An effective Chabauty-Kim theorem
Jennifer Balakrishnan Netan Dogra



Jennifer S. Balakrishnan, Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA
jbala@bu.edu

Netan Dogra, Department of Mathematics, Imperial College London, London SW7 2AZ, UK 
n.dogra@imperial.ac.uk

 The Chabauty–Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus.
We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a nonabelian generalisation of Coleman's effective Chabauty theorem.



An effective Chabauty–Kim theorem
    Netan Dogra
    December 30, 2023
=================================





§ INTRODUCTION

Chabauty's method <cit.> is one of the most powerful tools for studying the Diophantine geometry of curves of genus larger than 1. In its original form, it gives a proof of the Mordell conjecture for curves X/ of genus g whose Jacobians have Mordell-Weil rank less than g. The simple idea underlying the proof is to try to prove finiteness of the rational points of a curve X with Jacobian J by bounding the intersection of X(ℚ_p ) and the p-adic closure of J(ℚ) inside J(ℚ_p ). 

This paper concerns two subsequent refinements of Chabauty's argument. The first, due to Coleman, is an effective version in the sense of giving a bound on the number of rational points. This amounts to replacing `soft analysis' (finding, on each residue disk of X_ℚ_p, a nontrivial power series vanishing on X(ℚ)), with `hard analysis' (giving a bound on the number of zeroes of this power series). By bounding the number of zeroes of this power series, Coleman produces a bound on the size of X().

The second, due to Kim <cit.>, gives a generalisation of Chabauty's method which replaces the Jacobian with a nonabelian cohomology variety with values in (finite-dimensional quotients of) a motivic fundamental group in the sense of Deligne <cit.>. As explained in the next section, Kim's method  produces a decreasing sequence of subsets X( _p )⊃ X( _p )_1 ⊃ X( _p )_2 ⊃…⊃ X( ). Conjecturally, X( _p )_n =X( ) for all n≫ 0. However in general it is not known that X( _p )_n is eventually finite. By work of Coates and Kim <cit.>, we know unconditionally that X( _p )_n is finite for n≫ 0 when X is a curve whose Jacobian has complex multiplication. 
Recently, Ellenberg and Hast extended this result to give a new proof of Faltings' theorem for solvable covers of ℙ^1 <cit.>.

In this paper we only use the set X( _p )_2, which is much simpler to describe. In analogy with Coleman's original result, we bound the size of X( ), under certain technical conditions, by bounding the size of X( _p )_2.
Just as with the original effective Chabauty results, if one is careful, one can improve the bounds in various ways, but in the interest of simplicity, here we focus on the problem of finding an explicit bound on X(ℚ_p )_2 which is polynomial in the genus. 

To explain our conditions more precisely, we introduce some notation. Let X be a curve of genus g>1 over ℚ, with (X)=r. Define 

    ρ _f (J):= ( (X_))+ ( ( (X_ )^c=-1).


Our finiteness results will be dependent on one of the following conditions being satisfied:


  * Condition A :
r=g
and ρ _f (J) >1.

  * Condition B :
r=g and 

    H^1 _f (G_T ,H^2 _ét(X× X_, _p (1)))=0
.

For a generic curve X, the rank of the Néron-Severi group of J will be 1, and hence Condition A will not hold. However the condition that ρ _f (J)>1 still arises in many examples of interest. For example, if X is a non-trivial cover of a curve of higher genus, or more generally if J is isogenous to a product of two abelian varieties, then ρ _f (J)≥ρ (J) ≥ 2.
By contrast, it is very difficult to give examples when Condition B is satisfied; however, as explained in <cit.>, the latter part of Condition B is implied by a conjecture of Bloch and Kato <cit.>.

As in <cit.>, one may prove the finiteness of X(ℚ_p )_2 if Condition A or Condition B holds. 
For v a prime of bad reduction, we define n_v ∈ℤ_>0 to be the size of the image of X( _v ) under j_2,v (see the next section for a precise definition).

Let X/ be a curve of genus g > 1 with good reduction at a prime p ≥ 3, satisfying condition A or condition B.  Let κ _p =1+p-1/p-21/log (p). Then


  * # X( ) <κ_p (∏ _v∈ T_0 n_v )# X(𝔽_p )(16g^3+15g^2-16g+10).

  * If X is hyperelliptic and p≠ 2g+1, then

    # X( )< κ _p(∏ _v∈ T_0 n_v )  ((2g+2)# X(𝔽_p )+2g # W(𝔽_p )+8g^3 +64g^2 +20g +16),

where W is the subscheme of Weierstrass points.



As will be explained in the next section, one may obtain bounds on the local constants n_v in terms of the reduction data of the curve X at v. It seems difficult to avoid the bounds obtained by the nonabelian Chabauty method depending on how bad the reduction of X is at bad primes. For this reason, it is unclear the extent to which this theorem could directly be used to prove uniformity results in the manner of Stoll and Katz–Rabinoff–Zureick-Brown <cit.>.

However, in special cases, one can control the local factors to provide uniform bounds on the number of rational points of special families of curves. We illustrate this with the following corollary.


Let X be a smooth projective hyperelliptic curve of genus g with good reduction at 3 and potential good reduction at all primes. 
If the curve satisfies Condition A or Condition B, then

    # X( )<24g^3 + 228g^2 + 120g + 72.


An example of a hyperelliptic curve satisfying the hypotheses regarding the reduction type is given by 

    X:y^2 =x^n +k,

where n is a square-free positive integer prime to 6 and k is an integer prime to 3. If n is composite, then X also satisfies ρ (J)>1, and hence in this case the bound on the number of rational points will hold whenever r=g.

The method of proof of Theorem <ref> may also be used to bound the number of integral points on hyperelliptic curves, answering a question of <cit.>.

