3-g_2x-g_3=4(x-e_1)(x-e_2)(x-e_3). The roots e_i are
defined by g_2 and g_3 up to relabelling according to
sec:toolbox_ellipticfunctions:RootsAndInvariants. Simultaneously, Abel's
map together with eqn:rootsperiods implies


    ω_1/2=ω_2/2-ω_3/2=(∫_e_3^e_2dx/y)Λ ,


where the ω_i, or the fundamental parallelogram, respectively, are
chosen such that the periods are given by the integrals <cit.>


    ω_1=2∫_e_3^e_2dx/y ,     ω_2=2∫_e_3^e_1dx/y ,    ω_3=ω_2-ω_1=2∫_e_2^e_1dx/y .


Any other choice of labelling the roots will yield an integer linear
combination of the periods defined in
sec:toolbox_ellipticfunctions:Periods above, i.e.

    2∫_e_i^e_j   =m_ijω_1+n_ijω_2


with m_ij, n_ij∈. Hence, the choice of periods corresponds to
choosing different basis vectors for spanning the lattice Λ.
Correspondingly, the six possible labellings of the roots define six pairs of
periods (ω_1,ω_2), whereas the associated different tori are
isomorphic to a particular elliptic curve. 

The second, but related issue is that the complex plane may always be rescaled
by ω_1. Hence, only the ratio τ=ω_2/ω_1 matters when
dealing with the Λ-periodicity, i.e. the geometry of the torus.
Therefore, a torus is usually only defined by the modular parameter τ with
positive imaginary part (τ)>0 while the second period is chosen to be
one. Under scaling ω_1, the
Weierstrass ℘ function rescales as 


    ℘(z;1,τ)   =ω_1^2 ℘(ω_1z;ω_1,ω_2) .


Choosing τ in the upper half-plane means that three possible labellings of
the roots e_i are disregarded. The remaining three period ratios obtained
from the different labellings of the roots e_i are related to the τ in
the upper half plane by modular transformations. 

In summary, the Weierstrass invariants of an elliptic curve completely
define the torus up to modular transformations. Conversely, given two tori with
period ratios τ and τ' related by a modular transformation, the
Weierstrass equations of the projective elliptic curves obtained by
ξ_τ+,E and ξ_τ'+,E', respectively, are related by
a coordinate transformation of the form 


    x↦ a^2 x+b ,   y↦
      a^3y+ca^2 x+d ,  with  a,b,c,d∈ ,   a≠0 .


Two elliptic curves are called isomorphic if they are related in this way. 
For example, the transformation x↦ a^2x, y↦ a^3y only
changes the roots by the constant rescaling e_i↦ a^-2e_i, which is an
isomorphism on the complex plane (with respect to addition). Thus, the period
ratio τ modulo modular transformations uniquely defines the isomorphism
class of elliptic curves defined by ξ_τ+,E and vice versa.

Note that the computer algebra system [See e.g.2015arXiv151007818B for a guideline of the use and the choices of
's built-in conversions from the projective elliptic curve to the torus,
which is based on the conventions of <cit.>.] offers built-in functions
for translating from the projective formulation of elliptic curves to the torus.
In those functions, however, the ambiguities described above are chosen
implicitly. For example, the determination of the half periods ω_1/2,
ω_2/2 with the  function
 relies on a choice of the labellings
of the roots which is selected depending on the signs of the Weierstrass
invariants and the modular discriminant Δ=g_2^3-27 g_3^2.


In order to define a third formulation of the elliptic curve, let us
consider a torus defined by the modular parameter τ and define q=e^2π
iτ. The exponential map induces another isomorphism 


    ξ_τ,q:/(τ+)→^∗/q^ ,     z↦ e^2π i z .


where the codomain ^∗/q^ is called Tate curve [See
Katz73, appendix A.1.2, or Lur09, section 4.3, for a more
recent introduction to the Tate curve.] and is endowed with the multiplicative
group structure inherited by the exponential map from addition on the torus.
For example, the representatives z_1+n_1+ m_1τ and z_2+n_2+ m_2τ of
z_1 and z_2 modulo lattice displacements in /Λ, where n_i,
m_i∈, are mapped to the elements

    ξ_τ,q(z_i+n_i+ m_iτ)=e^2π i(z_i+n_i+ m_iτ) =e^2π
    iz_iq^m_i,
 which are representatives of t_1=e^2π i z_1
and t_2=e^2π i z_2, respectively, modulo integer powers of q.
Similarly, the sum z_1+z_2 modulo lattice displacements is mapped to the
product t_1 t_2 modulo integer powers of q on the Tate curve.

The description of elliptic functions on the Tate curve offers a connection to
rational functions on the Riemann sphere P^1 and, in particular, admits a
convenient tool to take the classical limit q→ 0. In order to
reveal this connection to functions on the Riemann sphere, let f:
P^1→ P^1 be a non-trivial rational function on the Riemann
sphere satisfying the condition 


    f(0)=f(∞)=1 ,


which will be justified in a moment. Note that this class of functions was
already discussed in the context of the classical Bloch relation in
ssec:dilogrel; the two approaches will be related below. For now,
recall from the discussion of the classical Bloch relation that this ensures
that f is of the form (<ref>), i.e.

    f(t)   =∏_i (t-a_i)^d_i ,


with 


    ∑_i d_i=0 ,    ∏_i a_i^d_i=1 .


Averaging f multiplicatively as follows over the Tate curve yields a function


    F(t)   =∏_l∈f(t q^l) ,


which obeys the transformed Λ-periodicity condition F(tq)=F(t), cf.sec:toolbox_ellipticfunctions:mapToTateCurve for the transformation
behaviour of lattice displacements under the isomorphism ξ_τ,q, and,
can therefore be called elliptic on the Tate curve. A discussion of the
properties of such elliptic functions on the Tate curve can be found in
BlochRelation.

The so far unexplained condition (<ref>)
can be justified as follows: on the one hand it ensures that in the limit
q→ 0 we recover f(t), on the other hand it implies the
condition (<ref>) on the zeros and
poles a_i of the elliptic generalisation F of f from
sec:toolbox_ellipticfunctions:FTateCurve after the application of the
isomorphism ξ_τ,q^-1 and the identification a_i=e^2π i
  A_i. [We generally denote an elliptic function by a capital Latin
  letter F (while functions on the Riemann sphere are denoted by small Latin
  letters) and its zeros and poles on the Tate curve by small letters a_i,
  while their images on the torus are denoted by the corresponding capital
  letters A_i. However, for a point on the torus, which corresponds to a
point t on the Tate curve, we usually write z_t. The same applies for a
given point P on the elliptic curve and its image z_P on the torus.] As we
will see in sec:toolbox:divisorFunction, these two conditions
(modulo lattice displacements) are not only necessary, but also sufficient to
be the zeros and poles of some elliptic function. Therefore, we can summarize
that the function F is the elliptic generalisation of f on the Tate curve
and all elliptic functions on the Tate curve can be obtained by this method up
to scaling.



 §.§ Elliptic multiple polylogarithms

There are several descriptions of elliptic generalisations of multiple
polylogarithms, so-called elliptic multiple polylogarithms. Based on the
fact that there is no elliptic function on the torus with just one simple pole,
such generalisations are either not meromorphic or not Λ-periodic.

However, in Feynman integral calculations one usually chooses to work with
meromorphic rather than single-valued functions. Motivated by this physical
reason, we focus on the holomorphic iterated integrals  on the torus
described in Broedel:2017kkb and relate some other notions of
(single-valued but non-holomorphic) elliptic multiple polylogarithms to these
iterated integrals in sec:connection:TatetoTorus. In analogy with
the multi-valuedness of the logarithm function, we still refer to these
holomorphic iterated integrals as elliptic multiple polylogarithms defined on
the torus.


Consider a torus with periods 1 and τ, where (τ)>0 as described
in sec:toolbox:ellipticFunction above, and denote


    t=e^2π i z ,     q=e^2π i τ  and    w=e^2π i α .


The holomorphic functions g^(n)(z,τ), which satisfy
g^(n)(z,τ)=g^(n)(z+1,τ), constitute the integration kernels of the holomorphic
iterated integrals  described in Broedel:2017kkb. They are
generated by the Eisenstein-Kronecker series <cit.>


    F(z,α,τ)=θ_1'(0,τ)θ_1(z+α,τ)/θ_1(z,τ)θ_1(α,τ)=1/α∑_n≥ 0g^(n)(z,τ)α^n ,


where θ_1(z,τ) is the odd Jacobi θ function. The corresponding
iterated integrals are defined via


    ([ n_1   … n_k; z_1   … z_k ]; z,τ)=∫_0^zdz'g^(n_1)(z'-z_1,τ) ([ n_2   … n_k; z_2   … z_k ]; z',τ) ,     (; z,τ)=1 .


The integration kernels g^(n)(z,τ) are not as abstract as they might
seem at first glance: they are closely related to the doubly-periodic kernels
introduced and used in BrownLev. Furthermore, the kernel
g^(1)(z,τ) can be expressed as 



    g^(1)(z,τ)   =ζ(z)-2η_1 z ,



where ζ(z) is the Weierstrass ζ function introduced in
eqn:WeierstrassZeta and η_1=ζ(1/2) is a quasi-period of the
elliptic curve (cf. sec:toolbox_ellipticfunctions:quasiperiod). For
n>1, the integration kernels can be expressed as polynomials of degree n in
g^(1)(z,τ) and the coefficients depend polynomially on ℘(z) and
℘'(z), where the first two examples are


    g^(2)(z)=1/2(g^(1)(z))^2-1/2℘(z) ,     g^(3)(z)=1/6(g^(1)(z))^3-1/2℘(z)g^(1)(z)-1/6℘'(z) .



More suitable for numerical evaluation is the description of the integration
kernels g^n(z,τ) by their q-expansions, which are stated in the
app:qExpansion.  Furthermore, since the Eisenstein-Kronecker series
satisfies the mixed heat equation 2π i ∂/∂τF(z,α,τ)=∂^2/∂ z∂αF(z,α,τ) <cit.>, the integration kernels solve the
partial differential equation


    2π i ∂/∂τ g^(n)(z,τ)   =n∂/∂ z g^(n+1)(z,τ) .


At this point, some facts about the regularisation of those iterated integrals
need to be mentioned. Considering the q-expansions
(<ref>)-(<ref>)
it is obvious that only the kernel g^(1) has a singularity at z=0. This
singularity is a simple pole, which renders the iterated integrals
([ n_1   … n_k; z_1   … z_k ];
z,τ) with z_k=0, n_k=1 singular. 

Employing the shuffle product of iterated integrals, any singular integral can
be rewritten such that the only singular terms are of the form
([ 1 … 1; 0 … 0 ]_n; z,τ). Those singular terms can then be regularised
in a way that preserves the shuffle algebra. Following the prescription
described in Broedel:2018iwv the logarithmic singularity at the lower integration boundary of the
integral ([ 1; 0 ]; z,τ) for z≠ 0 can be subtracted by defining its regularised value as follows:


    _reg([ 1; 0 ]; z,τ)   =lim_ϵ→ 0∫_ϵ^z dz'g^(1)(z',τ)+log(1-e^2π i ϵ)
       = log(1-e^2π i z)-π i z+4π∑_k,l>01/2π k(1-cos(2π k z))q^kl .


Note that while the original integral ([ 1; 0 ]; z,τ) vanishes at z=0 and is divergent at any other
value of z, the regularised version
_reg([ 1; 0 ];
z,τ) is finite at any z≠ 0, but has a logarithmic divergence at z=0.

The prescription can be easily generalized to iterated integrals with multiple
successive divergent entries: with the generalisation


    _reg([ 1 … 1; 0 … 0 ]_n; z,τ)   =1/n!(_reg([ 1; 0 ]; z,τ) )^n .


From here on – unless stated otherwise – we denote by
([ 1 … 1; 0 … 0 ]_n; z,τ) its regularised value and refer to the
unregularised version as follows


    _([ 1 … 1; 0 … 0 ]_n; z,τ)=∫_0^z dz'
      g^(1)(z',τ)_([ 1 … 1; 0 … 0 ]_n-1; z',τ) .



In sec:connection:TatetoTorus below, a particular class of the
iterated integrals  is discussed in detail, which is the one given by the
regularised elliptic polylogarithms of the form


    ([ 0 … 0 m; 0 … 0 0 ]_n; z,τ) ,


where n,m≥ 1. The numerical evaluation of this class of functions is
particularly simple, since their q-expansions can directly by given by n-fold
integration of the q-expansions (<ref>) and
(<ref>) of the integration kernels
g^(m)(z,τ) for m>1 and the q-expansion (<ref>)
of _reg([ 1; 0 ]; z,τ). The
results are given in equations (<ref>)-(<ref>).

The values of the (regularised) iterated integrals at z=1 are particularly
interesting since they can be used to define a class of elliptic multiple zeta
values <cit.>. Ordinary zeta values ζ_m,
for m>1, are defined as the values of the corresponding polylogarithms
evaluated at one


    ζ_m   =_m(1) .


Analogously, we consider the elliptic zeta values defined by evaluation at one
of the above class of elliptic polylogarithms


    ω_n(m;τ)   =([ 0 … 0 m; 0 … 0 0 ]_n; 1,τ) .


Note that this class of elliptic zeta values agrees with the definition of
elliptic multiple zeta values in Broedel:2014vla. Furthermore, the even
zeta values are related to the elliptic zeta values according to 


    ω_1(2m;τ)   =-2ζ_2m,


which can be seen from the q-expansion (<ref>)




 §.§ The divisor function



The last paragraph in sec:toolbox:ellipticFunction was devoted to
illuminating the relation between an elliptic function F on the Tate curve
and the corresponding rational function f on the Riemann sphere. As mentioned
at that point, this is closely connected to the formulation of the classical
Bloch relation in terms of divisors of such rational functions f discussed in
ssec:dilogrel. The combination of these two considerations leads to
the formulation of the elliptic Bloch relation using the concept of divisors of
elliptic functions.


