A geometric linear Chabauty comparison theorem
aa r X i v : . [ m a t h . N T ] F e b A GEOMETRIC LINEAR CHABAUTY COMPARISON THEOREM
SACHI HASHIMOTO AND PIM SPELIER
Abstract.
The Chabauty–Coleman method is a p -adic method for finding all rationalpoints on curves of genus g whose Jacobians have Mordell–Weil rank r < g . Recently,Edixhoven and Lido developed a geometric quadratic Chabauty method that was adaptedby Spelier to cover the case of geometric linear Chabauty. We compare the geometriclinear Chabauty method and the Chabauty–Coleman method and show that geometric linearChabauty can outperform Chabauty–Coleman in certain cases. However, as Chabauty–Coleman remains more practical for general computations, we discuss how to strengthenChabauty–Coleman to make it theoretically equivalent to geometric linear Chabauty. Introduction
Let C Q / Q be a smooth, proper, geometrically integral curve of genus g ≥ . Faltings’stheorem [Fal83] proves that the set of rational points C Q ( Q ) is finite. However, it does notprovide an explicit method for computing this finite set. Let J Q / Q to be the Jacobian of C Q , with Mordell–Weil rank r . Fix a prime p > of good reduction for C Q . The Chabauty–Coleman method is an explicit p -adic method for computing the set of rational points on C Q when r < g . The method computes a finite set of p -adic points C ( Z p ) CC containing therational points.In recent years, the Chabauty–Coleman method has been extended to lift the restriction r < g ; Balakrishnan, Besser, Müller, Dogra, Tuitman, and Vonk [BBM16, BD18, BD20,BDM +
19] developed the quadratic Chabauty method. Edixhoven and Lido proposed a par-allel geometric quadratic Chabauty method [EL19] that uses algebro-geometric methods andworks in torsors over the Jacobian instead of a certain Selmer variety.Spelier [Spe20] adapted the geometric method in Edixhoven–Lido to the linear case ofChabauty–Coleman. He outlined a theory of geometric linear Chabauty that parallels theChabauty–Coleman method. This geometric method works in the Jacobian itself instead ofits image under the logarithm in Q gp . Previously, Flynn’s method [Fly97] also leveraged theJacobian group law to perform Chabauty-type calculations, but relied on equations for J Q in projective space and was only practical for genus curves.The geometric linear Chabauty method computes a finite set of p -adic points C ( Z p ) GLC containing the rational points. Unlike the Chabauty–Coleman method, which uses compu-tations of p -adic power series and requires a certain p -adic precision based on their Newtonpolygons, geometric linear Chabauty can be performed modulo p n for any precision n ∈ Z > .Done modulo p , the computations are simply linear algebra (although they do not alwaysresult in an upper bound on the number of rational points).In this paper, we survey both methods, and we provide many examples of the new geomet-ric linear Chabauty method of Spelier. Our main result is a comparison theorem betweenthe geometric linear Chabauty and Chabauty–Coleman methods. In Theorem 5.1, we show SH was supported by National Science Foundation grant DGE-1840990. hat the geometric linear Chabauty method outperforms the Chabauty–Coleman method incertain cases. We have the inclusions: C Q ( Q ) ⊆ C ( Z p ) GLC ⊆ C ( Z p ) CC . Furthermore we give an explicit categorization of any excess points the Chabauty–Colemanmethod finds, C ( Z p ) CC \ C ( Z p ) GLC .However, because the geometric linear Chabauty method can be prohibitively difficultto implement, in Algorithm 5.6 we instead provide an upgrade for the Chabauty–Colemanmethod that makes it equivalent to geometric linear Chabauty. Finally, this paper makesan algorithmic improvement to the geometric linear Chabauty method, replacing Jacobianarithmetic over Z /p n Z with Coleman integration.We start by defining our notational conventions in Section 3.1. In Section 3.2 we introducethe geometric linear Chabauty method of Spelier. Section 3.5 reviews the Chabauty–Colemanmethod. We showcase the explicit linear algebra method for finding rational points on C Q in Section 4. The main theorem and discussion on comparison is found in Section 5.2. Acknowledgments
We are very grateful to Jennifer Balakrishnan for helpful comments during the preparationof this paper. We are also thankful to Bas Edixhoven for his generous advice.3.