Let X be a smooth projective hyperelliptic curve of genus g with good reduction at p≥ 3 and Mordell-Weil rank g. Suppose X has a rational Weierstrass point ∞, let Y:=X-∞, and let Y( ) denote the set of integral points of Y with respect to a minimal regular model. Then

    # Y( )< κ _p ( ∏ _v∈ T_0 m_v )(8g^3+44g^2-34g+9+(2g+1)# Y(𝔽_p )+(2g-1)# W(𝔽_p ))

if g>1 and 

    # Y( )<2κ _p ( ∏ _v m_v ) # Y( _p )

if g=1, where the m_v are local constants as in <cit.>, and W denotes the scheme of Weierstrass points not equal to ∞.



To explain the method of proof, we briefly recall Coleman's proof of effective Chabauty <cit.>. There, he gave a bound for the number of zeroes of G:=∫ω in a residue disk ]z[ for ω a global differential. 
This bound is derived from understanding some piece of the Newton polygon of G: specifically, from bounding the length of the slope -1 segment of the Newton polygon. By length of a segment, we take the usual convention: the length of the projection of the segment onto the x-axis. We recall the following classical result: 

Suppose the slope ≤ -1 segment of the Newton polygon has endpoint (n,m). Then G has at most n zeroes in B(0,|p|).


See e.g. <cit.>.

In particular, Coleman related the Newton polygon of G to the zeroes of ω mod p, which can be bounded by elementary algebraic geometry.

The idea of the proof in the depth 2 case is similar. We want to bound the number of zeroes of a non-algebraic power series G (from depth 2 Chabauty–Kim, see Proposition <ref>) in a residue disk ]z[, or equivalently, understand the slopes of its Newton polygon. We would like to reduce this to a question about the slopes of something algebraic, but as G involves double integrals, we have to replace simply taking the derivative by applying a more complicated differential operator 𝒟. We show in Section <ref> that for suitable `nice' differential operators, we can relate the Newton polygon of G to the zeroes of 𝒟(G).
We then want to find a 𝒟 that sends our power series G to some algebraic function whose zeroes we can bound mod p. We give constructions of 𝒟 in the general case, hyperelliptic case, and hyperelliptic and integral points case in the three subsequent sections.



 §.§ Acknowledgements

We thank Kevin Buzzard, Minhyong Kim and Jan Vonk for helpful suggestions.  Balakrishnan is supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant #550023.



§ EXPLICIT CHABAUTY–KIM AT DEPTH 2


We begin with a brief review of a few essential results from the Chabauty–Kim method <cit.>. Associated to a pointed curve (X,b) over , with a good reduction outside a finite set T_0, and a prime p of good reduction, we have a map

    j_n :X( )→ H^1 (G_T ,U_n )
 
where T:=T_0 ∪{p }, G_T is the Galois group of the maximal extension of  unramified outside T, and U_n is the maximal n-unipotent quotient of the _p pro-unipotent completion of π _1 ^ét(X_,b). We also have local maps 

    j_n,v:X( _v )→ H^1 (G_ _v,U_n )

for v in T_0  and 

    j_n,p:X( _p )→ H^1 (G_ _p ,U_n )

(for the definition of H^1 _f (G_ _p ,U_n ) see <cit.>). We define 

    X( _p )_n := j_n,p^-1(  _p (∩ _v∈ T_0  _v ^-1(X( _v ))) ) ⊂ X( _p ),

where _v is the localisation map from H^1 (G_T ,U_n ) to H^1 (G_v ,U_n ). By construction, the set of rational points 
X( ) is a subset of X( _p )_n for all n.

The behaviour of the maps j_n,v is fundamentally different depending on whether or not v=p. In the v≠ p case, we have the following theorem, due to Kim and Tamagawa <cit.>.
[Kim-Tamagawa]
Let v be a prime not equal to p. Then for all n, (j_v,n) is finite.

In fact, one can bound the image of j_n,v in terms of the reduction data of the curve as follows. Let L be a finite extension of K_v over which X_K_v acquires stable reduction. Let 𝒳/𝒪_L be a regular semistable model, and let V(𝒳_k_L) denote the set of irreducible components of the special fibre. 
Since the model is regular, specialisation induces a well-defined map

    r_v :X(K_v )→ V(𝒳_k_L).

For v a prime of bad reduction, we define n_v ∈ℤ_>0 to be the size of the image of X( _v ) under j_2,v in H^1 (G_v ,U_2 ). 
 
With notation as above,

    n_v ≤ (r_v ).


A detailed proof of this lemma will appear in <cit.>. However, for the sake of completeness, we briefly indicate the method of proof. First, if L|K is a finite extension, then it is easy to show that H^1 (G_K, U_n )→ H^1 (G_L ,U_n ) is injective, hence one reduces to the case where X has stable reduction. In this case, one can use the description of the action of G_L on π _1 ^ét, (v')(X_K,b) (the maximal prime-to-v quotient of π _1 ^ét(X_K,b)) in terms of the dual graph of a regular semistable model given in <cit.> to deduce that if points b_1 and b_2 lie on a common irreducible component of V(𝒳_k_L ), then the class of [π _1 ^ét,(v')(X_K;b_1 ,b_2 )] in H^1 (G_L ,π _1 ^ét,(v')(X_K,b_1 ) is trivial. This straightforwardly implies the lemma.

The finiteness of the maps j_n,v allows us to partition the set X( _p )_n as follows. We refer to a tuple α =(α _v )_v in 
∏ _v∈ T_0  (j_n,v) as a collection of local conditions, and define 

    X( _p )_α:= j_n,p^-1(  _p (∩ _v∈ T_0  _v ^-1(α _v )) ) ⊂ X( _p ).

By construction, X( _p )_n is the disjoint union of the X( _p )_α for α a collection of local conditions.
The bound in Theorem <ref> comes from a bound on # X( _p )_α in the case of n = 2, together with a bound on the number of local conditions, i.e. on the size of ∏ _v∈ T_0 j_2,vX( _v ).