The group of divisors (/Λ) of the torus /Λ is the
free abelian group _/Λ generated by the points on the torus
/Λ and similarly for the projective elliptic curve as well as the Tate curve,
which are related via the isomorphisms introduced in
sec:toolbox_ellipticfunctions:xiLEsec:toolbox_ellipticfunctions:toTatecurve
above. Hence, a generic divisor is a finite sum of the form 


    ∑_i n_i (z_i)∈(/Λ) ,  ∑_i n_i
    (P_i)∈(E())  or  ∑_i n_i (t_i)∈(^∗/q^) ,


respectively, with n_i∈, z_i∈/Λ,
ξ_Λ,E(z_i)=P_i∈ E() and ξ_q,E(z_i)=t_i∈^∗/q^. Analogously to the case of rational functions on the
Riemann sphere, cf. eqn:defdivisorRiemannSphere, and according to the
general definition (<ref>), the divisor of an elliptic function
F captures the structure of the zeros and poles of F and is defined by


    (F)   =∑_z∈ P_Λ_z(F)(z)∈(/Λ)


where the sum runs over all points in the fundamental domain P_Λ of /Λ.

According to the identification of elliptic functions on
the torus with rational functions on the projective elliptic curve and elliptic
functions on the Tate curve alluded to above, the divisor
(<ref>) of an elliptic function F can be translated
by the usual isomorphisms to the projective formulation and the Tate curve via 


    (F)=∑_P∈ E()_P(F)(P)∈(E())


and


    (F)=∑_t∈^∗/q^: |q|<|t|≤ 1_t(F)(t)∈(^∗/q^) ,


where the orders of the rational function F(x,y) and the elliptic function on
the Tate curve F(t) are defined by the order of the elliptic function F(z)
on the torus at the corresponding points.
For two divisors D=∑_i d_i (P_i)
and E=∑_j e_j (Q_j), a new divisor


    D^-=∑_i d_i (-P_i)


and the binary product


    D∗ E   =∑_i,jd_i e_j (P_i+Q_j)


can be defined, such that a divisor of the form


    η^κ_F   =(F)∗(κ-F)^- ,


can be associated to any rational function F, and similarly on the torus and
the Tate curve, respectively. In the above definition, κ∈ is a
scaling parameter, which needs to equal one in the classical situation
described in ssec:dilogrel.
The divisor η_F^κ associated to an elliptic function F plays an
important role in later sections of this article and in particular in the
formulation of the elliptic Bloch relation in sec:toolbox:ellBloch.

The fact that two elliptic functions are equal up to scaling if they have the
same zeros and poles (counted with multiplicities) translates to the condition
of having the same divisors. On the other hand, a divisor D is said to be
principal, if there exists an elliptic function F such that
D=(F). Now, we can properly rephrase the last two sentences in
sec:toolbox:ellipticFunction: It turns out that a divisor D is
principal if and only if it is of the form D=∑_i d_i (A_i) where


    ∑_i d_i =0 ,    ∑_i d_i A_i∈Λ .


A proof of this equivalence can be outlined as follows (cf. deShalit):
the necessary implication follows from the conditions
(<ref>) on the zeros
and poles of an elliptic function. In order to prove sufficiency, first note that any
divisor D=∑_i d_i (A_i) satisfying equations
(<ref>) can be written
as a linear combination of divisors of the form (A_1)+(A_2)-(0)-(A_1+A_2).
Now, consider elliptic functions of the form


    F_λ(z)   = (1-λ)℘'(z)-℘'(-A_1-A_2)/℘(z)-℘(-A_1-A_2)+λ .


In deShalit it is shown that one can always find a complex parameter
λ such that the divisor associated to the above function reads:


    (F_λ)   = (A_1)+(A_2)-(0)-(A_1+A_2)


and D can indeed be written as a divisor of an elliptic function
D=∑_je_j(F_λ_j)=(∏_jF_λ_j^e_j), since the divisor function satisfies (F_1
F_2)=(F_1)+(F_2) for two elliptic functions F_1 and F_2.
Alternatively, the elliptic function F such that (F)=D can be
constructed by means of the Weierstrass σ function as in
sec:toolbox_ellipticfunctions:FandSigma.



 §.§ The elliptic Bloch relation



After having introduced the mathematical background for elliptic curves and
elliptic functions in the previous subsections, the elliptic version of Bloch's
dilogarithm identity (<ref>) can be discussed: in
order to do so, an elliptic generalisation of the Bloch-Wigner function 
defined in eqn:BlochWigner is required. Since the Bloch-Wigner function
satisfies (0)=(∞)=0, the elliptic generalisation on the Tate curve
in terms of an infinite product as in
sec:toolbox_ellipticfunctions:FTateCurve is not applicable. However, an
additive average over the Tate curve yields


    (t,q)   =∑_l∈(tq^l) .


This function  is referred to as the elliptic Bloch-Wigner function
<cit.>. It inherits some symmetry properties from the
classical Bloch-Wigner function , in particular the inversion relation


    (t^-1,q)   =-(t,q)


and the duplication relation 

    (t^2,q)   =2((t,q)+(t√(q),q)+(-t,q)+(-t√(q),q))


from sec:invitation:BlochGroup:symDsec:invitation:BlochGroup:dupD,
respectively. The elliptic version of Bloch's dilogarithm identity
(<ref>) is that for any elliptic function F,
any κ∈ and the divisors (F)=∑_i d_i (a_i) and
(κ-F)=∑_j e_j (b_j) expressed on the Tate curve, the
following identity holds


    ∑_i,jd_i e_j(a_i/b_j,q)   =0 ,


which takes the form


    (η_F^κ,q)   =0 ,


when expressed in terms of η^κ_F, the divisor defined in
sec:toolbox_divisors:etaF. Here again  is extended by linearity to
the group of divisors. The above identity is referred to as the
elliptic Bloch relation.  

Bloch proves this statement starting from the classical case with a rational
function f satisfying f(0)=f(∞)=1, approximating its elliptic
generalisation F on the Tate curve, constructed according to
sec:toolbox_ellipticfunctions:FTateCurve, by F_N=∏_|l|≤
Nf(tq^l) and an error estimation as N→∞
<cit.>. Note that in contrast to the classical case, cf. the
second equation in (<ref>), the constant κ
defined in sec:toolbox_divisors:etaF does not need to equal one. But
since any scaling of the elliptic function F is allowed, this condition is
redundant and the elliptic Bloch relation can be stated without loss of
generality with κ=1. In the classical limit q→ 0 the Tate
curve ^∗/q^ degenerates to ^∗, and, simultaneously,
the elliptic Bloch-Wigner function degenerates to its classical version, the
Bloch-Wigner function.  Finally, for an elliptic function on the Tate curve of
the form


    F=lim_N→∞F_N


with F_N and f (scaled) as before, the elliptic Bloch relation degenerates to the
classical Bloch relation, cf. (<ref>),


    ((F)∗(1-F)^-,q)→∑_i,j_a_i(f)_b_j(1-f)(a_i/b_j) .


However, the above limit, i.e. the transition from elliptic to classical
and vice-versa, is subtle, as it can be seen by Bloch's careful proof of the
elliptic Bloch relation in terms of the classical Bloch relation. Another
hint for this subtlety is the following: If on the left-hand side in the limit
(<ref>) instead of 1-F, the difference κ-F for
κ≠ 1 is chosen, the left-hand side still vanishes identically
according to sec:toolbox_ellBloch:ellBlochRelation. But the right-hand
side of (<ref>) in general only vanishes for 1-f, but not for
κ-f. Therefore, in such a case the elliptic Bloch relation does not
degenerate to its classical analogue.


Analogously to the classical Bloch group ℬ_2(), Zagier and Gangl
define the group of functional relations 𝒞_2(E) in the construction
of the elliptic Bloch group
ℬ_2(E)=𝒜_2(E)/𝒞_2(E) as a subgroup of the group
𝒜_2(E) of "allowable" elements in the free abelian group generated
by points on the elliptic curve E, the precise meaning of allowable is
reviewed in <cit.>. It is generated by the elliptic Bloch
relation (<ref>), the inversion
relation (<ref>) and the duplication
relation (<ref>), which are expected to form a
full set of relations for the elliptic Bloch-Wigner function on the points in
𝒜_2(E) <cit.>. Thus, in contrast to the classical case
discussed in sec:review:BG, where the five-term identity is
sufficient to generate the subgroup of functional relations for the dilogarithm
on the points in 𝒜_2(), the elliptic analogue may require a
larger class of functional relations generated by the elliptic Bloch relation.
Using the construction of elliptic functions on the Tate curve described in
sec:toolbox:ellipticFunction, the class of functional relations of
the elliptic Bloch-Wigner function is parametrised by rational functions on the
Riemann sphere and the complex number κ. We refer to this procedure to
generate functional identities for the elliptic Bloch-Wigner function as Zagier
and Gangl's method. In the following three subsections we discuss some examples
and show explicit calculations which use the above concepts and in particular
the elliptic Bloch relation.



  §.§.§ First example: a divisor on y^2=4x^3-4x+1


Let us consider the following example[This is the example
E_37:y^2-y=x^3-x in ZagierGangl. However, we directly work in the
Weierstrass form, which can be obtained from the original example by the
coordinate transformation y↦y+1/2.] of ZagierGangl to
approve the elliptic Bloch relation. Take the elliptic curve with Weierstrass
equation y^2=4x^3-4 x+1, i.e. g_2=4 and g_3=-1, and the rational
function  on E(). The three zeros of F are
P=0-1, P_1=1-1 and  and since
F(℘(z),℘'(z))=℘'(z)+1/2, the (pull-back of the) rational
function F as an elliptic function on the torus has a triple pole at the
lattice points, such that on the elliptic curve _∞(F)=-3. Using
the group addition on the elliptic curve described in
app:chord-tangent, one obtains P_1=2P and P_2=-3P and more generally


-3P   =-1-1 ,   -2P   =11,    -P   =01 , 

P   =0-1,   2P   =1-1 ,    3P   =-11 , 

4P   =25 ,    5P   =1/41/4 ,    6P   =6-29 .


Therefore, the divisor of F on the projective elliptic curve is 


    (F)   =(P)+(2P)+(-3P)-3(∞) .


Similarly, the divisor of 1-F is given by


    (1-F)   =(-P)+(-2P)+(3P)-3(∞) ,


such that the associated divisor η_F^1 of F(x,y)=y+1/2 defined
in sec:toolbox_divisors:etaF is


    η^1_F   =(-6P)-6(-3P)+2(-2P)+2(-P)+9(∞)-6(P)-5(2P)+2(3P)+(4P) .


The roots of the elliptic curve 
are[We have chosen to display the first ten digits of
numbers only. Of course, all calculations have been performed with much higher
precision.]


    e_1=0.8375654352 ,     e_2=0.2695944364 ,     e_3=-1.1071598716 ,


such that according to equations (<ref>)
the periods of the corresponding tori are given by


    ω_1=2.9934586462   ,    ω_3=2.4513893819i ,    ω_2=ω_1+ω_3


with the period ratio


    τ=ω_2/ω_1=1 + 0.8189153991i .


The point z on the torus with lattice Λ=ω_1+ω_2 which
corresponds to P=0-1 is determined by Abel's map
(<ref>)


    z̃_P   = ∫_0^∞dx/√(4(x-e_1)(x-e_2)(x-e_3))
       =2.0638659408 + 1.2256947056i .


The rescaled point corresponding to P in the fundamental parallelogram of the
torus defined by τ is


    z_P=z̃_P/ω_1 (+τ)=0.6894586481 + 0.4094577022 i


which maps to 


    t_P=e^2π i z_P=-0.0283399159 - 0.0708731874 i


on the Tate curve, while the parameter q takes the value


    q=e^2π i τ=0.0058261597 .


For practical purposes, let us define the following approximation of (t,q):


    _k^(t,q)   =∑_l=-k^k(tq^l),


which allows to control the accuracy of convergence depending on the number of
terms 2k+1. According to the elliptic Bloch relation, our example η_F^1
from sec:toolbox_divisors:ex is expected to satisfy


    -8(t_P,q)-7(t_P^2,q)+8(t_P^3,q)+(t_P^4,q)-(t_P^6,q)   =0 ,


where we already used the inversion relation
(<ref>) to simplify the evaluation of the
divisor η_F^1. 

Using the approximation (<ref>) for numerical evaluation of the
above equation, we find agreement up to 10^-7 already for k=10.

For other permutations of labelling the roots e_i the elliptic Bloch relation holds as
well, as can be tested numerically. 




  §.§.§ Second example: lines on the projective elliptic curve


As a second example, consider a line on the projective elliptic curve, i.e. a rational
function of the form


    L_a,b,c(x,y)   =ax+by+c


with a or b not equal to zero, x and y satisfying y^2=4x^3-g_2
x-g_3. The poles of L_a,b,c are located at ∞ with multiplicities
2 if b=0 and 3 otherwise (They correspond to the double and triple pole
of x=℘(z) and y=℘'(z), respectively. See the discussion around
sec:toolbox_divisors:DivFTateCurve.). The zeros of L_a,b,c can be
determined explicitly as algebraic functions of the coefficients a, b and
c, they satisfy the cubic equation 


    (a/bx+c/b)^2=4x^3-g_2x-g_3 ,     y=-a/bx-c/b .


Similarly, the zeros of 1-L_a,b,c satisfy


    (a/bx+c-1/b)^2=4x^3-g_2x-g_3 ,     y=-a/bx-c-1/b .