Background
Set-up.
Let C Q / Q be a smooth, proper, geometrically integral curve of genus g ≥ .Fix p > a prime of good reduction for C Q and let C/ Z ( p ) be a smooth model for the curveover the local ring. Then(1) C ( Z ( p ) ) = C Q ( Q ) so the problem of determining Q -points on C Q can be replaced by the problem of determining Z ( p ) -points on C .Let J/ Z ( p ) be the Jacobian of C and suppose that the Mordell–Weil rank r of J Q ( Q ) or,equivalently, J ( Z ( p ) ) is less than g . We use M to denote the p -adic closure of the Mordell–Weil group J ( Z ( p ) ) in J ( Z p ) . Let r ′ ≤ r be the rank of M as a Z p -module; we assume wehave computed r ′ elements of J ( Z ( p ) ) that topologically generate M . We also assume C ( Z ( p ) ) is non-empty and fix forever a basepoint b ∈ C ( Z ( p ) ) . Let b ∈ C ( F p ) denote the reduction of b modulo p .For X a scheme, R a local ring with residue field F p , and Q ∈ X ( F p ) , let X ( R ) Q denotethe residue disk over Q ; we use the same notation for the residue disks of M .We will need a description of X ( R ) Q the residue disk of a smooth scheme over Z p . Forthis, we use the following lemma from [Spe20] that can be applied to an affine chart of X containing Q . Lemma 3.1 ([Spe20, Lemma 2.2]) . Let X be a smooth affine scheme over Z p of relativedimension d , let Q be an F p -point of X , and let t , . . . , t d be parameters of X at Q , i.e.elements of the local ring O X,Q such that the maximal ideal is given by ( p, t , . . . , t d ) . Define e t i := t i /p . Then evaluation of e t , the vector ( e t , . . . , e t d ) , gives a bijection e t : X ( Z p ) Q → ( Z p ) d . n fact, this is shown in a geometric fashion by giving a bijection between X ( Z p ) Q and e X pQ ( Z p ) , an open affine subscheme of the blowup of X at Q . Then the coordinate ring of e X pQ ( Z p ) has p -adic completion equal to the ring of convergent power series Z p (cid:10)e t , . . . , e t d (cid:11) = { f ∈ Z p [[ e t , . . . , e t d ]] : for all n ≥ , f ∈ Z p [ e t , . . . , e t d ] + ( p n ) } . Evaluating the e t i yields a bijection e X pQ ( Z p ) → Z dp by the formula e X pQ ( Z p ) = Hom( O e X pQ , Z p ) = Hom( Z p (cid:10)e t , . . . , e t d (cid:11) , Z p ) = A d Z p . Remark . Lemma 3.1 works equally well modulo p n , giving a bijection X ( Z /p n Z ) Q ≃ ( Z /p n − Z ) d . The geometric linear Chabauty method.