 §.§ Local structure at p

The power series G in the introduction is from the following result of <cit.>:
[<cit.>]
Let X/ be a curve of genus g > 1.  Suppose X satisfies condition A or condition B.  Let ω _0 ,… ,ω _2g-1∈ H^0 (X,Ω (D)) be differentials of the second kind forming a basis of H^1 _(X), where D is an effective divisor.  Then, for all local conditions α, there are constants a_ij and a_i, a differential of the third kind η, and a function h∈ H^0 (X,𝒪(2D)) such that

    X(ℚ_p )_α⊂{ z∈ X(ℚ_p ):G(z)=0 },
 where 
    G(z):=∑ _0≤ i,j <2g a_ij∫ ^z _b ω _i ω _j +∑ _0≤ i<2ga_i ∫ ^z _b  ω _i +∫ ^z _b η +h(z).



Hence to prove Theorem <ref>, it is enough to prove the following: for any curve X of genus g>1, one can choose an effective divisor D and a collection of differentials of the second kind ω _0 ,… ,ω _2g-1∈ H^0 (X,Ω (D)) with the following properties:


  * The ω _i form a basis of H^1 _(X).

  * For any differential of the third kind η∈ H^0 (X,Ω(D)) and h∈ H^0 (X,𝒪(2D)), and any constants a_i and a_ij, the number of zeroes of G(z) is at most  

    κ _p # X(𝔽_p )(16g^3+15g^2-16g+10).






 §.§ Proof of Corollary <ref>

In this subsection we prove that Theorem <ref> implies Corollary <ref>.

Let v≠ p be a prime of potential good reduction. Then, for all n,the map

    j_n,v:X( _v )→ H^1 (G_v ,U_n )

is trivial.


Let K_w | _v be an extension over which X acquires good reduction.
Then the map

    j_n,w:X(K_w )→ H^1 (G_w ,U_n )

has trivial image.
Recall from <cit.> that, given a profinite group G, closed normal subgroup H, and G-group A, we get an exact sequence of pointed sets

    H^1 (G/H,A^H ) → H^1 (G,A)⟶ H^1 (H,A).

We apply this when G=G_v, H=G_w. We claim U_n ^G_w =1. To see this, note that it is enough to show that the graded pieces U_n [i] of U_n with respect to the central series filtration satisfy U_n [i]^G_w =1, which follows from the fact that U_n [i] is an unramified representation of G_w of weight -i.

Hence we deduce that the restriction map

    H^1 (G_v ,U_n )⟶ H^1 (G_w,U_n)

is injective. The lemma thus follows from commutativity of the diagram

    (m) [matrix of math nodes, row sep=3em,
    column sep=3em, text height=1.5ex, text depth=0.25ex]
    X(K_v)     H^1 (G_v ,U_n)  
    
     X(L_w)     H^1 (G_w,U_n ) 
    ;
    [->]
    (m-1-1) edge[auto] node[auto] j_n,v (m-1-2)
    edge[auto] node[auto]  (m-2-1)
    (m-1-2) edge[auto] node[auto]  (m-2-2)
    (m-2-1) edge[auto] node[auto] j_n,w (m-2-2);


Now let X be as in Corollary <ref>. Then all the n_v are 1. The corollary follows from taking p=3 and using the Hasse–Weil estimate for # X( _3 ).



§ BOUNDING THE NUMBER OF ZEROES VIA A DIFFERENTIAL OPERATOR

In this section, we explain how to bound the zeroes of a power series G by finding a bound on 𝒟(G) for a suitably `nice' (in a way we will make precise shortly) differential operator 𝒟. The construction of a nice differential operator in the case when G is the Coleman function from Proposition <ref> will be given in the next section.

We begin by fixing notation. We denote by v the p-adic valuation homomorphism ℚ_p ^×→ℤ.  We fix a point b and a rational function x which is a uniformizing parameter at b. We let ]b[ denote the tube (or residue disk) of b, i.e., the set of points reducing to b mod p. Given an analytic function F on ]b[, we let N_b (F) denote the number of ℂ_p-valued zeroes of F in ]b[.

Let C_i denote the function 
ℚ_p [ [x] ]→ℚ_p
sending a power series to its x^i coefficient.
By a differential operator, we will simply mean an element of the noncommutative ring ℚ_p [ [x] ][d/dx]. By an algebraic differential operator, we will mean a differential operator in the image of K(X)[d/dx]. The order of a differential operator will refer to its  degree as a polynomial in d/dx, when given in the form ∑_i=0^N a_i (d/dx)^i, for a_i ∈ℚ_p [ [x] ]. 

A differential operator 𝒟=∑ _i=0^N g_i d^i /dx^i ∈ℚ_p [ [x] ][d/dx] is nice if all the g_i are in ℤ_p [ [x] ], and g_N is in ℤ_p [ [x] ]^×. 

The main result of this section is the following proposition, which shows that one may use nice differential operators to bound the zeroes of power series, in analogy with Coleman's use of differentiation.

Let G be a power series in ℚ_p [ [x] ]. Let 𝒟 be a nice differential operator of order N. Suppose 𝒟(G) is an algebraic function with no poles on ]b[. Then the number of zeroes of G in ]b[ is at most 
κ _p (N_b (𝒟(G))+N). 


The proof of this proposition will occupy the remainder of the section.


Let F∈ℚ_p [ [x] ] come from a nonzero element of ℚ_p (X) without poles in ]b[. Then { v (C_i (F)) :i≥ 0 } is bounded below, and the least i such that v (C_i (F)) attains this bound is less than or equal to N_b (F).


There is some λ in K such that λ F reduces to a nonzero rational function on X_𝔽_p. Since F has no poles in ]b[, the reduction mod p of λ F is the _p (x)-adic expansion of λ F thought of as a rational section of X_𝔽_p, hence the least i such that the minimum of v (C_i (F)) is attained is just the order of _p (λ F).

Now let G be a power series with 𝒟(G) ∈ H^0 (X,𝒪(D)) as in  Proposition <ref>. Let M denote the length of the slope ≤ -1 part of the Newton polygon. Write 𝒟 as 
∑_i=1^N g_i (d/dx)^i, where g_N ∈ℤ_p [ [x] ]^×. Recall the following well-known lemma:

For any n_1 ≤ n_2,

    v (n_2 !/n_1 !)≤log _p (n_1 )+n_2 -n_1/p-1.