These cubic equations can be solved by radicals, such that
(L_a,b,c) and (1-L_a,b,c) depend
algebraically on the coefficients a, b and c. Furthermore, since the
group addition on the projective formulation () of the elliptic curve
also only involves algebraic operations, as can be seen from the explicit
equations in the app:chord-tangent, the resulting divisor
η_L_a,b,c^1 expressed on the projective elliptic curve is algebraic in a, b
and c. However, applying Abel's map and translating the solutions to the Tate
curve, where the elliptic Bloch-Wigner
relation (<ref>) is defined (so far),
generally turns the zeros into integral expressions of the variables a, b
and c. Thus, in order to obtain a functional relation with algebraic
arguments, the elliptic Bloch-Wigner relation has to be expressed on the torus
and ultimately on the projective elliptic curve, which is done in
sec:connection.

Alternatively, instead of solving for the zeros starting from a particular
choice of parameters a,b and c, one could as well choose three zeros
directly and obtain another three from 372373. However, since there
are only three free parameters (the lines L_a,b,c and 1-L_a,b,c have
the same slope), i.e. three roots which determine the remaining roots in terms
of at least one non-linear equation, this still involves some non-trivial
algebraic dependencies of the arguments in the final functional relation
induced by the elliptic Bloch-Wigner relation on the Tate curve. 

Let us illustrate the above argumentation by an example: if e.g. the zeros
P_1, P_2 of L_a,b,c and one zero Q_1 of 1-L_a,b,c are called
x_1y_1,x_2y_2 and x_3y_3, respectively, the third zero of
L_a,b,c is P_3=-P_1-P_2 according to the definition of addition in
app:chord-tangent and thus algebraic in x_i,y_i, i=1,2. 
This ensures that the x-coordinates of the divisor
of L_a,b,c are very simple. One of the two remaining zeros, Q_2 and Q_3,
of 1-L_a,b,c is defined by Q_3=-Q_1-Q_2. But the last zero, Q_2, is still
determined by a quadratic equation in terms of Q_1, such that mapping the
divisor to the Tate curve yields again a non-algebraic functional relation. 



  §.§.§ Third example: the five-term identity

As a last example, consider f given by sec:invitation:Bloch:5termf
which is the rational function generating the classical five-term identity when
inserted in the classical Bloch relation (<ref>).
It satisfies f(0)=f(∞)=1, such that its elliptic generalisation on the
Tate curve F(t)=∏_l∈f(tq^l)
(cf. sec:toolbox_ellipticfunctions:FTateCurve) with the following
associated divisor can be formed


    (F)   =(a)+(a')+(bb')-(b)-(b')-(aa') ,


where all variables have been defined after sec:invitation:Bloch:5termf.
Since the elliptic Bloch relation (η^1_F,q)=0 degenerates to the
classical one (η_F)=0 for q→ 0, it can be expected that
the elliptic Bloch relation evaluated for F generates an elliptic analogue of
the five-term identity <cit.>: it is an elliptic
dilogarithm identity generated by the same rational function f on the Riemann
sphere, which implies the five-term identity.
In order to write it down explicitly, the zeros and poles of 1-F need to be
known. While the poles are the same as the ones of F, finding the zeros of
1-F is a major obstacle, which was already encountered in the example with
the lines in the previous paragraph.  While the cubic equation for the zeros of
1-F in the line example could be solved by radicals, the current situation
involves a quintic equation in the x-coordinate of the projective elliptic
curve, which cannot be solved in general.  Correspondingly, an elliptic
analogue of the five-term identity can in general not be written down
explicitly in terms of algebraic arguments.  

The quintic equation is obtained as follows: following the argumentation at the
end of sec:toolbox:divisorFunction, there exist
λ_a,λ_b∈, such that


    (F_λ_a/F_λ_b)   =(F) ,


where F_λ is a rational function on the projective elliptic curve of the form
(<ref>) and the divisor of F is
expressed on the projective elliptic curve via the usual isomorphisms 


    a∈^∗/q^↦ξ_τ,q^-1(a)=A∈/(τ+)↦ξ_τ+,E(A)=P_a=x_ay_a∈ E() .


Performing the translation, F can be expressed on the projective elliptic curve as the
rational function


    F(x,y)=1/κF_λ_a(x,y)/F_λ_b(x,y)=(1-_a)(y-y_1/aa')(x-x_1/bb')+_a(x-x_1/bb')(x-x_1/aa')/(1-_b)(y-y_1/bb')(x-x_1/aa')+_b(x-x_1/bb')(x-x_1/aa')


for some scaling factor ∈, x_a=℘(A) and y_a=℘'(A). The poles
of 1-F are the same as the ones of F, i.e. P_a, P_a' and P_bb'.
The zeros of 1-F are determined by κ F__b-F__a=0, which translates by
the Weierstrass equation to the quintic equation
mentioned above. Since a and b are variables as well as based on the fact
that _a and _b depend non-trivially on a and b the resulting
quintic equation is not solvable by radicals in general. Even though the elliptic analogue
of the five-term identity generated by the elliptic Bloch relation can not be
written down explicitly, it may however be described implicitly as
above.


In summary, the elliptic Bloch relation
(<ref>) generates many (conjecturally
all) functional relations of the elliptic
Bloch-Wigner function. But for most of these relations, the relevant divisor
η_F^1 can not be expressed as a linear combination of variables depending
algebraically on each other. The most notable exceptions are the divisors
η_L_a,b,c^1 generated by lines expressed on the projective elliptic curve. It
is this situation, it would still be possible to explicitly write down
functional relations. However, they are by no means nice and elucidating and we
will thus refrain from doing so.  Instead, all relations are going to be
formulated on the torus in order to be contrasted with relations between
elliptic polylogarithms on the torus introduced in ssec:EMP.



§ ELLIPTIC MULTIPLE POLYLOGARITHMS: CONNECTING TWO LANGUAGES



The aim of this section is to translate the elliptic Bloch relation
(<ref>) from the Tate curve to the torus
and to the projective elliptic curve, respectively. From the previous section, it is
known how elliptic functions and their divisors can be translated between the
three descriptions of an elliptic curve. Hence, we are left with the
translation of the elliptic Bloch-Wigner function , defined in
sec:toolbox_Bloch:DE, to the iterated integrals  on the torus, which
will be performed in sec:connection:TatetoTorus.  Moreover, a
further translation will allow to express the Bloch-Wigner function in the
projective formulation of the elliptic curve. 

In sec:connection:HigherEllPolylogs we show how these translations
can be generalised to two families of elliptic polylogarithms of higher weight,
both of which include the elliptic Bloch-Wigner function. Finally, in
sec:BlochRelationTorus we combine
our results and write down the elliptic Bloch relation on the torus and on the
projective elliptic curve explicitly. Moreover, we discover some holomorphic functional
relations on the torus which imply the elliptic Bloch relation and thereby give
an interpretation of the elliptic Bloch relation in terms of the elliptic
symbol calculus. 



 §.§ The elliptic dilogarithm: from the Tate curve to the torus



We begin with establishing a connection between the iterated integrals
 defined in sec:toolbox_eMPL:Gammas above and the sum


    _n,m(t,s,q)   =-(_n,m(t,s,q)-(-1)^n+m_n,m(t^-1,s^-1,q)),


where the objects 


    _n,m(t,s,q)   = ∑_k,l>0t^k/k^ns^l/l^mq^kl


have been introduced and described in Adams:2016xah. 

In the end, it will turn out that the value _n,-m(t,1,q)=_n,-m(e^2
    π i z,1,q) of _n,-m defined on the Tate curve is, up to
    polynomials in z, equal to the n-fold iterated integral of the
    integration kernel g^(m+1)(z,τ), i.e.    ([   0   … m+1;   0   …   0 ]_n; z,τ), which is an iterated integral
    defined on the torus.
    
In order to show this, the case m=0 is discussed first, for which the definition


    _n(t,s,q)   =
    -(1/2_n(t)-(-1)^n1/2_n(t^-1))+_n,0(t,s,q)


turns out to be useful. In terms of the variables t, q and w defined in
(<ref>), the Eisenstein-Kronecker series
(<ref>) can be rewritten as <cit.>


    F(t,w,q)   
    =-2π i(t/1-t+1/1-w+∑_k,l>0(t^k w^l-t^-kw^-l)q^kl),


such that from the limit
g^(1)(z,τ)=lim_α→ 0(F(z,α,τ) -
g^(0)(z,τ)/α) a straightforward calculation
implies[This calculation has been pointed out in ref.<cit.> and motivated to consider the generalisations for
E_n,-m(t,1,q) with n,m>0 described in the following parts of this subsection. Similar considerations can be found in Levin.
]


    _0(t,1, q)   =1/2π ig^(1)(z,τ) .


The iterated integrals  on the torus may be recovered using the partial
differential equation


    ∂/∂ z_n(z,1,τ)   = 2π i _n-1(z,1,τ) ,


where the function _n is pulled-back to the torus by the exponential map.
This leads to the following integral representation of _1(t,1,q)


    _1(t,1,q)   =lim_ϵ→ 0∫_ϵ^zdz'∂/∂ z'_1(z',1,τ)+_1(e^2π i ϵ,1,q)
       =([ 1; 0 ]; z,τ)-2_1,0(1,1,q)+π i/2 ,


where ([ 1; 0 ]; z,τ) is the
  regularised integral (see ssec:EMP). Note that the logarithmic singularity of
_([ 1; 0 ];
z,τ)=∫_0^z dz' g^(1)(z',τ) cancels the singular contribution
_1(1) of _1(1,1,q), leaving only a phase shift π i/2
caused by the different directions of the paths approaching the singularity of
_1(1). For n>1, there is no singularity at all if the regularised
iterated integrals are used, since for n>1


    _n(1,1,q)   =
     -(1/2(1-(-1)^n)_n(1)+(1-(-1)^n)_n,0(1,1,q))
       =
     0   n even
     -ζ_n-2_n,0(1,1,q)   n odd


is finite as well. This can be seen by considering Ell:EBW:Enm for s,t=1: 


    _n,0(1,1,q)= ∑_k>0q^k/(1-q^k)k^n=-2i∑_k>0e^kπ i τ/sin(kπτ)k^n .


Fortunately, the calculation of the above series can be circumvented by
considering the integral representation of _2 on the torus: taking into
account that _2(1,1, q)=0, a representation of _2(t,1, q) can be
obtained by the following calculation


    _2(t,1, q)   =
    2π i ∫_0^zdz'_1(z',1,τ)
       =2π i ([ 0 1; 0 0 ]; z,τ)+2π i (π i/2-2_1,0(1,1,q))z .


Evaluation at z=1 of Ell:EBW:E2Gamma together with
sec:connecting_eMPL:En1 yields the value of _1,0(1,1,q) in terms
of the regularised iterated integrals


    2  _1,0(1,1,q)   =ω_2(1;τ)+π i/2 ,


such that 


    _2(t,1, q)   =
    2π i ∫_0^zdz'_1(z',1,τ)
       =2π i (([ 0 1; 0 0 ]; z,τ)-ω_2(1;τ)z) .


Turning back to the functions _n(t,1,q), one finds recursively that for
n≥ 1


    _n(t,1,q)
       =(2π i)^n-1([ 0 … 0 1; 0 … 0 0 ]_n; z,τ)+P_n(z,q) ,


where P_n(z,q) is the polynomial of degree n-1 in z


    P_n(z,q)   =-(2π i)^n-1ω_2(1;τ)z^n-1/(n-1)!+∑_j=2^n(2π i)^n-j_j(1,1,q)z^n-j/(n-j)!.


In sec:connecting_eMPL:EnandGamma, the whole z dependence of
_n(t,1,q) is expressed solely in terms of (polynomials of) the regularised
iterated integrals  with at most weight one, since
z=([ 0; 0 ]; z,τ). The integration
constants _j(1,1,q), given in (<ref>) and
appearing in the polynomial P_n(z,q), can be expressed as a linear
combination of elliptic zeta values. The result can be obtained recursively by
evaluation of sec:connecting_eMPL:EnandGamma at one. The full calculation
is shown in the app:eMZV and results in the explicit expression for
n>1


    _n(1,1,q)   =
    -(1/2(1-(-1)^n)_n(1)+(1-(-1)^n)_n,0(1,1,q))
       =
    (2π i)^n-1∑_k=0^(n-1)/2d_2k+1ω_n+1-2k(1;τ)   n odd
    
    0   n even,


cf. app:intConst:EnExplicit, where d_k is the sequence defined by 


    d_k=
    -1   k=1
    
    0   k even
    
    -d_1/k!-d_3/(k-2)!-…-d_k-2/3!   k odd,


such that e.g.

    d_1=-1 ,     d_3=1/3! ,     d_5=1/5!-1/3!3! ,     d_7=1/7!-1/5!3!-1/3!5!+1/3!3!3! .


Therefore, the polynomial P_n(z,q) can be rewritten in terms of elliptic zeta
values as


    P_n(z,q)   =(2π i)^n-1∑_j=0^⌊n-1/2⌋∑_k=0^jd_2k+1ω_2j+2-2k(1;τ)z^n-1-2j/(n-1-2j)!


and the sums _n(t,1,q) for n≥ 2 can entirely be expressed by means of
the elliptic polylogarithms on the torus


    _n(t,1,q)
       =(2π i)^n-1( ([ 0 … 0 1; 0 … 0 0 ]_n; z,τ)+∑_j=0^⌊n-1/2⌋∑_k=0^jd_2k+1ω_2j+2-2k(1;τ)z^n-1-2j/(n-1-2j)!) .