We recall an idea of Chabauty provingthe finiteness of rational points on certain curves of genus g ≥ . Theorem 3.3 ([Cha41]) . Let AJ b : C ( Z p ) → J ( Z p ) denote the Abel–Jacobi map induced bythe basepoint b . Then AJ b ( C ( Z p )) ∩ M is finite (and therefore C ( Z ( p ) ) is). The Geometric linear Chabauty method makes Theorem 3.3 explicit by computing the set AJ b ( C ( Z p )) ∩ M exactly. To start, we break up the set into a union of residue disks. Fix Q ∈ C ( F p ) and consider the set AJ b ( C ( Z p ) Q ) ∩ M Q − b in J ( Z p ) . We study the closure of theMordell–Weil group and the image of the curve under Abel–Jacobi separately.To describe M Q − b , we simply need to know whether it is empty; if not, fix a choice of T ∈ J ( Z ( p ) ) Q − b ; if it is, then AJ b ( C ( Z p ) Q ) ∩ M Q − b is empty, so we can sieve out this residuedisk. Indeed, this is a reformulation of the Mordell–Weil sieve at the prime p , as discussedin Section 3.4.Now we identify J ( Z p ) Q − b with Z gp by Lemma 3.1. Note that this identification does notpreserve the additive structure. Then AJ b ( C ( Z p ) Q ) is cut out by convergent power series f , . . . , f g − in the ring of convergent p -adic power series Z p h z , . . . , z g i [Spe20, Remark 2.6](for the right choice of parameters, we may assume the f i are linear), and the inclusion M Q − b → J ( Z p ) Q − b , identifying the former with Z r ′ p , is given by g power series κ , . . . , κ g ∈ Z p h x , . . . , x r ′ i [Spe20, Theorem 3.1]. All in all, we get the diagram(2) Z g − p M Q − b J ( Z p ) Q − b C ( Z p ) Q κ f AJ b λ Q GLC . he coordinates λ i for i = 1 , . . . , g − of λ Q GLC consist of the pullbacks of f , . . . , f g − along κ . They are given by composing convergent power series that are affine linear mod p and thus themselves are given by convergent power series that are affine linear mod p . Inthis diagram, the vertical sequence is exact in the sense that ker f = AJ b ( C ( Z p ) Q ) . That isthe key behind the following proposition. Proposition 3.4 ([Spe20, Theorem 4.1]) . Let Q be an F p -point of C such that there existsan element T ∈ J ( Z ( p ) ) Q − b . The zero set Z ( λ Q GLC ) is equal to M Q − b ∩ AJ b ( C ( Z p ) Q ) =( M + T ) ∩ AJ b ( C ( Z p ) Q ) . Thus λ Q GLC consists of the equations we will use to compute Chabauty’s finite set explicitly.
Definition 3.5.
Let(3) C ( Z p ) GLC := [ Q s.t. J ( Z p ) Q − b = ∅ Z ( λ Q GLC ) be the geometric linear Chabauty set.In practice, the λ i can only be calculated in finite p -adic precision, where, because theyare given by convergent power series, they become polynomials. Although one can say quitea lot about the degrees of these polynomials, this method is especially fruitful modulo p ,where the λ i become affine linear polynomials. To give an upper bound on Z ( λ Q GLC ) , one canuse the following theorem. Proposition 3.6 ([EL19, Theorem 4.12]) . Let A = Z p h x , . . . , x r ′ i / ( λ , . . . , λ g − ) and ¯ A := F p [ x , . . . , x r ′ ] / ( λ , . . . , λ g − ) its reduction modulo p . Assume ¯ A is finite. Then ¯ A is Artinianand so ¯ A ≃ Q m ∈ MaxSpec( ¯ A ) ¯ A m .We have the following upper bound on | Hom Z p ( A, Z p ) | and hence on the number of pointsin C ( Z ( p ) ) Q : X m dim F p ¯ A m ≥ | Hom Z p ( A, Z p ) | ≥ C ( Z ( p ) ) Q where the sum is taken over m such that ¯ A/m ¯ A = F p . By Proposition 3.6, as long as ¯ A is finite-dimensional, it suffices to compute λ i modulo p to obtain upper bounds for C ( Z ( p ) ) . The λ i are affine linear modulo p , so ¯ A can only befinite dimensional if it is F p or the zero ring, which happens if the linear system of equations { λ i ≡ p for all i } has respectively one or zero solution(s). This observation enablesthe following reformulation of Proposition 3.6. Corollary 3.7.