Using Legendre's formula of v(n!) = n - s(n)/p-1, where s(n) is the sum of digits in base p, it follows that

    v(n_2!/n_1!) = n_2-n_1/p-1 + s(n_1)-s(n_2)/p-1   ≤n_2-n_1/p-1 + s(n_1)/p-1
       ≤n_2-n_1/p-1 + (p-1)log_p(n_1)/p-1
       =n_2-n_1/p-1 + log_p(n_1).




Let M be the length of the slope ≤ -1 part of the Newton polygon of G. Suppose that M>1, and i ≤ M satisfies 

    v(C_i (G)) ≤ v(C_M (G))+v (M!/i!).

Then i ≥κ _p ^-1M.


Since i≤ M and M is the length of the slope ≤ -1 part of the Newton polygon, we have 

    M-i ≤ v(C_i (G))-v(C_M (G)) ≤ v (M!/i!) .

This implies 
log _p (i)+(M-i)/(p-1) ≥ M-i,
by the previous lemma. Using the inequality  log _p (i)≤ i/log (p), we get 

    κ _p i ≥ M.


Given a power series F, let S(F)={ i ≥ 0: v(C_i (F))=min{ v(C_j (F)):j ≥ 0 }} if this minimum exists, and take S to be empty otherwise.

We now prove a key lemma which gives a quantitative relation between the Newton polygon of G and the Newton polygon of 𝒟(G), when 𝒟 is a nice differential operator. The idea of the proof is as follows. 
Let s denote the minimum of v(C_i (𝒟(G))). We would like to say that if the valuation of C_M (G) is smaller than the valuation of C_i (G) for all i<M, then the valuation of C_M-N(𝒟(G)) is smaller than that of C_i(𝒟(G)) for all i<M-N (and hence s≥ M-N). 

This is not quite true, because when we apply (d/dx) ^N to C_M (G)x^M, we increase the valuation by v(M!/(M-N)!), so it may happen that there is some cancellation. However, for such cancellation to occur, there must be an M_1 <M for which v(C_M_1  (G)) is within v(M!/(M-N)!) of v(C_M (G)). Similarly, if v(C_M_1 -N(𝒟(G))) is not smaller than v(C_i (𝒟(G))) for all i<M_1 -N, then there must be some M_2 <M_1 such that v(C_M_2 (G)) is close to v(C_M_1 (G)), and so on giving a sequence M,M_1 ,… until we get to M_n ≤ s+N. By construction, the v(C_M_i(G)) are `close together', but since M is the endpoint of the slope ≤ -1 part of the Newton polygon they are also `far apart', and comparing these two conditions gives the lemma. 

Note that, without any additional conditions on 𝒟 or G, to prove a result of the form “𝒟(G) has small slopes implies G has small slopes"  it is necessary to assume p>2 (consider, for example, the case 𝒟=(d/dx) -1 and G=exp (x)+1). Note that, by Lemma <ref>, if F is algebraic without poles on ]b[, then min S(F) ≤ _b( _p (F)). Hence the following lemma implies Proposition <ref>.


Let p>2, and let M be the length of the slope ≤ -1 part of the Newton polygon of G.   Suppose S(𝒟(G)) is nonempty.
Then

    M < κ _p (N+min S(𝒟(G))).




For integers i≤ j, let 

    q(i,j):={[ v ( i!/(j-N)!),     if  i≥ j-N,;              0,      otherwise. ].

 For k≥ 0, let 

    T(k)={0≤ i≤ k:v (C_i (G))+q(i,k) ≤ v (C_k (G))+q(k,k) }.


Clearly T(M) contains M. Note that the set of i∈ℤ_≥ 0 for which 
    v(C_M-N(𝒟(C_i(G)x^i )))≤ v(C_M-N(𝒟(C_N(G)x^N )))
 is a subset of T(M). Hence if T(M)={M}, then 
    v (C_N-M(𝒟(G)))=v (C_M (G))+q(M,M),
 and either M≤ N or 
N-M=min S(𝒟(G)). In both cases the lemma follows.

Suppose now that T(M) has cardinality larger than 1. We define a decreasing sequence M_0 ,… ,M_n of positive integers as follows. Let M_0 :=M, and define M_1 =min T(M_0 ). If T(M_1 )={ M_1 }, this is the end of the sequence, otherwise we define M_2 as the minimum, and so on. Let M_n be the last term in the sequence.

We claim M_n ≥ M-p-1/p-2log _p (M_n).
To see this, note that for each i, we have 

    v(C_M_i+1(G))+q(M_i+1,M_i) ≤ v(C_M_i (G))+q(M_i ,M_i ).

Hence 

    v(C_M_n(G))-v(C_M_0(G)) ≤∑ _i (q(M_i ,M_i )-q(M_i ,M_i+1)) ≤ v(M_0 !/M_n !).

Since they lie in the slope ≤ -1 part of the Newton polygon, this implies that M_n and M=M_0 satisfy the inequality
M-M_n ≤ v(M! /M_n !), which by Lemma <ref> is less than or equal to  log _p (M_n)+(M-M_n )/(p-1). Thus we deduce 

    M_n ≥ M-p-1/p-2log _p (M_n).


Hence, by Lemma <ref>, we have 

    M-p-1/p-2log _p (M_n) ≤ N+N_b (𝒟(G)).

On the other hand, since M≥ 2 we have the elementary estimate 

    log _p (M_n) < M_n/log (p).