Employing similar calculations, it is possible to relate iterated integrals
 of weight higher than one to the -functions.
The q-expansions (<ref>) and
(<ref>) of g^(m+1) for m>0 lead to


    _0,-m(t,1,q)   = m!/(2π i)^m+1(g^(m+1)(z,τ)+(1+(-1)^m+1)ζ_m+1)


and therefore, since _n,m satisfies the same partial differential equation as _n,


    ∂/∂ z_n,m(z,1,τ)   =2π i _n-1,m(z,1,τ) ,


the following relations can be identified: for n=1, m>0


    _1,-m(t,1,q)   = ∫_0^zdz' ∂/∂ z'_1,-m(z',1,τ)+_1,-m(1,1,q)
       = m!/(2π i)^m([ m+1;   0 ]; z,τ)+m!/(2π i)^m (1+(-1)^m+1)ζ_m+1z +_1,-m(1,1,q) ,


for n=2,m>0


    _2,-m(t,1,q)   = ∫_0^zdz' ∂/∂ z'_2,-m(z',1,τ)+_2,-m(1,1,q)
       = m!/(2π i)^m-1([   0 m+1;   0   0 ]; z,τ)+m!/(2π i)^m-1  (1+(-1)^m+1)ζ_m+1z^2/2
       =+2π i _1,-m(1,1,q)z +_2,-m(1,1,q) .


A recursion leads to the general formula for n>0,m>0


    _n,-m(t,1,q)   =m!(2π i)^n-m-1([   0   …   0 m+1;   0   …   0   0 ]_n; z,τ)+P_n,m(z,q) ,


where 


    P_n,m(z,q)   =m!(2π i)^n-m-1(1+(-1)^m+1)ζ_m+1z^n/n!+∑_j=1^n(2π i)^n-j_j,-m(1,1,q)z^n-j/(n-j)! .


As in the case m=0, evaluation of _n,-m(t,1,q) given in
sec:connecting_eMPL:EnmandGamma and the fact that _n,-m(1,1,q)
vanishes for n+m even leads to an expression of the integration constants
_n,-m(1,1,q) in terms of elliptic zeta values. The calculation is shown
in app:eMZV and the result is given in app:intConst:EnmExplicit,
i.e.

    _n,-m(1,1,q)   =m!(2π i)^n-1-m∑_k=0^⌊n/2⌋d_2k+1ω_n+1-2k(m+1;τ)    n+m odd
    
    0   n+m even,


such that



    _n,-m(t,1,q)
       =m!(2π i)^n-1-m(([   0   …   0 m+1;   0   …   0   0 ]_n; z,τ)+∑_j=0^⌊n/2⌋∑_k=0^ j d_2k+1ω_2j+1-2k(m+1;τ)z^n-2j/(n-2j)!)    m odd
    
    m!(2π i)^n-1-m(([   0   …   0 m+1;   0   …   0   0 ]_n; z,τ)+∑_j=0^⌊n-1/2⌋∑_k=0^ j d_2k+1ω_2j+2-2k(m+1;τ)z^n-1-2j/(n-1-2j)!)   m even.



For example and latter purposes, we find in particular the relations 


    _1,0(t,1,q)
       =([ 1; 0 ]; z,τ)-ω_2(1;τ)+1/2(_1(t)+_1(t^-1)),


    _2,0(t,1,q)
       =2π i (([ 0 1; 0 0 ]; z,τ)-ω_2(1;τ)z) +1/2(_2(t)-_2(t^-1) )

and 


    _1,-1(t,1,q)   =1/2π i(([ 2; 0 ]; z,τ)-ω_1(2;τ) z)
       =1/2π i([ 2; 0 ]; z,τ)+1/π iζ_2 z .


Thus, we have established a direct connection between the functions _n,-m
on the Tate curve and the iterated integrals of the form
([ 0 … m; 0 … 0 ]_n; z,τ) for n,m>0, which are defined on the torus.

On the other hand, the elliptic
Bloch-Wigner function ^ can be rewritten in terms
of the above examples _1,0, _2,0 and _1,-1. This involves the identities


    ∑_l>0(_2(tq^l)-_2(t^-1q^l))   =-_2,0(t,1,q)

and


    ∑_l>0log(|tq^l|)_1(tq^l)-∑_l>0log(|t^-1q^l|)_1(t^-1q^l)   =-log(|t|)_1,0(t,1,q)-log(|q|)_1,1(t,1,q) ,


which follow straightforwardly from the definition (<ref>) of
_n,m(t,s,q). Therefore, the value of ^ can be expressed in terms of
the iterated integrals  on the torus as follows


    ^(t,q)   =∑_l>0(_2(tq^l)-_2(t^-1q^l))-∑_l>0(log(|tq^l|)_1(tq^l)-log(|t^-1q^l|)_1(t^-1q^l))
       =+(t)
       =-(_2(t,1,q))+log(|t|) (_1(t,1,q))+log(|q|) (_1,-1(t,1,q))
       =(τ) (([ 2; 0 ]; z,τ))+2π( ([ 1 0; 0 0 ]; z,τ))-2π(z) (([ 1; 0 ]; z,τ))
       =+2(z)(π(ω_2(1;τ))+ζ_2(τ)),

where the q-independent term (t) is absorbed in the second equality by
going from _n,m(t,1,q) to _n(t,1,q) according to
Ell:EBW:BarEnDef. The logarithmic factors with the absolute values of t
and q, respectively, yield contributions of the imaginary parts of z and
τ, respectively. The final expression explicitly involving the real part
of z is obtained by using
equations (<ref>)-(<ref>) and the identity
(z_1 z_2)+(z_1)(z_2)=(z_1)(z_2), where z_1,z_2∈, for
the last equality above.

The translation of the elliptic Bloch-Wigner function  from the torus, as
given by sec:connection_eMPL:DEandGamma, to the projective elliptic curve
is based on the results in Broedel:2017kkb.  The iterated integrals
 on the torus can be expressed via the isomorphism ξ_τ+,E
in terms of some iterated integrals on the projective elliptic curve, which are
defined as follows 


    _3([ n_1   … n_k; c_1   … c_k ]; x,e⃗)=∫_0^xdx'φ_n_1(c_1;x',a⃗) _3([ n_2   … n_k; c_2   … c_k ]; x',e⃗) ,     _3(; x,e⃗)=1 ,


with c_i∈∪{∞}, e⃗=(e_1,e_2,e_3) is the vector of the
roots[Note that we use slightly different conventions than in
Broedel:2017kkb, where the defining cubic equation of the projective
curve is written in standard form y^2=(x-a_1)(x-a_2)(x-a_3) in contrast to
our notation which only involves the Weierstrass form.] of the Weierstrass
equation and the integration kernels φ_n(c;x,e⃗) are defined
according to the construction of Broedel:2017kkb. For example, the
differential φ_0(0,x,e⃗) dx is simply the holomorphic differential
dx/y which itself is the differential dz on the torus


    φ_0(0,x,e⃗) dx= dx/y=d℘(z)/℘'(z)=dz .


The integration kernels φ_n(∞;x,e⃗) for n≥ 1 are defined
as follows: first, define the integral of x/y with an additional term as
follows


    Z_3(x_P,e⃗)   =-∫_e_1^x_Pdx(x/ y+2η_1/y).


This defines the kernel for n=1


    φ_1(∞;x,e⃗)   =1/yZ_3(x,e⃗) .


The kernels for higher n are defined by some polynomials Z_3^(n), which
are of degree n in Z_3(x) with the coefficients being polynomials in x
and y and that do not have any poles in x. For example in the case of
n=2, the integration kernel is defined as


    φ_2(∞;x,e⃗)=1/yZ_3^(2)(x,e⃗)=1/y(1/8Z_3(x,e⃗)^2-x/2).


The (explicit) construction of Z_3^(n) is exactly the same as the
construction of g^(n)(z,τ) as a polynomial in g^(1)(z,τ) with
polynomial coefficients in ℘(z) and ℘'(z), see Broedel:2017kkb.
This leads to a very close relation between the kernels
φ_n(∞;x,e⃗) and g^(n)(z,τ). For n=0, we first
rewrite[Note that for this calculation, we choose the sign of y=±√(4x^3-g_2x-g_3) in Abel's map
(<ref>) such that we indeed obtain
sec:connecting_eMPL:Z3g1 and not the negative of the right-hand side,
i.e. Z_3(x,e⃗)=-g^(1)(z,τ).]


    Z_3(x,e⃗)=ζ(z)-2η_1 z=g^(1)(z,τ)


using sec:toolbox_eMPL:g1Zeta, such that


    φ_1(∞;x,e⃗) dx   =g^(1)(z,τ) dz .


Thus, the construction of Z_3^(n) ensures that the same result holds for n≥ 1


    φ_n(∞;x,e⃗) dx   =g^(n)(z,τ) dz ,


which is all that is needed to rewrite . With z_0 being a zero of ℘
such that ℘'(z_0)>0, the identification x=℘(z) and from the equations
(<ref>), (<ref>) and
(<ref>) for the differentials, the iterated integrals
in sec:connection_eMPL:DEandGamma can be expressed as follows on the
projective elliptic curve


    ([ 1 0; 0 0 ]; z,τ)   =_3([ 1 0; ∞ 0 ]; x,e⃗)+([ 1 0; 0 0 ]; z_0,τ) ,




    ([ 2; 0 ]; z,τ)   =_3([ 2; ∞ ]; x,e⃗)+([ 2; 0 ]; z_0,τ) ,



    ([ 1; 0 ]; z,τ)   =_3([ 1; ∞ ]; x,e⃗)+([ 1; 0 ]; z_0,τ) ,


as well as


    z   =_3([ 0; 0 ]; x,e⃗)+z_0 .


Therefore, the elliptic Bloch-Wigner function takes the following form on the
projective elliptic curve 


    ^(t,q)
       =(τ) (([ 2; 0 ]; z,τ))+2π( ([ 1 0; 0 0 ]; z,τ))-2π(z) (([ 1; 0 ]; z,τ))
       =+2(z)(π(ω_2(1;τ))+ζ_2(τ))
       =(τ) (_3([ 2; ∞ ]; x,e⃗)+([ 2; 0 ]; z_0,τ))+2π(_3([ 1 0; ∞ 0 ]; x,e⃗)+([ 1 0; 0 0 ]; z_0,τ))
       =-2π(_3([ 0; 0 ]; x,e⃗)+z_0) (_3([ 1; ∞ ]; x,e⃗)+([ 1; 0 ]; z_0,τ))
       =+2(_3([ 0; 0 ]; x,e⃗)+z_0)(π(ω_2(1;τ))+ζ_2(τ)) .


The constant terms involving the iterated integrals on the torus evaluated at
z_0 and 1, respectively, drop out once the elliptic Bloch relation
(<ref>) is formed.



 §.§ Higher elliptic polylogarithms


The translation procedure from the Tate curve to the torus described in the
previous section is applicable to elliptic generalisations of higher
polylogarithms. In this subsection we present two such families, both of which
include the elliptic Bloch-Wigner function, and show how they can be expressed
in terms of the elliptic integrals  on the torus. These families of
functions are not independent, the first one is actually a subclass of the
second.

The first construction of higher elliptic polylogarithms is based on the
averaging process over the Tate curve which was used to define the elliptic
Bloch-Wigner function in sec:toolbox_Bloch:DE. The single-valued
polylogarithms that are to be averaged were first described by Ramakrishnan
<cit.> and generalise the Bloch-Wigner function to higher
orders. They are defined by


    ℒ_n(t)   =ℛ_n(∑_k=0^n-12^k B_k/k!log^k(|t|)_n-k(t)),


where ℛ_n denotes the imaginary or real part if n is even or odd,
respectively, and B_k the k-th Bernoulli number. The Bloch-Wigner function
 is obtained for n=2, and these functions also satisfy a
similar inversion relation as , namely


    ℒ_n(t^-1)   =(-1)^n-1ℒ_n(t) .


The elliptic generalisation used in ZagierGangl and proposed in
Zagier, as linear combinations of the more general class described
below, is


    ℒ_n^(t,q)   =∑_l∈ℤℒ_n(tq^l)
       =∑_k=0^n-12^k B_k/k!ℛ_n(∑_l>0log^k(|tq^l|)_n-k(tq^l)+(-1)^n-1∑_l>0log^k(|t^-1q^l|)_n-k(t^-1q^l))
       =+ℒ_n(t) ,


such that in particular ℒ_2^=. By a similar calculation as for
, which e.g. involves the identity 


    ∑_l>0log^k(|tq^l|)_n-k(tq^l)+(-1)^n-1∑_l>0log^k(|t^-1q^l|)_n-k(t^-1q^l)
       =-∑_m=0^k kmlog^k-m(|t|)log^m(|q|)_n-k,-m


generalising equations (<ref>)
and (<ref>), the elliptic polylogarithms
ℒ_n^ turn out to be related to the functions _n,-m according
to


    ℒ_n^(t,q)   =-∑_m=0^k km2^k B_k/k!log^k-m(|t|)log^m(|q|)ℛ_n(_n-k,-m)+ℒ_n(t) .


Just like in the dilogarithmic case of n=2, this result can immediately be
expressed in terms of the iterated integrals on the torus and the projective
curve using the results of the previous section. 


The more general class of single-valued elliptic polylogarithms,
introduced in Zagier and used in DHoker:2015wxz in the context
of modular graph functions for one-loop closed string amplitudes, can be
constructed from the single-valued sum


    _a,b(t)   =(-1)^a-1∑_n=a^a+b-1n-1a-1(-2log(|t|))^a+b-1-n/(a+b-1-n)!_n(t)
       =+(-1)^b-1∑_n=b^a+b-1n-1b-1(-2log(|t|))^a+b-1-n/(a+b-1-n)!_n(t),


which satisfies _a,b(t)=_b,a(t), where the overline
denotes complex conjugation. The functions ℒ_n above are linear
combinations of _a,b and hence, a subclass of the latter <cit.>.
For example, 


    _1,2(t)=2i(t)+2log(|t|)log(|1-t|) ,


such that the Bloch-Wigner function can be written as
(t)=1/4i(_1,2(t)-_2,1(t)).  The elliptic
generalisation is similar to the previous average over the Tate curve and given
by <cit.>


    ^_a,b(t,q)   =∑_l≥ 0_a,b(tq^l)+(-1)^a+b∑_l>0_a,b(t^-1q^l)+(4π(τ))^a+b-1/(a+b)!B_a+b(u) ,


where B_n is the n-th Bernoulli polynomial and z=uτ+v with u,v∈ [0,1]. 
For example, the elliptic Bloch-Wigner function can be expressed as


    (t,q)   =-1/2(^_2,1(t,q)) .