Assume M / ( p ) → J ( Z /p Z ) is injective with image M . If in everyresidue disk of J ( Z /p Z ) there is at most one intersection between the image M of J ( Z ( p ) ) and AJ b ( C ( Z /p Z )) , then | C ( Z ( p ) ) | ≤ | M ∩ AJ b ( C ( Z /p Z )) | ≤ | C ( F p ) | . In particular, if the (not necessarily homogeneous) linear system of equations { λ i ≡ p for all i } in Proposition 3.6 has zero or one solution, there is respectively zero or at mostone point in C ( Z ( p ) ) Q . Definition 3.8.
We say that the Mordell–Weil group is of good reduction (modulo p ) if themap M / ( p ) → J ( Z /p Z ) is injective. This is equivalent to asking that the dimension of thekernel and cokernel are constant on Spec Z p . Otherwise, we say that it is of bad reduction . emark . This method is independent of the choice of b ∈ C ( Z ( p ) ) . Choosing a differentbasepoint b ′ shifts the image of the curve by b − b ′ . Equivalently, it shifts M by b ′ − b . But b ′ − b is an element of the Mordell–Weil group, so the method is basepoint-independent.3.3. The modulo p method.
We now describe how to translate geometric linear Chabautymodulo p into F p -linear algebra. For each Q ∈ C ( F p ) , we can either find T ∈ J ( Z ( p ) ) Q − b , orthere is no rational point in the residue disk C ( Z p ) Q ; this is explained in greater detail inSection 3.4. Fix a choice of T . We want to calculate AJ b ( C ( Z p ) Q ) ∩ M Q − b by finding theaffine linear polynomials λ i mod p of Proposition 3.4. To calculate these linear polynomialsmodulo p it suffices to work in residue disks of J ( Z /p Z ) . In this section, we assume theMordell–Weil group is of good reduction.Choosing a parameter t Q for C at Q gives a bijection e t Q : C ( Z /p Z ) Q ∼ −→ F p ; we write Q µ forthe point mapping to µ . In the same way, by choosing parameters, we have an isomorphism J ( Z /p Z ) ≃ F gp as groups. After translation by − T , we see C ( Z /p Z ) Q embeds as a -dimensional affine subspace of J ( Z /p Z ) by the map C ( Z /p Z ) Q → J ( Z /p Z ) , sending x x − b − T .Write M Q − b = T + M . Letting M denote the image of M in J ( Z /p Z ) ; then M ∼ = F r ′ p and we see that (AJ b ( C ( Z /p Z ) Q ) ∩ M Q − b ) − T is exactly AJ b ( C ( Z /p Z ) Q − b − T ) ∩ M .Now, let D Q ⊂ J ( Z /p Z ) be the one-dimensional subspace(4) D Q := { Q µ − Q ν : µ, ν ∈ F p } , and let v := Q − b − T . We can rephrase Corollary 3.7 purely in terms of linear algebra: let φ denote the linear map φ : D Q ⊕ M → J ( Z /p Z ) arising from the embeddings D Q , M ⊂ J ( Z /p Z ) , then(5) | (AJ b ( C ( Z /p Z ) Q ) ∩ M Q − b ) | = | φ − ( v ) | . Remark . If we know there is a rational point P in the residue disk C ( Z p ) Q , then we cantake t Q to be a parameter at P , and choose T = P − b to get v = 0 and hence | φ − (0) | onthe right side of this equation. (In general, | φ − (0) | is always an upper bound on | φ − ( v ) | .)3.4. The Mordell–Weil sieve.