 §.§ Example: integral points on elliptic curves


Before describing a general method for constructing suitable nice differential operators,  we illustrate how Proposition <ref> can be used to prove effective versions of known finiteness results in the quadratic Chabauty method by considering the simplest possible case: that of integral points on a rank 1 elliptic curve. By work of Kim <cit.>, we know that
integral points on rank 1 elliptic curves are contained in the zeroes of 

    G(z)=∫ ^z _t ω _0 ω _1 +a ∫ ^z _t ω _0 ω _0 +b_i

for some constants a,b_i ∈ _p, where the number of b_i is determined by the Tamagawa numbers at bad primes. In this case, finding a differential operator is quite simple: if we take 𝒟= ( d/ω _0 ) ^2, then

    𝒟(G)=x+a,

and we deduce that # X( _p )_2 < 2κ _p ( ∏ _v m_v ) # (E-O)( _p ). 



§ DIFFERENTIAL OPERATORS FOR RATIONAL POINTS: GENERAL CASE

To use Proposition <ref> to bound X( _p )_2, it remains to give a construction of a nice differential operator 𝒟 such that 𝒟(G) is an algebraic function whose divisor can be controlled when G is the iterated integral function from Proposition <ref>.
The construction of the operator 𝒟 is elementary.  First we make some preliminary notes about calculating the divisor of 𝒟(F) when F and 𝒟 are algebraic.

Let D=∑ n_i P_i be an effective divisor and let F(x) be a function in H^0 (X,𝒪(D)). Suppose dx is an algebraic differential with divisor W-W', with W,W' effective, and W=∑ m_i Q_i. Define D_0 :=∑ P_i and W_0 :=∑ Q_i. 
Then, for all j>0,

    d^j F/dx^j ∈ H^0 (X,𝒪(jW+(j-1)W_0+D+jD_0 )).

In particular d^j F/dx^j ∈ H^0 (X,𝒪((2j-1)W+(j+1)D).


When j=1, this follows from the fact that
the differential dF has poles only in the support of ]b[ and has a pole of order n_i+1 at P_i.  The differential dx only has zeroes at W, each of order 1. The general case follows by induction.


We now restrict to our specific case of interest. Fix a point z in X( _p ). Let D be an effective divisor on X whose support is disjoint from ]z[. Let ω _0 ,… ,ω _2g-1∈ H^0 (X,Ω ^1  (D) ) be a set of differentials of the second kind forming a basis of H^1 _(X). 
Let x∈ℚ_p [ [t] ] be a formal parameter at some point z_0 ∈ ]z[, such that dx is algebraic with divisor D_1 -D_0, (where D_1 and D_0 are effective). Let
f_i := ω _i /dx ∈ H^0 (X,𝒪(D+D_1 ) ). Finally, let η be a differential in H^0 (X,Ω ^1 (D)), and let

    G(z):=∑ a_ij∫ ^z _b ω _i ω _j +∑ a_i ∫ ^z _b  ω _i +∫ ^z _b η +h(z)

be the Coleman function from Proposition <ref>.


The first step in constructing a differential operator satisfying the hypotheses of Proposition <ref> is to reduce to constructing a nice differential operator which kills all the f_i.

Suppose 𝒟_1 =∑ _i=0^N g_i (d/dx)^i is a nice differential operator of degree N, with coefficients in H^0 (X,𝒪(E)), for an effective divisor E such that 

    𝒟_1 (f_i )=0

for all i. Then 𝒟 :=𝒟_1 d/dx is a nice differential operator with 
    𝒟(G)∈    H^0 (X,𝒪(E+3(N-1)D_1 +(N+3)D)), N ≥ 4 
        H^0(X, 𝒪(E +(2N+1)D_1 + (N+3)D)), N = 2, 3.




The operator 𝒟 is nice because its leading coefficient is the same as that of 𝒟_1. We deal with the ∫ω _i , ∫η ,h and ∫ω _i ω _j terms of G separately. First, we have 

    𝒟(∫ω _i )=𝒟_1 (f_i )=0.

For ∫η, d/dx(∫η )=η /dx ∈ H^0 (X,𝒪(D+D_1 )). Thus by Lemma <ref>, 𝒟(∫η )∈ H^0 (X,𝒪(E+(2N+1)D_1 +(N+1)D)).
For h, by Lemma <ref> we have, for all k>0,  

    d^k h/dx^k ∈ H^0 (X,𝒪((2k-1)D_1 +(k+2)D),

hence  𝒟(h)∈ H^0 (X,𝒪((2N+1)D_1 +(N+3)D+E)).  

Finally,

    𝒟(∫ω _i ω _j )     =𝒟_1 (f_i ∫ω _j ) 
        = ∑ _k≤ Ng_k (d/dx)^k (f_i ∫ω _j ) 
        = ∑ _k≤ Ng_k ∑ _0≤ m≤ kkm(d/dx)^m (f_i )(d/dx)^k-m( ∫ω _j ) 
        = 𝒟_1 (f_i )∫ω _j +∑ _k≤ N g_k ∑ _0≤ m<kkm(d/dx)^m (f_i )(d/dx)^k-m-1 (f_j ) 
        = ∑ _k≤ N g_k ∑ _0≤ m<kkm(d/dx)^m (f_i )(d/dx)^k-m-1 (f_j )

since 𝒟(f_1 )=0. By Lemma <ref>, for all k≤ N, and all m<k,

    (d/dx)^m (f_i )(d/dx)^k-m-1(f_j ) ∈ H^0 (X,𝒪((2k-4)D_1 +(k+1)D_2 ) ),

where D_2 :=D_1 +D. Hence 𝒟(∫ω _i ω _j )∈ H^0 (X,𝒪(3(N-1)D_1 +(N+1)D+E)).





 §.§ Finding 𝒟_1: the general case

By the previous lemma, to get a bound on the number of zeroes of G, we need to construct a nice differential operator (with algebraic coefficients we can control) which annihilates all the f_i :=ω _i /dx. In general, given m functions F_1 ,… ,F_m, it is an elementary exercise to construct a nontrivial differential operator of order at most m which annihilates all the F_i. Hence the nontrivial question is how to find a nice differential operator. 