In order to express the functions ^_a,b in terms of _n,m, the
relevant prefactor in ^_a,b(t,q) for the translation has to be determined.
This is the factor obtained by plugging the right-hand side of the definition
(<ref>) of _a,b into sec:toolbox_eMPL:DEab and
pushing the sum over l to the logarithmic functions depending on this
summation index, i.e.

    ∑_l> 0(log(|tq^l|)^a+b-1-n_n(tq^l)+(-1)^a+blog(|t^-1q^l|)^a+b-1-n_n(t^-1q^l))
       =-∑_m=0^a+b-1-na+b-1-nmlog(|t|)^a+b-1-n-mlog(|q|)^m_n,-m(t,1,q)


where we used sec:connection:sumLogToE. This leads to an expression of
^_a,b(t,q) as a linear combination of terms of the form _n,-m and
complex conjugates thereof, such that, according to the previous section, it is
indeed a linear combination of (powers of) the iterated integrals  and
their complex conjugates. The explicit result is rather lengthy and can be
found in app:DEab. In particular, it matches the result for 
given in sec:connection_eMPL:DEandGamma. 


Let us make a comment about the K-theoretic use of the elliptic Bloch-Wigner
function  in the construction of a regulator map :
K_2(E)→ in equation (8.1.1) of BlochRelation, where
K_2(E) is the second K-group associated to an elliptic curve E over .
The non-elliptic version of the map  generalised to higher K-groups is of
particular interest in the formulation of the conjectures of ZagierGangl,
which relate the Dedekind zeta function ζ_F(m) of a number field F to
special values of the single-valued polylogarithms ℒ_m, and which
are also used in the description of the m-th Bloch group. The elliptic
version  can be used in the construction of the second elliptic Bloch
group, see e.g. ZagierGangl, and its imaginary part is the elliptic
Bloch-Wigner function . In order to describe its real part, let 


    (t)   =log(|t|)log(|1-t|) ,


such that the real part of the regulator map  is given by 


    ^(t,q)   =∑_l≥ 0(t q^l)-∑_l >0(t^-1 q^l) .


Comparing equations (<ref>) and
(<ref>) as well as the definitions of their elliptic
generalisations (<ref>) and
(<ref>), leads to the conclusion that


    ^(t,q)   =1/2(^_1,2(t,q))+(4π(τ))^2/6B_3(u) .


Therefore, according to sec:toolbox_eMPL:DEandDEab, the regulator map
 equals one half of ^_1,2 up to the last term in
sec:connecting_eMPL:JE, such that, as for its imaginary part, i.e. the
elliptic Bloch-Wigner function, the whole regulator map  can immediately
be translated to the iterated integrals on the torus and the projective elliptic curve,
as described above. 



 §.§ The elliptic Bloch relation on the torus


The connections between the different notions of elliptic (multiple)
polylogarithms found in the previous subsections <ref>
and <ref> can be exploited to translate and to
compare various concepts and structures among them. In this section we show how the
elliptic Bloch relation (<ref>)
translates to the torus, discover more general relations thereon and hence,
provide an alternative proof of the elliptic Bloch relation. In doing so, we
will show, how the Bloch relation can be interpreted in terms of differentials of 
iterated integrals or, more generally, in terms of the elliptic symbol calculus
introduced in Broedel:2018iwv.

Let F be an elliptic function on the Tate curve with the following divisor


    (F)=∑_i d_i (a_i) ,    ∑_i d_i =0 ,    ∏_i a_i^d_i=1 .


Formulated on the torus the above equation translates into, 


    (F)=∑_i d_i (A_i) ,    ∑_i d_i =0 ,    ∑_i d_iA_i=0 .


where a_i=e^2π i A_i.  Using
sec:toolbox_ellipticfunctions:FandSigma, one can express F in terms of
a product of Weierstrass σ functions


    F(z)=s_A∏_iσ(z-A_i)^d_i=s_Aexp(∑_i d_i ∫_0^z-A_idz' ζ(z'))


for some scaling s_A∈^∗ of F. Similarly,
for a given ∈^∗, κ-F can be represented by[Note
that here, the e_j do not denote the roots of a Weierstrass equation, but the
orders of the zeros and poles of the elliptic function κ-F.]


    -F(z)=s_B∏_jσ(z-B_j)^e_j=s_Bexp(∑_j e_j ∫_0^z-B_jdz' ζ(z')),


where s_B∈^∗. For notational convenience, let us split the set of
zeros and poles of F and -F, denoted by I and J, respectively, into
the zeros of F, I'={A_i|d_i>0}, the zeros of -F, J'={B_j|e_j>0},
and the common set of poles K={A_i|d_i<0}={B_j|e_j<0}. 
Using these
conventions, the elliptic Bloch relation
(<ref>) can be rewritten by means of
sec:connection_eMPL:DEandGamma as 




    0   =∑_i,jd_i e_j (a_i/b_j,q)
       =-2π∑_i,jd_i e_j (( ([ 1 0; 0 0 ]; A_i-B_j,τ))
                      + (τ/2π i) (([ 2; 0 ]; A_i-B_j,τ))
                     -(A_i-B_j) (([ 1; 0 ]; A_i-B_j,τ)) ),



where B_j is given by b_j=e^2π i B_j and the summation indices (i,j)
run over I× J, unless mentioned otherwise. 

We give an alternative proof of equation
(<ref>), which we refer to as the
elliptic Bloch relation on the torus, in the following paragraphs by showing
that the sums over the single iterated integrals  occurring in the above
formula vanish separately (and for the first two also their imaginary parts,
yielding two holomorphic analogues of the elliptic Bloch relation). Note that
since we are interested in generating functional equations we consider the
zeros and poles A_i and B_j as well as the scaling factors s_A and s_B
to be (not independent) variables, e.g. depending on variable coefficients of
the rational function on the elliptic curve that determine F, cf. the
examples in sec:toolbox:ellBloch. 

Let us start with the first term of the elliptic Bloch relation on the torus,
eqn:EBRtorusa: naturally, the zeros and poles satisfy the constraints
∑_i d_i A_i=0 and ∑_j e_j B_j=0 as functional identities. Hence, the
functional identity


    =-F(A_i)=s_B∏_jσ(A_i-B_j)^e_j


holds for i∈ I', such that taking the total differential of both sides and
using sec:toolbox_eMPL:g1Zeta, i.e. ζ(z)=g^(1)(z,τ)+2η_1
z, as well as the representations (<ref>) and (<ref>) the
differential equation


    ∑_j e_j  g^(1)(A_i-B_j)d(A_i-B_j)   = -dlog(s_B)-c_1∑_j e_j B_j dB_j


can be obtained. For k∈ K, a functional identity involving the residue instead
of the infinite value -F(A_k) can be used for a similar calculation: since
by convention σ'(0)=1, the residue of -F at A_k is


    _A_k(-F)   =s_B∏_j≠ kσ(A_k-B_j)^e_j,


which implies that


    ∑_j≠ ke_j g^(1)(A_k-B_j)d(A_k-B_j)   =dlog(_A_k(-F))-dlog(s_B)-c_1∑_je_jB_jdB_j.


Two similar differential equations for sums over I can be found, the first
one starting from =F(B_j), where j∈ J',


    ∑_i d_i  g^(1)(A_i-B_j)d(A_i-B_j)   = -dlog(s_A)-c_1∑_i d_i A_i dA_i.


With k∈ K and using that _A_k(F)=-_A_k(-F), the last such
differential equation turns out to be


    ∑_i≠ kd_i g^(1)(A_k-A_i)d(A_k-A_i)   =dlog(_A_k(-F))-dlog(s_A)-c_1∑_id_iA_idA_i.


Going through an elaborate calculation, whose details we have outsourced to
app:vanish, the four differential
equations (<ref>),
(<ref>),
(<ref>) and
(<ref>) can be combined into the differential
equation 


    ∑_i,jd_i e_j (A_i-B_j)g^(1)(A_i-B_j,τ)d(A_i-B_j)   =0 .


For integration paths with dτ=0, the differential of the iterated
integral ([ 1 0; 0 0 ]; z,τ) is given
by


    d([ 1 0; 0 0 ]; z,τ)   =zg^1(z,τ)dz .


Accordingly, sec:connection_Bloch:vanishingSumzg1 implies that


    ∑_i,jd_i e_j ([ 1 0; 0 0 ]; A_i-B_j,τ)   =c_2


for some constant c_2∈. In general, the zeros and poles of F are only
constrained by ∑_i d_i A_i=0=∑_i d_i, thus, it may be assumed that they
can be split in a way such that the divisor of F consists of triplets with
two of them being unconstrained and the third one being given by
A_3=-A_1-A_2. An alternative way of saying this is that divisors of the form
(A_1)+(A_2)-(0)-(A_1+A_2) span the set of principal divisors, which was
encountered in sec:toolbox:divisorFunction, cf.sec:toolbox_ellipticfunctions:spanningFlambda. Thus, by continuity, the
above equation can be evaluated at the point where all A_i=0 to determine


    c_2=∑_j e_j ([ 1 0; 0 0 ]; -B_j,τ)∑_i d_i=0.


Therefore, we find a holomorphic analogue of the elliptic Bloch relation


    ∑_i,jd_i e_j ([ 1 0; 0 0 ]; A_i-B_j,τ)   =0.



Similar arguments apply for the term (<ref>) involving the
iterated integral z([ 1; 0 ]; z,τ) in
  the elliptic Bloch relation on the torus
  (<ref>). Let i∈ I' and write


    =-F(A_i)=s_Bexp(∑_j e_j ∫_0^A_i-B_jdz g^(1)(z,τ)+c_1/2∑_j e_j B_j^2),


such that


    ∑_j e_j ([ 1; 0 ]; A_i-B_j,τ)   =log()-log(s_B)-c_1/2∑_j e_j B_j^2-2π i m_1,


for some m_1∈, which holds for
([ 1; 0 ]; A_i-B_j,τ) being
  the regularised or unregularised iterated integral, because the factor
  ∑_j e_j=0 cancels the logarithmic singularity. For k∈ K and with
  σ(z)=s_C exp(∫_z_0^z dz' ζ(z')) such that
  σ'(0)=1, the same calculation as before leads to


    _A_k(-F)   =s_Bs_Cexp(∑_j≠ k e_j ∫_z_0^A_k-B_jdz ζ(z) )
       =s_Bs_Cexp(∑_j≠ k e_j ∫_0^A_k-B_jdz g^(1)(z,τ)+c_1/2∑_je_jB_j^2 +∫_0^z_0dz ζ(z) )


which implies that


    ∑_j≠ k e_j ∫_0^A_k-B_jdz g^(1)(z,τ)   =log(_A_k(-F))-log(s_C)-log(s_B)-c_1/2∑_j e_j B_j^2
       =-∫_0^z_0dz ζ(z)-2π i m_2,


where m_2∈, and analogously for the sum over I∖{k}


    ∑_i≠ k d_i ∫_0^A_k-A_idz g^(1)(z,τ)   =log(_A_k(F))-log(s_C)-log(s_A)-c_1/2∑_i d_i A_i^2
       =-∫_0^z_0dz ζ(z)-2π i m_3,


for m_3∈. A similar result holds for j∈ J',


    ∑_i d_i ([ 1; 0 ]; A_i-B_j,τ)   =log()-log(s_A)-c_1/2∑_i d_i A_i^2-2π i m_4,


where m_4∈. Since log(_A_k(F))=log(_A_k(1-F))+iπ,
equations (<ref>) and
(<ref>) lead to


    ∑_j e_j ([ 1; 0 ]; A_k-B_j,τ)-∑_i d_i ([ 1; 0 ]; A_k-A_i,τ)
       =-iπ(1+2m_2-2m_3) + log(s_A)+c_1/2∑_i d_i A_i^2-log(s_B)-c_1/2∑_j e_j B_j^2.


Finally, using the equations (<ref>),
(<ref>),
(<ref>) and
(<ref>) all together, the identities


    ∑_i,jd_i e_j (A_i-B_j)(([ 1; 0 ]; A_i-B_j,τ))   =0


and 


    ∑_i,jd_i e_j (A_i-B_j)(([ 1; 0 ]; A_i-B_j,τ))   =0.


can be obtained, see app:vanish for the calculation. 

Now, we are left with the term (<ref>) involving
([ 2; 0 ]; z,τ). Let us take the
  partial derivative of sec:connection_Bloch:IprimezGamma10SumJ with
  respect to τ and use the partial differential
  sec:toolbox_eMPL:gnHeatEq of the integration kernel, i.e. 2π i
  ∂/∂τg^(1)(z,τ)=∂/∂
  zg^(2)(z,τ), to find


    ∑_j e_j g^(2)(A_i-B_j,τ)   =-2π i∂/∂τc_1/2∑_j e_j B_j^2-2π i∑_j e_jg^(1)(A_i-B_j,τ)∂/∂τ(A_i-B_j),


valid for i∈ I'. A similar result holds for j∈ J'


    ∑_i d_i g^(2)(A_i-B_j,τ)   =-2π i ∂/∂τc_1/2∑_i d_i A_i^2-2π i∑_i d_ig^(1)(A_i-B_j,τ)∂/∂τ(A_i-B_j)


and for k∈ K 


    ∑_j e_j g^(2)(A_k-B_j)-∑_i d_i g^(2)(A_k-A_i)
       =-2π i∂/∂τc_1/2∑_i d_i A_i^2-2π i∂/∂τc_1/2∑_j e_j B_j^2
       =-2π i∑_j e_j g^(1)(A_k-B_j)∂/∂τ(A_k-B_j)+2π i∑_i d_i g^(1)(A_k-A_i) ∂/∂τ(A_k-B_j).