First we introduce some notation. Let F_1 ,… ,F_2g be elements of a formal power series algebra K[ [x] ]. Let S be  subset of ℤ_≥ 0 of size 2g+1.
Write S={ n_1 ,… n_2g+1} with n_i <n_i+1. Let A(S,F_1 ,… ,F_2g) denote the 2g × (2g+1) matrix with entries in K[ [x] ] whose (i,j)th entry is 1/n_j !d^n_j/dx^n_j(F_i ). Let A^(j)(S,F_1 ,… ,F_2g) denote the 2g × 2g matrix obtained by deleting the jth column. Let 𝒟=𝒟_S,F_1 ,… ,F_2g∈ K[ [x] ][d/dx] denote the differential operator

    𝒟_S,F_1 ,… ,F_2g:=∑ _i=1^2g+1(-1)^i+1n_2g+1!/n_i ! (A^(i))d^n_i /dx^n_i .

We first note that 𝒟 is always a differential operator which annihilates the F_i, then show that the set S can be chosen so that 𝒟 is nice.

For any choice of S, and all i,

    𝒟(F_i )=0.



For any power series f,

    𝒟_S,F_1 ,… ,F_2g(f) = n_2g+1! (
    [         1/n_1 !d^n_1 /dx^n_1 (F_1 )                                   …  1/n_2g+1! d^n_2g+1/dx^n_2g+1(F_1 );                                   ⋮                                   ⋱                                   ⋮;        1/n_1 !d^n_1 /dx^n_1 (F_2g )                                   … 1/n_2g+1! d^n_2g+1/dx^n_2g+1(F_2g );           1/n_1 !d^n_1 /dx^n_1 (f )                                   …    1/n_2g+1! d^n_2g+1/dx^n_2g+1(f );                                     ]).

When f=F_i, the matrix does not have full rank.


We now apply this construction in our case of interest. Let D,D_0 ,D_1 ,ω _i ,f_i be as defined earlier in this section.

There exists an S∈ℤ_≥ 0^2g+1 with max S ≤ (D)+2g-1  such that 𝒟_S,f_0 ,… ,f_2g-1  is nice. 



By construction, for any choice of S, the differential operator 𝒟_S,f_0 ,… ,f_2g-1 lies in ℤ_p [ [x] ][d/dx], hence the only nontrivial condition is that the leading coefficient is in ℤ_p ^×. Note that 

    1/n_i !d^n_i /dx^n_i f|_x=0=C_n_i (f),

hence requiring that the leading coefficient is in ℤ_p ^× is equivalent to requiring that 

    (C_i (f_j ))_1≤ i≤ 2g,0≤ j ≤ 2g-1∈ℤ_p ^× .

Therefore, by definition, the least N such that there exists a subset S with max S≤ N+1 for which 𝒟_S,f_0 ,… ,f_2g-1 is nice is exactly the least N such that the f_i remain 𝔽_p-linearly independent after reduction mod (p,x^N).  
Suppose that for all subsets S of { 0,… ,N } of size 2g+1, 𝒟_S,f_0,… ,f_2g-1 is not nice. Then there is a nontrivial 𝔽_p-linear combination of _p ω _0 ,… , _p ω _2g-1 which has a zero of order N. This gives an element of H^0 (X_𝔽_p ,Ω ^1 (D)) with a zero of order N, which completes the proof.


To complete the proof of Theorem <ref>, it remains to estimate the degree of the coefficients of a nice 𝒟:=𝒟_S,f_0 ,… ,f_2g-1. We follow the construction of basis differentials in <cit.>. Let P be a non-Weierstrass point of X(ℚ_p ) whose mod p reduction is different from b, and h∈ℚ_p (X) a non-constant function in H^0 (X,𝒪((g+1)P)). Let ω _0 ,… ,ω _g-1 be a basis of H^0 (X,Ω ^1 ). Define ω _i+g=hω _i for 0≤ i≤ g-1. Then (ω _i )_0≤ i ≤ 2g-1 gives a basis of H^1 _(X). 

Let ω∈ H^0 (X,Ω^1 ) be a differential which does not vanish mod p. Let x denote the formal parameter on ]z[ obtained by integrating ω. Let D_1 =(ω ). As above, define f_i =ω _i /dx ∈ H^0 (X,𝒪(D_1 +(g+1)P)). 

	Since D=(g+1)P above, we have an S={ n_1 ,… ,n_2g+1} such that 𝒟_1 =𝒟_S,f_0 ,… ,f_2g-1 is nice and max S ≤ 3g. Hence the differential operator 𝒟:=𝒟_1 d/ω has order at most 3g+1. To apply Lemma <ref>, it remains to estimate the degrees of the coefficients of 𝒟_S,f_0 ,… ,f_2g-1.  By Lemma <ref>, we have, for all k≥ 0, 

    ( d^k /dx^k ) f_i ∈ H^0 (X,𝒪(2kD_1 +(g+k+1)P).

The kth coefficient of 𝒟 is hence a sum of functions in 

    H^0 (X,𝒪(∑ _1≤ i≤ 2g+1, i≠ k(2n_i D_1 +(g+n_i +1)P))).

Note that 

    ∑ _i ≠ kn_i ≤∑ _i=g+1^3gi = 4g^2 +g.

Hence the coefficients of 𝒟_1 are in 

    H^0 (X,𝒪((8g^2 +2g)D_1 +3g(2g+1)P)).


Applying Lemma <ref> with g ≥ 2, N=3g, 
E = (8g^2 +2g)D_1 +3g(2g+1)P,
and D =(g+1)P
we find that 

    𝒟(G) ∈    H^0 (X,𝒪((8g^2 +2g)D_1 +3g(2g+1)(P)+3(3g-1)D_1 +3(g+1)^2 P)) 
        = H^0 (X,𝒪((8g^2 +11g-3)D_1 +3(3g^2+3g+1)P)).

Hence 𝒟(G) has degree at most 

    (8g^2 +11g-3)(2g-2) +3(3g^2+3g+1) = 16g^3+15g^2-19g+9.

Applying Proposition <ref>, we deduce that on each residue disk, X( _p )_α has at most 

    κ_p(16g^3+15g^2-16g+10)
 points.