The equations (<ref>),
(<ref>) and
(<ref>) imply that for paths with dτ=0
the differential equation


    d ∑_ijd_i e_j ([ 2; 0 ]; A_i-B_j,τ)
       =0


holds, the explicit calculation is shown in the app:vanish. By
the same argument as for sec:connection_Bloch:BlochRelationGamma1000, we
therefore find another functional identity which can be interpreted as a
holomorphic analogue of the elliptic Bloch relation on the torus


    ∑_i,jd_i e_j ([ 2; 0 ]; A_i-B_j,τ)=0.


To summarise, we managed to express the elliptic Bloch relation
(<ref>) in terms of iterated
integrals on the torus. 

Let us comment on the two holomorphic functional equations
(<ref>) and
(<ref>) respectively, in terms of the
iterated integrals  on the torus which have the same structure as the
original elliptic Bloch relation: in the language of BlochRelation, it
turns out that the iterated integrals
([ 1 0; 0 0 ]; z,τ) and
([ 2; 0 ]; z,τ) are Steinberg
  functions. However, we have to be careful when using these functional
  identities: these iterated integrals are multi-valued and in order to
  reproduce
  sec:connection_Bloch:BlochRelationGamma1000sec:connection_Bloch:BlochRelationGamma20
  they have to be evaluated on the representatives of the zeros and poles of
  F and -F which satisfy ∑_i d_i A_i=0=∑_j e_j B_j, and not only
  such that these sums lie in the lattice Λ. These equations have been
  obtained by differential calculus of iterated integrals, which is simply the
  symbol calculus of an iterated integral with depth 1. Thus, together with
  sec:connection_Bloch:BlochRelationRezReGamma10 we provide an
interpretation of the elliptic Bloch relation using the elliptic symbol
calculus of the iterated integrals  on the torus. 




 §.§ The elliptic Bloch relation in the projective formulation


By means of equations
(<ref>)-(<ref>), the
elliptic Bloch relation (<ref>) can
also be expressed on the projective elliptic curve


    0   =∑_i,jd_i e_j (a_i/b_j,q)
       =-2π∑_i,jd_i e_j (( ([ 1 0; 0 0 ]; A_i-B_j,τ))+ (τ/2π i) (([ 2; 0 ]; A_i-B_j,τ))
       =-(A_i-B_j) (([ 1; 0 ]; A_i-B_j,τ)) )
       =-2π∑_i,jd_i e_j (( _3([ 1 0; ∞ 0 ]; x_ij,e⃗))+ (τ/2π i) (E_3([ 2; ∞ ]; x_ij,e⃗))
       =-(_3([ 0; 0 ]; x_i,e⃗)-_3([ 0; 0 ]; x_j,e⃗)) (_3([ 1; ∞ ]; x_ij,e⃗)) ),


where x_i=℘(A_i), x_j=℘(B_j) and x_ij=℘(A_i-B_j). Similarly, the holomorphic functional relations (<ref>) and (<ref>) translate to 


    ∑_i,jd_i e_j  _3([ 1 0; ∞ 0 ]; x_ij,e⃗)   =0


and


    ∑_i,jd_i e_j  _3([ 2; ∞ ]; x_ij,e⃗)   =0.





§ CONCLUSIONS

In this article, we have investigated the elliptic Bloch-Wigner function 
in order to obtain functional relations of the iterated integrals
 on the torus and especially to formulate an elliptic analogue
of the five-term identity on the torus. This analysis led to several results:



  
  * The elliptic Bloch-Wigner function , which is usually defined on
    the Tate curve, has been translated into the language of iterated elliptic
    integrals  on the torus. This was the precondition for the application
    of the elliptic symbol calculus.

  
  * We have been extending the translation to the torus for two additional
    classes of functions:   the first class are the sums _a,b^
    <cit.> on the Tate curve defined in sec:toolbox_eMPL:DEab.
    These functions play a crucial role in the calculation of modular graph
    functions <cit.>. The final formulæ can be obtained by
    combining app:eqDEab with equations (<ref>),
    (<ref>), (<ref>) and
    (<ref>).  The representation of functions
    _a,b^ in terms of 's on the torus allows for series expansions
    and therefore the investigation of relations between different modular
    graph functions. In particular, those representations might shed some light
    on the explicit construction of a representation of a single-valued map for
    genus-one string amplitudes as suggested in Broedel:2018izr. 

    The second class is a particular subclass of the former and is given by the functions ℒ_n^, defined
    in sec:connection:LEm, which are based on Ramakrishnan's single-valued
    polylogarithms: their representation on the torus follows from a combination
    of sec:connection:LEm with equations (<ref>),
    (<ref>), (<ref>) and
    (<ref>).

  
  * Once representations on the torus do exist, it is just one further step
    to translate <cit.> those into representations in the
    projective formulation of the elliptic curve. In particular, we have chosen
    to express the elliptic Bloch-Wigner function in terms of iterated
    integrals _3 on the projective elliptic curve. For the two general classes of
    functions mentioned above this can be done in a straightforward manner as
    well.  
  
    The above expressions on the Tate curve and on the torus can be extended to
    the projective elliptic curve as well. In the case of the elliptic Bloch-Wigner
    function , we have found sec:connection_eMPL:DEandE3, which
    expresses the value of  on the Tate curve, the torus and the projective
    curve.





  
  * Employing the above translations, we have taken the elliptic Bloch
    relation (<ref>), which is defined in
    terms of the elliptic Bloch-Wigner function , from the Tate curve to
    the torus and the projective elliptic curve: the result is noted in
    sec:connection_Bloch:BlochRelationProjCurve. 

  
  * The investigation of the elliptic Bloch relation on the torus led to
	  the holomorphic analogues of the elliptic Bloch relation given by
	  equations (<ref>) and
	  (<ref>) as well as the
	  non-holomorphic equations
	  (<ref>) and
	  (<ref>). Since
	  validity of those equations can be proved using the elliptic symbol
	  calculus on the torus, we hereby found an alternative proof and
	  interpretation of the elliptic Bloch relation.

  
  * Translating the elliptic Bloch relation even one step further to the
    projective elliptic curve yields a possibility to write down functional equations in
    terms of algebraic arguments. However, this is possible only when
    restricting the parametrising rational function to lines.  Beyond lines,
    the complexity of the calculation of zeros of the parametrising rational
    function prevents the corresponding functional relation from being purely
    algebraic in the arguments. 

    In particular, this applies to the elliptic analogue of the five-term
    identity, i.e. the functional relation induced via the elliptic Bloch
    relation by the elliptic generalisation
    (<ref>) of the rational function
    (<ref>) parametrising the classical five-term
    identity.
    
    In general, due to the complexity of Abel's map, it can not be expected,
    that generic functional relations generated by the Bloch relation may be
    formulated explicitly in terms of algebraic arguments on the torus or the
    Tate curve, respectively. 



In the classical case, the five-term identity is conjectured to generate all
other functional identities between the dilogarithm.  It would be interesting
to investigate, whether a similar conjecture can be formulated for the elliptic
case. As described at the end of sec:toolbox:ellBloch, Gangl and
Zagier state <cit.> that the elliptic Bloch relation, the symmetry
and the duplication relation are expected to generate all the functional
relations of the elliptic Bloch group associated to the elliptic Bloch-Wigner
function. However, the construction of higher elliptic Bloch groups and in
particular the corresponding group of functional relations awaits further
investigation. 
 
In particular it would be nice, if this
conjecture came along with a geometric interpretation of the elliptic
analogue of the five-term identity, similar, but for sure more complicated,
to the classical case.  

While the elliptic Bloch relation is capable of providing a large class
of functional relations, it is not clear, whether there might exist other
structures or similar mechanisms generating further relations, for example
based on functions beyond the elliptic Bloch-Wigner function. Based on the
experience gained from the current implementation and the investigations in
this article, a straightforward generalisation seems unlikely. 

In the context of Feynman diagrams leading to elliptic polylogarithms, the main
recent focus was usually on evaluating the integrals. Despite the lack of a
dedicated application of the functional relations investigated in this article,
we hope that our results will facilitate simplifications and unravel new
structures in further elliptic Feynman calculations to come. 







 §.§ Acknowledgments


We are grateful to Herbert Gangl and Nils Matthes for the very careful reading
and their helpful comments on the draft of this article. Furthermore, we would
like to thank Rob Klabbers for various interesting discussions. JB would like
to thank the Institute for Theoretical Studies at ETH Zürich for organisation
of the program "Modular forms, periods and scattering amplitudes", where part
of this work was done. AK would like to thank the IMPRS for Mathematical and
Physical Aspects of Gravitation, Cosmology and Quantum Field Theory, of which
he is a member and which renders his studies possible.  Furthermore, AK is
supported by the Swiss Studies Foundation, to which he would like to express
his gratitude. 








§ APPENDIX








§ GROUP ADDITION ON E()

The geometric picture of the addition on the elliptic curve is that two
distinct points P_1=x_1y_1 and P_2=x_2y_2 with y_1≠±
y_2 form a line which intersects the elliptic curve y^2=4x^3-g_2 x-g_3 at a
third point -P_3=x_3-y_3. The sum P_3=P_1+P_2 is defined as being
the projection of -P_3=x_3-y_3 to the negative y-coordinate
P_3=x_3y_3. Thus, two points with their y-coordinate being of the
opposite sign are indeed the inverse of each other with ∞=[0:1:0]
being the unit element since the line defined by P_3 and -P_3 intersects
the elliptic curve only at infinity. The algebraic description is the
following. For P_1 and P_2 as above, the line intersecting them is given by
y=λ x+μ, where


    λ=y_2-y_1/x_2-x_1 ,    μ=y_1x_2-y_2x_1/x_2-x_1 .


The x-coordinate of the third point -P_3=x_3-y_3 intersecting the line
and the elliptic curve is the third solution (besides x_1 and x_2) of the
cubic equation (λ x+μ)^2=4x^3-g_2x-g_3, which is in terms of x_1
and x_2 


    x_3=-x_1-x_2+λ^2/4 .


The y coordinate of P_3 is then simply the negative of the y coordinate determined by the line and x_3,


    y_3=-λ x_3-μ .


The last case we need to consider is if the points P_1 and P_2 are
identical and not the unit element, i.e. P_1=P_2=P=x_Py_P. For
y_P≠ 0, the above description of taking the line intersecting P_1 and
P_2 degenerates to taking the tangent on the elliptic curve at P. The sum
2P=P+P=x_2Py_2P is then again the projection of the second point
lying on this tangent and the elliptic curve with respect to the
x-coordinate. Algebraically, this corresponds to 


    λ=12 x_P^2-g_2/2 y_P ,    μ = y_P-λ x_P


and 


    x_2P=-2x_P+λ^2/4 ,     y_2P=-λ x_2P-μ


as before. In the case of y_P=0, the point P is inverse to itself, such
that P+P=P-P=∞. These addition rules exactly agree with the well-known
addition formula of the Weierstrass ℘ function 


    ℘(x_1+x_2)   =-℘(x_1)-℘(x_2)+1/4(℘'(x_2)-℘'(x_1)/℘(x_2)-℘(x_1))^2


for x_1≠ x_2 and similar for its derivative. This ensures that
ξ_Λ,E defined in sec:toolbox_ellipticfunctions:xiLE is indeed
a homomorphism.



§ Q-EXPANSION OF INTEGRATION KERNELS AND ELLIPTIC POLYLOGARITHMS


Starting from the q-expansion of the Jacobi θ function, the
q-expansion of the integration kernels are obtained via the generating
Eisenstein-Kronecker series <cit.> and are given by


    g^(0)(z,τ)    =1 ,



    g^(1)(z,τ)   =π(π z)+ 4π∑_k,l>0sin(2π k z)q^kl


and for m>0 by


    g^(2m)(z,τ)   =-2ζ_2m-2(2π i)^2m/(2m-1)!∑_k,l>0cos(2π k z) l^2m-1q^kl ,


as well as by


    g^(2m+1)(z,τ)   =-2 i (2π i)^2m+1/(2m)!∑_k,l>0sin(2π k z) l^2mq^kl .


The (n-1)-fold integration of the regularised integral (<ref>), i.e.

    _reg([ 1; 0 ]; z,τ)
       = log(1-e^2π i z)-π i z+4π∑_k,l>01/2π k(1-cos(2π k z))q^kl ,


and the n-fold integration of the above integration kernels g^(m)(z,τ)
for m>1 can be determined analytically. This yields the following efficient
method to write down their q-expansion and, hence, for their numerical
evaluation. The central observation is that for n≥ 0 the 2n-fold
integration of sin(2 π k z) with k∈ is given by 


    ∫_0^zdz_1∫_0^z_1dz_2…∫_0^z_2n-1dz_2nsin(2π k z_2n)   =(-1)^n/(2 π k)^2nsin(2 π k z)
       =+∑_j=1^n (-1)^n-j/(2 π k)^2n+1-2jz^2j-1/(2j-1)!


and the (2n+1)-fold integration by


    ∫_0^zdz_1∫_0^z_1dz_2…∫_0^z_2ndz_2n+1sin(2π k z_2n+1)   =(-1)^n+1/(2 π k)^2n+1cos(2 π k z)
       =+∑_j=0^n (-1)^n-j/(2 π k)^2n+1-2jz^2j/(2j)! .


A similar result holds for the iterative integration of cos(2 π k z),


    ∫_0^zdz_1∫_0^z_1dz_2…∫_0^z_2n-1dz_2ncos(2π k z_2n)   =(-1)^n/(2 π k)^2ncos(2 π k z)
       =+∑_j=0^n-1(-1)^n+1-j/(2 π k)^2n-2jz^2j/(2j)!


and


    ∫_0^zdz_1∫_0^z_1dz_2…∫_0^z_2ndz_2n+1cos(2π k z_2n+1)   =(-1)^n/(2 π k)^2n+1sin(2 π k z)
       =+∑_j=1^n (-1)^n-j/(2 π k)^2n+2-2jz^2j-1/(2j-1)! .


Combining the above results yields the following q-expansions of the elliptic
polylogarithms of the form ([ 0 … 0 m; 0 … 0 0 ]_n; z,τ) for n≥ 1:



    ([ 0 … 0 1; 0 … 0 0 ]_2n; z,τ)   =-1/(2π i)^2n-1_2n(e^2π i z)+∑_j=1^2n-1ζ_j+1/(2 π i)^jz^2n-1-j/(2n-1-j)!-π i z^2n/(2n)!
       =+(-1)^n 4π∑_k,l>01/(2 π k)^2n(sin(2 π k z)+∑_j=1^n(-1)^j/(2 π k)^1-2jz^2j-1/(2j-1)!)q^kl ,



    ([ 0 … 0 1; 0 … 0 0 ]_2n+1; z,τ)   =-1/(2π i)^2n_2n+1(e^2π i z)+∑_j=1^2nζ_j+1/(2 π i)^jz^2n-j/(2n-j)!-π i z^2n+1/(2n+1)!
       =+(-1)^n+1 4π∑_k,l>01/(2 π k)^2n+1(cos(2 π k z)+∑_j=0^n(-1)^j+1/(2 π k)^-2jz^2j/(2j)!)q^kl



and for m>1 and n≥ 0



    ([  0  …  0 2m;  0  …  0  0 ]_2n; z,τ)   =-2ζ_2mz^2n/(2n)!
       =+(-1)^n+12(2π i)^2m/(2m-1)!∑_k,l>01/(2 π k)^2n(cos(2 π k z)+∑_j=0^n-1(-1)^1+j/(2 π k)^-2jz^2j/(2j)!) l^2m-1q^kl ,



    ([  0  …  0 2m;  0  …  0  0 ]_2n+1; z,τ)   =-2ζ_2mz^2n+1/(2n+1)!
       =+(-1)^n+12(2π i)^2m/(2m-1)!∑_k,l>01/(2π k)^2n+1(sin(2 π k z)+∑_j=1^n (-1)^j/(2 π k)^1-2jz^2j-1/(2j-1)!) l^2m-1q^kl ,



as well as



    ([    0    …    0 2m+1;    0    …    0    0 ]_2n; z,τ)   =(-1)^n+12 i (2π i)^2m+1/(2m)!∑_k,l>01/(2π k)^2n(sin(2 π k z)+∑_j=1^n (-1)^j/(2 π k)^1-2jz^2j-1/(2j-1)!) l^2mq^kl ,



    ([    0    …    0 2m+1;    0    …    0    0 ]_2n+1; z,τ)   =(-1)^n2 i (2π i)^2m+1/(2m)!∑_k,l>01/(2π k)^2n+1(cos(2 π k z)+∑_j=0^n (-1)^1+j/(2 π k)^-2jz^2j/(2j)!) l^2mq^kl



where, in the above formula, we denote the integration kernels by
([ 0 … 0 m; 0 … 0 0 ]_0; z,τ)=g^(m)(z,τ).






§ INTEGRATION CONSTANTS AS ELLIPTIC ZETA VALUES


This section is dedicated to the calculation of the integration constants from
sec:connection:TatetoTorus, i.e.

    _n,-m(1,1,q)   =-(1-(-1)^n+m)_n,-m(1,1,q)


for n≥ 1 and m≥ 0, where


    _n,m(1,1,q)   = ∑_k,l>01/k^n1/l^mq^kl ,

in terms of the elliptic zeta values 


    ω_n(m;τ)   =([ 0 … 0 m; 0 … 0 0 ]_n; 1,τ)


defined in sec:toolbox:ezv.

Let us begin with the case m=0, where we consider for n≥ 2


    _n(1,1,q)=
    -1/2(1-(-1)^n)ζ_n+_n,0(1,1,q)=
    0   n even
     -ζ_n-2_n,0(1,1,q)   n odd


as defined in Ell:EBW:BarEnDef. While _n(1,1,q) vanishes for even
n, for odd n≥ 3 the integration constants _n(1,1,q) will turn out to
be linear combinations of
ω_1(1;τ),ω_3(1;τ),…,ω_n(1;τ), which we derive
similarly as the result for n=1 given by sec:connecting_eMPL:ELi1


    2  _1,0(1,1,q)   =ω_2(1;τ)+π i/2 .


In order to do so, let n≥ 4 be even. In this case, the recursion given by
sec:connecting_eMPL:EnandGamma evaluated at one, which is based on the
partial differential equation (<ref>),


    ∂/∂ z_n(z,1,τ)   = 2π i _n-1(z,1,τ) ,


takes the explicit form


    0   =_n(1,1,q)
       =2π i∫_0^1 dz_0 E_n-1(z_0,1,τ)
       =(2π i)^2∫_0^1 dz_0∫_0^z_0dz_1 E_n-2(z_1,1,τ)+2π i _n-1(1,1,q)
       =(2π i)^4∫_0^1 dz_0∫_0^z_0dz_1∫_0^z_1dz_2∫_0^z_2dz_3 E_n-4(z_3,1,τ)
       =+(2π i)^3/3!_n-3(1,1,q)+2π i _n-1(1,1,q)
       =(2π i)^n-2∫_0^1 dz_0∫_0^z_0dz_1…∫_0^z_n-4dz_n-3 E_2(z_n-3,1,τ)
       =+(2π i)^n-3/(n-3)!_3(1,1,q)+…+2π i _n-1(1,1,q) .


Plugging in


    _2(t,1, q)   =2π i (([ 0 1; 0 0 ]; z,τ)-ω_2(1;τ)z)


from Ell:EBW:E2Gamma, and solving for _n-1(1,1,q) leads to the recursive formula 


    _n-1(1,1,q)   =(2π i)^n-2(ω_2(1;τ)/(n-1)!-ω_n(1;τ))-(2π i)^n-4/(n-3)!_3(1,1,q)-…-(2π i)^2/3!_n-3(1,1,q) .


The first examples are n=4


    _3(1,1,q)   =(2π i)^2(ω_2(1;τ)/3!-ω_4(1;τ)) ,


n=6


    _5(1,1,q)   =(2π i)^4(ω_2(1;τ)/5!-ω_6(1;τ))-(2π i)^2/3!_3(1,1,q)
       =(2π i)^4(ω_2(1;τ)/5!-ω_6(1;τ))-(2π i)^2/3!(2π i)^2(ω_2(1;τ)/3!-ω_4(1;τ))
       =(2π i)^4((1/5!-1/3!3!)ω_2(1;τ)+1/3!ω_4(1;τ)-ω_6(1;τ))


and n=8


    _7(1,1,q)   =(2π i)^6(ω_2(1;τ)/7!-ω_8(1;τ))-(2π i)^4/5!_3(1,1,q)-(2π i)^2/3!_5(1,1,q)
       =(2π i)^6(ω_2(1;τ)/7!-ω_8(1;τ))-(2π i)^4/5!(2π i)^2(ω_2(1;τ)/3!-ω_4(1;τ))
       =-(2π i)^2/3!(2π i)^4((1/5!-1/3!3!)ω_2(1;τ)+1/3!ω_4(1;τ)-ω_6(1;τ))
       =(2π i)^6((1/7!-1/5!3!-1/3!5!+1/3!3!3!)ω_2(1;τ)
       =+(1/5!-1/3!3!)ω_4(1;τ)+1/3!ω_6(1;τ)-ω_8(1;τ)).


This recursive structure can be expressed explicitly in terms of the series


    d_k=
    -1   k=1
    
    0   k even
    
    -d_1/k!-d_3/(k-2)!-…-d_k-2/3!   k odd


with the final result being for any natural number n≥ 1


    E_2n+1(1,1,q)   =(2π i)^2n∑_k=0^nd_2k+1ω_2n+2-2k(1;τ) ,


which can be checked inductively as follows: first, note that the series d_k begins with


    d_1=-1 ,     d_3=-1/3! (-1)=1/3! ,     d_5=-1/5!(-1)-1/3!1/3!=1/5!-1/3!3!


and


    d_7=1/7!-1/5!1/3!-1/3!(1/5!-1/3!3!)=1/7!-1/5!3!-1/3!5!+1/3!3!3! ,


such that for n=1,2,3 the explicit formula (<ref>) is
indeed in agreement with the first three examples
(<ref>)-(<ref>). For the general case, let
n>1 and assume that the explicit formula (<ref>)
holds for n-1, such that the recursive formula
(<ref>) implies


    _2n+1(1,1,q)   =(2π i)^2n(ω_2(1;τ)/(2n+1)!-ω_2n+2(1;τ))
       =-(2π i)^2n-3/(2n-1)!_3(1,1,q)-…-(2π i)^2/3!_2n-1(1,1,q)
       =(2π i)^2n(ω_2(1;τ)/(2n+1)!-ω_2n+2(1;τ))
       =-(2π i)^2n-2/(2n-1)!((2π i)^2∑_k=0^1d_2k+1ω_4-2k(1;τ))-…
       =-(2π i)^2/3!((2π i)^2n-2∑_k=0^n-1d_2k+1ω_2n-2k(1;τ))
       =(2π i)^2n(∑_l=0^n-1∑_k=0^l(-d_2k+1)/(2n+1-2l)!ω_2l+2-2k(1;τ)-ω_2n+2(1;τ))
       =(2π i)^2n(∑_k=0^n-1∑_l=k^n-1(-d_2k+1)/(2n+1-2l)!ω_2(l-k)+2(1;τ)-ω_2n+2(1;τ))
       =(2π i)^2n(∑_k=0^n-1∑_m=0^n-k-1-d_2k+1/(2(n-m)+1-2k)!ω_2m+2(1;τ)-ω_2n+2(1;τ))
       =(2π i)^2n(∑_m=0^n-1(∑_k=0^n-m-1-d_2k+1/(2(n-m)+1-2k)!)ω_2m+2(1;τ)-ω_2n+2(1;τ))
       =(2π i)^2n(∑_m=0^n-1d_2(n-m)+1ω_2m+2(1;τ)-ω_2n+2(1;τ))
       =(2π i)^2n∑_m=0^nd_2(n-m)+1ω_2m+2(1;τ)
       =(2π i)^2n∑_k=0^nd_2k+1ω_2n+2-2k(1;τ) ,


where we used the definition (<ref>) of d_2n+1 for n>1, i.e.

    d_2n+1   =∑_k=0^n-1-d_2k+1/(2n+1-2k)! .


This calculation proves the explicit formula (<ref>).


For m≠ 0, the two trivial cases, where the _n,-m(1,1,q) vanish by
definition, are n and m both being either even or both being odd and have
to be distinguished. Starting with the former and using the partial
differential equation (<ref>)


    ∂/∂ z_n,m(z,1,τ)   =2π i _n-1,m(z,1,τ) ,


a similar recursion formula as above, which corresponds to the evaluation of
sec:connecting_eMPL:EnmandGamma at one, can be obtained for even m≥
1, n≥ 4


    0   =_n,-m(1,1,q)
       =2π i∫_0^1 dz_0 E_n-1,-m(z_0,1,τ)
       =(2π i)^2∫_0^1 dz_0∫_0^z_0dz_1 E_n-2,-m(z_1,1,τ)+2π i _n-1,-m(1,1,q)
       =(2π i)^4∫_0^1 dz_0∫_0^z_0dz_1∫_0^z_1dz_2∫_0^z_2dz_3 E_n-4,-m(z_3,1,τ)
       =+(2π i)^3/3!_n-3,-m(1,1,q)+2π i _n-1,-m(1,1,q)
       =(2π i)^n-2∫_0^1 dz_0∫_0^z_0dz_1…∫_0^z_n-4dz_n-3 E_2,-m(z_n-3,1,τ)
       =+(2π i)^n-3/(n-3)!_3,-m(1,1,q)+…+2π i _n-1,-m(1,1,q) .


This can be solved for _n-1,-m(1,1,q) using the result from equation (<ref>) for m even, i.e.

    _2,-m(t,1,q)
       = m!/(2π i)^m-1([   0 m+1;   0   0 ]; z,τ)+2π i _1,-m(1,1,q)z  ,


which leads to


    _1,-m(1,1,q)
       =- m!/(2π i)^mω_2(m+1;τ)


upon evaluation at one, such that


    _n-1,-m(1,1,q)   =m!(2π i)^n-m-2(ω_2(m+1;τ)/(n-1)!-ω_n(m+1;τ))
       =-(2π i)^n-4/(n-3)!_3,-m(1,1,q)-…-(2π i)^2/3!_n-3,-m(1,1,q) .


This evaluates e.g. for n=4 to 


    _3,-m(1,1,q)   =m!(2π i)^2-m(ω_2(m+1;τ)/3!-ω_4(m+1;τ))


and for n=6 to



    _5,-m(1,1,q)   =m!(2π i)^4-m(ω_2(m+1;τ)/5!-ω_6(m+1;τ))-(2π i)^2/3!_3,-m(1,1,q)
       =m!(2π i)^4-m(ω_2(m+1;τ)/5!-ω_6(m+1;τ))
       =-(2π i)^2/3!m!(2π i)^2-m(ω_2(m+1;τ)/3!-ω_4(m+1;τ))
       =m!(2π i)^4-m((1/5!-1/3!3!)ω_2(m+1;τ)+1/3!ω_4(m+1;τ)-ω_6(m+1;τ)) .