§ DIFFERENTIAL OPERATORS FOR RATIONAL POINTS: HYPERELLIPTIC CASE



 §.§ The hyperelliptic case: non-Weierstrass disks

In this subsection, we prove the second part of Theorem <ref>. Let X be a hyperelliptic curve of genus g with good reduction at p≠ 2g+1 and Mordell-Weil rank g. The assumptions on p imply that X has a smooth model over ℤ_p of the form 

    y^2 =f(x)=x^2g+2+ a_2g+1x^2g+1 + ⋯ + a_0.
 
Let ω _i be the differential x^i dx/y. We take as a basis of H^1 _ (X) a subset of the -span of the differentials ω _i :=x^i dx/y, 0≤ i≤ 2g. Hence, changing notation somewhat, we may write G in the form

    G(z)=∑ _0≤ i,j ≤ 2ga_ij∫ ^z _b ω _i ω _j +∑ _0≤ i≤ 2ga_i ∫ ^z _b ω _i +h(z).

Let ∞:=∞^+ + ∞^- denote the degree two divisor of the two points ∞^+, ∞^- above infinity.  Since all the ω _i are in H^0 (X,Ω ^1 ((g+1)∞ )), h lies in H^0 (X,𝒪(2(g+1)∞ )).

First let 𝒟_0 be the differential operator d/ω _0. Then define G_1 := 𝒟_0 G. Hence 

    G_1 =∑ a_ijx^i ∫ω _j +∑ a_i x^i +h_1

where h_1 :=yd/dxh.
Define 𝒟_1 =(d/dx)^2g+1, and 𝒟=𝒟_1 𝒟_0.

Let W denote the degree 2g+2 divisor of Weierstrass points.
Then the power series 𝒟(G) lies in H^0 (X,𝒪((g+1)∞ +(4g+1)W)).


We have 𝒟(∫ω _i )=0 for all i, so it will be enough to prove this for h and for ∫ω _i ω _j.  Note that we have ÷(dx) = W - 2∞, ÷(y) = W - (g+1)∞, and so ÷(ω_0) = (g-1)∞. For h, we use Lemma <ref> to deduce 
    𝒟_0 (h)∈ H^0 (X,𝒪((3g+2)∞)).

Since dx has divisor W - 2∞,
if F ∈ H^0 (X,𝒪(n∞ +mW)), a direct computation gives

    dF/dx∈{[ H^0 (X,𝒪((n-1)∞ +(m+2)W)),                       m>0,;      H^0 (X,𝒪((n-1)∞ +W)),                       m=0.;                            ].

Hence 𝒟(h) ∈ H^0 (X,𝒪((g+1)∞ +(4g+1)W)). 

For ∫ω _i ω _j, we have 

    𝒟(∫ω _i ω _j)     = 𝒟_1 (x^i ∫ω _j ) 
        = ∑ _k=0^i2g+1ki!/(i-k)!x^i-k(d/dx)^2g-k+1∫ (ω _j ) 
        = ∑ _k=0^i2g+1ki!/(i-k)!x^i-k(d/dx)^2g-k(x^j/y).


Now we have 
    x^j/y∈ H^0 (X,𝒪((j-g-1)∞ +W),
 so 

    ( d/dx) ^(2g-k)(x^j/y) ∈ H^0 (X,𝒪((j-3g+k-1)∞ +(4g-2k+1)W)),
 which gives that  
    x^i-k( d/dx) ^(2g-k)(x^j/y) ∈ H^0 (X,𝒪((i+j-1-3g)∞ +(4g-2k+1)W)),
 and thus each summand of (<ref>) is an element of H^0 (X,𝒪((g-1)∞ +(4g+1)W)).


Since the degree of ((g+1)∞ +(4g+1)W) is 2g+2+(4g+1)(2g+2)=8g^2 +12g+4, applying Proposition <ref>, and summing over all non-Weierstrass residue disks, we find that the number of points of X( _p )_α which reduce to non-Weierstrass points away from infinity is at most 
    κ_p((8g^2 +12g+4)+(2g+2) # (X-W-∞)(𝔽_p )).

For the two points at infinity, we may apply the same analysis with the equation at infinity 

    y^2 =a_0 x^2g+2+a_1 x^2g+1+… +1.

We deduce that the number of points of X( _p )_α which reduce to non-Weierstrass points is at most 
    κ_p((16g^2 +24g+8)+(2g+2) # (X-W)(𝔽_p )).




 §.§ The hyperelliptic case: Weierstrass points


The computation at Weierstrass disks is carried out in a manner similar to the method developed in Section <ref>. The essential difference is that, instead of trying to find a new nice differential operator 𝒟_1 annihilating the 2g functions { f_0 ,… ,f_2g-1} for each residue disk ]b[, we find a differential operator 𝒟_1 which annihilates the 2g+1 functions ω _i /ω _0 ,  (0≤ i≤ 2g) at all Weierstrass disks. The price paid for this is that the degree is slightly larger.
 Let B∈ M_2g+1 (K(X)) denote the matrix 
    B=(1/(2i)!
    ( d/ω _0 ) ^2i x^j )_0≤ i,j≤ 2g.
 



For all Weierstrass points z=(α ,0) ∈ X(𝔽_p ), (B)∈𝒪_𝒳,z^×.


Clearly (B) is defined at z, so it is sufficient to prove it is nonzero at z. First note that, since x-α has a zero of order 2 at z, a linear combination of 1,x,… ,x^2g can have a zero of order at most 4g. Hence the (2g+1) × (4g+1) matrix (d^i /ω _0 ^i x^j |_z)_0≤ i≤ 4g,0≤ j≤ 2g has rank at least 2g+1. On the other hand, for all odd j, d^j /ω _0 ^j x^i is an odd function with respect to the hyperelliptic involution, and hence vanishes at z. Hence B|_z is invertible in M_2g+1(𝔽_p ).