Since this recursion is the same as the one for m=0 up to the factor m!(2π
i)^-m and the higher elliptic zeta values, the explicit formula solving this
recursion corresponds to the previous formula given in
app:intConst:EnExplicit and can immediately be written down and proven as
before. The result is that for any natural number n≥ 0 and even m≥ 1 


    E_2n+1,-m(1,1,q)   =m!(2π i)^2n-m∑_k=0^na_2k+1ω_2n+2-2k(m+1;τ) .




The remaining case is m≥ 1, n≥ 3 both odd. The recursive formula can
be obtained as before


    0   =_n,-m(1,1,q)
       =2π i∫_0^1 dz_0 E_n-1,-m(z_0,1,τ)
       =(2π i)^2∫_0^1 dz_0∫_0^z_0dz_1 E_n-2,-m(z_1,1,τ)+2π i _n-1,-m(1,1,q)
       =(2π i)^n-1∫_0^1 dz_0∫_0^z_0dz_1…∫_0^z_n-3dz_n-2 E_1,-m(z_n-2,1,τ)
       =+(2π i)^n-2/(n-2)!_2,-m(1,1,q)+…+2π i _n-1,-m(1,1,q) .


As above, we can plug in _1,-m(t,1,q) given by Ell:EBW:E1mGamma for m odd, i.e.

    _1,-m(t,1,q)
       = m!/(2π i)^m( ([ m+1;   0 ]; z,τ)+2 ζ_m+1z)  ,


and solve for _n-1,-m(1,1,q), which yields
the recursive formula


    _n-1,-m(1,1,q)   =m!(2π i)^n-m-2(-2ζ_m+1/n!-ω_n(m+1;τ))
       =-(2π i)^n-3/(n-2)!_2,-m(1,1,q)-…-(2π i)^2/3!_n-3,-m(1,1,q) .


Evaluation of app:intConst:E1mOddRecursive at one, or considering the q-expansion (<ref>), leads to the following connection between the even zeta values and the elliptic zeta values


    ω_1(m+1;τ)   =-2ζ_m+1 ,


such that the above recursion can be expressed in the more familiar form 


    _n-1,-m(1,1,q)   =m!(2π i)^n-m-2(ω_1(m+1;τ)/n!-ω_n(m+1;τ))
       =-(2π i)^n-3/(n-2)!_2,-m(1,1,q)-…-(2π i)^2/3!_n-3,-m(1,1,q) .


This yields for n=3


    _2,-m(1,1,q)   =m!(2π i)^1-m(ω_1(m+1;τ)/3!-ω_3(m+1;τ))


and for n=5



    _4(1,1,q)   =m!(2π i)^3-m(ω_1(m+1;τ)/5!-ω_5(m+1;τ))-(2π i)^2/3!_2,-m(1,1,q)
       =m!(2π i)^3-m(ω_1(m+1;τ)/5!-ω_5(m+1;τ))
       =-(2π i)^2/3!m!(2π i)^1-m(ω_1(m+1;τ)/3!-ω_3(m+1;τ))
       =m!(2π i)^3-m((1/5!-1/3!3!)ω_1(m+1;τ)+1/3!ω_3(m+1;τ)-ω_5(m+1;τ)) .



Thus, the explicit solution can combinatorially be deduced as the ones above,
which leads for n≥ 1 a natural number and m≥ 1 odd to 


    _2n,-m(1,1,q)   =m!(2π i)^2n-1-m∑_k=0^nd_2k+1ω_2n+1-2k(m+1;τ) .



The above results (<ref>) and
(<ref>) for m≠ 0 can conveniently be
summarised in one single formula: the values _n,-m(1,1,q) for n,m≥ 1
can be expressed as the following linear combinations of elliptic zeta values


    _n,-m(1,1,q)   =m!(2π i)^n-1-m∑_k=0^⌊n/2⌋d_2k+1ω_n+1-2k(m+1;τ)    n+m odd
    
    0   n+m even.







§ _A,B^ ON THE TORUS

In the following equation the functions ^_a,b, defined in
sec:toolbox_eMPL:DEab, are related to the sums _n,-m as introduced
in Ell:EBW:Enm:


    ^_a,b(t,q)
       =∑_l≥ 0_a,b(tq^l)+(-1)^a+b∑_l>0_a,b(t^-1q^l)+(4π(τ))^a+b-1/(a+b)!B_a+b(u)
       =(-1)^a-1∑_n=a^a+b-1n-1a-1(-2)^a+b-1-n/(a+b-1-n)!
       =∑_l> 0(log(|tq^l|)^a+b-1-n_n(tq^l)+(-1)^a+blog(|t^-1q^l|)^a+b-1-n_n(t^-1q^l))
       =+(-1)^b-1∑_n=b^a+b-1n-1b-1(-2)^a+b-1-n/(a+b-1-n)!
       =∑_l> 0(log(|tq^l|)^a+b-1-n_n(tq^l)+(-1)^a+blog(|t^-1q^l|)^a+b-1-n_n(t^-1q^l))
       =+_a,b(t) +(4π(τ))^a+b-1/(a+b)!B_a+b(u)
       =(-1)^a∑_n=a^a+b-1n-1a-1(-2)^a+b-1-n/(a+b-1-n)!
       =∑_m=0^a+b-1-na+b-1-nmlog(|t|)^a+b-1-n-mlog(|q|)^m_n,-m(t,1,q)
       =+(-1)^b∑_n=b^a+b-1n-1b-1(-2)^a+b-1-n/(a+b-1-n)!
       =∑_m=0^a+b-1-na+b-1-nmlog(|t|)^a+b-1-n-mlog(|q|)^m_n,-m(t,1,q)
       =+_a,b(t) +(4π(τ))^a+b-1/(a+b)!B_a+b(u) .


The sums _n,-m can be written in terms of the iterated integrals on the
torus as shown in sec:connection:TatetoTorus. This provides an
explicit translation of _a,b^ on the Tate curve to the elliptic
integrals  on the torus.







§ VANISHING SUMS OVER INTEGRATION KERNELS


In this section we show some explicit calculations used in the main part of
this article. First, let us show how we can get from equations
(<ref>),
(<ref>),
(<ref>) and
(<ref>) to
sec:connection_Bloch:vanishingSumzg1, i.e.

    ∑_i,jd_i e_j (A_i-B_j)g^(1)(A_i-B_j,τ)d(A_i-B_j)   =0 .


In order to apply the initial equations, the sum has to be split correctly, the
equations have to be plugged in, and the sum pulled together again, such that
∑_i d_i A_i=0=∑_j e_j B_j can be used. Explicitly, this is the
following calculation


    ∑_i,jd_i e_j (A_i-B_j)g^(1)(A_i-B_j,τ)d(A_i-B_j)
       =∑_i∈ I'd_iA_i∑_j∈ J e_jg^(1)(A_i-B_j,τ)d(A_i-B_j)+∑_k∈ Kd_kA_k∑_j∈ J∖{k} e_jg^(1)(A_k-B_j,τ)d(A_k-B_j)
       =-∑_j∈ J'e_jB_j∑_i∈ I d_ig^(1)(A_i-B_j,τ)d(A_i-B_j)-∑_k∈ Kd_kA_k∑_i∈ I∖{k} d_ig^(1)(A_k-A_i,τ)d(A_k-A_i)
       =∑_i∈ I'd_iA_i∑_j∈ J e_jg^(1)(A_i-B_j,τ)d(A_i-B_j)-∑_j∈ J'e_jB_j∑_i∈ I d_ig^(1)(A_i-B_j,τ)d(A_i-B_j)
       =+∑_k∈ Kd_kA_k(∑_j∈ J∖{k} e_jg^(1)(A_k-B_j,τ)d(A_k-B_j)-∑_i∈ I∖{k} d_ig^(1)(A_k-A_i,τ)d(A_k-A_i))
       =∑_i∈ I'd_iA_i(-dlog(s_B)-c_1∑_j∈ J e_j B_j dB_j)-∑_j∈ J'e_jB_j(∑_i∈ I -dlog(s_A)-c_1∑_i∈ I d_i A_i dA_i)
       =+∑_k∈ Kd_kA_k(dlog(s_A)-dlog(s_B)+c_1∑_i∈ Id_iA_idA_i-c_1∑_j∈ Je_jB_jdB_j)
       =-∑_i∈ Id_iA_i(dlog(s_B)+c_1∑_j∈ J e_j B_j dB_j)+∑_j∈ Je_jB_j(∑_i∈ I dlog(s_A)+c_1∑_i∈ I d_i A_i dA_i)
       =0 .


A similar calculation leads from
equations (<ref>),
(<ref>),
(<ref>) and
(<ref>) to the equation


    ∑_i,jd_i e_j (A_i-B_j)([ 1; 0 ]; A_k-B_j,τ)
    
       =-iπ(2m_1∑_i∈ I'd_i A_i-2m_4∑_j∈ J'e_j  B_j+(1+2m_2-2m_3)∑_k∈ Kd_k  A_k) ,


which implies sec:connection_Bloch:BlochRelationRezReGamma10 upon taking
the real part. sec:connection_Bloch:BlochRelationImzReGamma10 can be
obtained by the same calculation with (A_i-B_j) being replaced by
(A_i-B_j). Note that the following sum is valid for the regularised as
well as for the unregularised version of
([ 1; 0 ]; z,τ). With this in mind, let
  us calculate (<ref>) and split the sum as before to
  find


    ∑_i,jd_i e_j (A_i-B_j)([ 1; 0 ]; A_k-B_j,τ)
       =∑_i∈ I'd_i  (A_i)∑_j∈ J e_j([ 1; 0 ]; A_i-B_j,τ)+∑_k∈ Kd_k  (A_k)∑_j∈ J∖{k} e_j_([ 1; 0 ]; A_k-B_j,τ)
       =-∑_j∈ J'e_j  (B_j)∑_i∈ I d_i([ 1; 0 ]; A_i-B_j,τ)-∑_k∈ Kd_k  (A_k)∑_i∈ I∖{k} d_i_([ 1; 0 ]; A_k-A_i,τ)
       =∑_i∈ I'd_i  (A_i)(-2π i m_1+log()-log(s_B)-c_1/2∑_j e_j B_j^2)
       =-∑_j∈ J'e_j  (B_j)(-2π i m_4+log()-log(s_A)-c_1/2∑_i d_i A_i^2)
       =+∑_k∈ Kd_k  (A_k)(-iπ(1+2m_2-2m_3) -log()+ log(s_A)
       ============+c_1/2∑_i d_i A_i^2+log()-log(s_B)-c_1/2∑_j e_j B_j^2)
       = (∑_i∈ Id_i A_i)(log()-log(s_B)-c_1/2∑_j∈ J e_j B_j^2)
       =-e_j  (∑_j∈ JB_j)(log()-log(s_A)-c_1/2∑_i∈ I d_i A_i^2)
       =-iπ(2m_1∑_i∈ I'd_i A_i-2m_4∑_j∈ J'e_j  B_j+(1+2m_2-2m_3)∑_k∈ Kd_k  A_k)
       =-iπ(2m_1∑_i∈ I'd_i A_i-2m_4∑_j∈ J'e_j  B_j+(1+2m_2-2m_3)∑_k∈ Kd_k  A_k) .


The last calculation of this kind is the step getting from
equations (<ref>),
(<ref>) and
(<ref>) to
sec:connection_Bloch:vanishingSumdGamma20, i.e.

    d ∑_i,jd_i e_j ([ 2; 0 ]; A_i-B_j,τ)
       =0 .


Here, we have to apply the above splitting of the sum twice to obtain for dτ=0


    d ∑_i,jd_i e_j ([ 2; 0 ]; A_i-B_j,τ)
       =∑_i,jd_i e_j g^(2)(A_i-B_j,τ)d(A_i-B_j)
       =∑_i∈ I'd_i dA_i∑_j∈ J e_j g^(2)(A_i-B_j,τ)-∑_j∈ J'e_j dB_j∑_i∈ I d_i g^(2)(A_i-B_j,τ)
       =+∑_k∈ Kd_k dA_k(∑_j∈ J e_j g^(2)(A_k-B_j,τ)-∑_i∈ I d_i g^(2)(A_k-A_i,τ))
       =∑_i∈ I'd_i dA_i(-2π i∂/∂τc_1/2∑_j∈ J e_j B_j^2-2π i∑_j∈ J e_jg^(1)(A_i-B_j,τ)∂/∂τ(A_i-B_j))
       =-∑_j∈ J'e_j dB_j(-2π i ∂/∂τc_1/2∑_i∈ I d_i A_i^2-2π i∑_i∈ I d_ig^(1)(A_i-B_j,τ)∂/∂τ(A_i-B_j))
       =+∑_k∈ Kd_k dA_k(-2π i∂/∂τc_1/2∑_i∈ I d_i A_i^2-2π i∂/∂τc_1/2∑_j∈ J e_j B_j^2
       =-2π i∑_j∈ J e_j g^(1)(A_k-B_j)∂/∂τ(A_k-B_j)+2π i∑_i∈ I d_i g^(1)(A_k-A_i) ∂/∂τ(A_k-B_j))
       =∑_ijd_i e_j ∂/∂τ(A_i-B_j)g^(1)(A_i-B_j,τ)d(A_i-B_j)
       =0 ,


where for the last equality, we split the sum once again and proceed as in the
calculation of app:vanish:1stCalculation.



















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