 
We deduce that  we can apply the construction of Section <ref> with A taken to be the (2g+1)× (2g+2) matrix 
    ( ( d/ω _0  ) ^i x^j ) _0≤ j≤ 2g,i=0,2,4,… ,4g,4g+1.
 The function ( d/ω _0 )^i x^j lies in H^0 (X,𝒪((gi+j)∞ ) ), hence the differential operator 𝒟_1 =𝒟_S, ω_0/ω_0, ω_1/ω_0, …, ω_2g/ω_0
has coefficients in 
    H^0 (X,𝒪(( g∑ _i=1^2g2i )+g(4g+1)+∑ _j=0^2gj))=H^0 (X,𝒪(4g^3 +8g^2+2g)∞ )).

Define 𝒟:=𝒟_1 𝒟_0, where 𝒟_0 :=(d/ω _0 ).
Applying Lemma <ref> with  
E=(4g^3 +8g^2+2g)∞,
N=4g+1, D_1 =(g-1)∞, D=(g+1)∞, we deduce

    𝒟(G)∈  H^0 (X,𝒪((4g^3 +24g^2 -2g+4)∞ )).

The number of points of X( _p )_α on all Weierstrass disks is hence, by Proposition <ref>, at most 

    κ _p ((4g+2)# W(𝔽_p )+2(4g^3+24g^2-2g+4)).
 Combining with the bounds from the non-Weierstrass residue disks in the previous subsection, we find that 

    # X( _p )_α≤κ _p ((2g+2)# X(𝔽_p )+2g # W(𝔽_p )+8g^3 +64g^2 +20g +16).





§ INTEGRAL POINTS FOR HYPERELLIPTIC CURVES

The proof of the g>1 case of Theorem <ref> follows a similar strategy to the previous section.
Let X be a hyperelliptic curve of genus g>1 with equation

    y^2 =f(x)=x^2g+1+a_2gx^2g + ⋯  + a_0

and suppose the rank of the Jacobian of X is equal to g. Then the set X(ℤ_p )_2 is partitioned into a disjoint union of sets X(ℤ_p )_α. 
Each set X(ℤ_p )_α is contained in the set of zeroes of a Coleman function of the form

    G(z)=∑ _0≤ i,j <2g a_ij∫ ^z _b  ω _i ω _j +∑ _0≤ i<2g a_i ∫ ^z _b ω _i +h(z),

where h∈ H^0 (X,𝒪(4g ∞ ) ), with ∞ now denoting the divisor of degree 1 consisting of the unique point at infinity. Let W denote the degree 2g+1 divisor of Weierstrass points away from infinity and define 𝒟_0 :=d/ω _0. For non-Weierstrass points, we take the differential operator to be 

    𝒟=( d/dx) ^2g𝒟_0 .

Since ω _0 has a zero of order (2g-2) at ∞, we have

    dh/ω _0 ∈ H^0 (X,𝒪((6g-1)∞ ) ).

Similar to the case of an even degree model, since dx has divisor W-3∞, if F is in H^0 (X,𝒪(nW+m∞ ) ) then 

    dF/dx∈{[ H^0 (X,𝒪((n+2)W+(m-2)∞ ) ),                         n>0;      H^0 (X,𝒪(W+(m-2)∞ ) ),                        n=0.;                             ].


Hence 𝒟(h) is in H^0 (X,𝒪((2g-1)∞ +(4g-1)W)).

For the remaining term, note that 

    ( d/dx) ^k (x^j/y) ∈ H^0 (X,𝒪((2k+1)W+(2j-2k-2g-1)∞ ) ).

Since 

    𝒟(∫ω _i ω _j )    = ( d/dx) ^2g(x^i ∫ω _j ) 
        = ∑ _0≤ k<2g2gki!/(i-k)!x^i-k( d/dx) ^2g-k-1(x^j/y),

equation (<ref>) implies

    x^i-k( d/dx) ^2g-k-1(x^j/y) ∈ H^0 (X,𝒪((4g-2k-1)W+(2j+2i-6g+1)∞ ) ).

Hence 𝒟(∫ω _i ω _j ) lies in H^0 (X,𝒪((4g-1)W+(2g-3)∞ )).  Arguing as in Section <ref>, we deduce that the number of integral points of X which reduce to non-Weierstrass points mod p is bounded by

    κ _p ( ∏ _v∈ T_0 m_v )(8g^2+4g-4+(2g+1)# (Y-W)(𝔽_p )).



 §.§ Differential operators at Weierstrass points

Let B∈ M_2g(K(X)) be the matrix ( 1/(2i)!( d/ω _0 ) ^2i x^j ) _0≤ i,j<2g. 
As in Section <ref>, we may show (B) is a unit at all points in W, and hence construct a differential operator 𝒟_1 from the matrix 

    A= ( 1/(i)!( d/ω _0 ) ^i x^j ) _0≤ j<2g,i=0,2,… ,4g-2,4g-1.

Since ω _0 has a zero of order (2g-2) at ∞, we find that 
    ( d/ω _0 ) ^i x^j  ∈ H^0 (X,𝒪((i(2g-1)+2j)∞ )).

We deduce that the coefficients of 𝒟_1 lie in
H^0 (X,𝒪(8g^3+4g^2-6g+1)∞ ).
Define 𝒟:=𝒟_1 𝒟_0, where 𝒟_0 :=(d/ω _0 ).
Applying Lemma <ref> with E =(8g^3+4g^2-6g+1)∞, D_1 =2(g-1)∞, D=2g ∞ and N=4g-1, we deduce that 𝒟(G) lies in 
H^0 (X,𝒪((8g^3+36g^2-38g+13)∞ )) 
Hence by Proposition <ref>, we find that the number of integral points reducing to Weierstrass points is bounded by 

    ( ∏ _v m_v ) κ _p (8g^3+36g^2-38g+13+4g # W( _p )).

We deduce a bound for the total number of integral points of 

    κ _p ( ∏ _v∈ T_0 m_v )(8g^3+44g^2-34g+9+(2g+1)# Y(𝔽_p )+(2g-1)# W(𝔽_p )).




alpha 