For each residue disk C ( Z p ) Q , either there exists T ∈ J ( Z ( p ) ) Q − b , or there are no rational points in the disk. Assuming generators of the Mordell–Weil group are known, the existence of T can be checked by a simple calculation in J ( F p ) .This calculation can be thought of as a Mordell–Weil sieve [BS10, Sik15] at the single prime p . The Mordell–Weil sieve is a more general technique that produces information aboutcongruence conditions of rational points for subvarieties of abelian varieties.To determine whether T exists, we consider the diagram(6) C ( Z ( p ) ) J ( Z ( p ) ) C ( F p ) J ( F p ) AJ b αβ . For Q ∈ C ( F p ) , if β ( Q ) is not in the image of α , then we say Q fails the Mordell–Weil sieve (at p ). In this case, the residue disk C ( Z p ) Q cannot contain a rational point. Otherwise, Q passes the Mordell–Weil sieve (at p ). .5. The Chabauty–Coleman method.
We briefly outline the Chabauty–Coleman methodfor producing a finite set of p -adic points C ( Z p ) CC ⊂ C ( Z p ) that contains the rational points C ( Z ( p ) ) . For more details and other perspectives on the method, we refer the reader to[Col85, Wet97, MP12].Fix a basepoint b ∈ C ( Z ( p ) ) and consider the inclusion of the curve into the Jacobian AJ b : C ( Z p ) → J ( Z p ) via the Abel–Jacobi map. Coleman [Col85, Theorem 2.11] defined a p -adic integral on the curve C . The Coleman integral on regular one-forms agrees with thelogarithm on J ( Z p ) interpreted as a p -adic Lie group via the equality J ( Q p ) = J ( Z p ) . Werecall some properties of the logarithm here. Remark . Much of the literature on formal groups and the logarithm works with Q p -vector spaces. We will need results about Z p -modules. Most results carry over; for details onthe Z p -module case we reference [Spe20, Section 3]; for the Q p -vector space case see [Hon70]or [Bou89, III §7].Recall H ( J Z p , Ω J Z p ) is a free Z p -module of rank g . For any element j ∈ J ( Z p ) , we havean element(7) log( j ) := 1 p Z j ∈ Hom Z p ( H ( J Z p , Ω J Z p ) , Q p ) , sending a differential ω to the logarithm /p R j ω . The resulting map log : j /p R j is ahomomorphism of abelian groups. Remark . The value of the logarithm in (7) is defined to be /p the value of the usualColeman integral. We divide by p to renormalize: the value R j ω is always divisible by p if j ∈ J ( Z p ) . Proposition 3.13 ([Spe20, Lemma 3.7]) . Recall that we assume p > ; then the logarithm in-duces an isomorphism of abelian groups on the kernel of reduction J ( Z p ) ∼ −→ H ( J Z p , Ω J Z p ) ∨ ,where the dual is taken in the category of Z p -modules.Write m := Ann J ( F p ) . In particular, the integral /p R j := 1 /p · (1 /m · R mj ) lands in thesubmodule Hom Z p ( H ( J Z p , Ω J Z p ) , (1 / Ann J ( F p )) · Z p ) . As Ω J Z p is locally free, log : J Z /p n Z ( Z /p n Z ) → H ( J Z /p n Z , Ω J Z /pn Z ) ∨ ⊗ Z /p n − Z ,j /p Z j is an isomorphism given by lifting j to Z p , taking log , then reducing modulo p n − . If | J ( F p ) | is invertible in Z p , this even extends to a morphism(8) J Z /p n Z ( Z /p n Z ) → H ( J Z /p n Z , Ω J Z /pn Z ) ∨ ⊗ Z /p n − Z . Choosing a basis ( ω i ) gi =1 of H ( J Z p , Ω J Z p ) and dualizing, we get log : J ( Z p ) → Z gp , reducingto log : J ( Z /p n Z ) → ( Z /p n − Z ) g . Fix a point R ∈ J ( F p ) . For j ∈ J ( Z p ) R , the logarithm log( j ) has a convergent powerseries expansion [Spe20, Lemma 3.7]. Let t , . . . , t g be local parameters of J at R andexpand ω i ( t , . . . , t g ) = P gi =1 f i ( t , . . . , t g ) dt i , with f i ∈ Z p [[ t , . . . , t g ]] . y formally integrating, ω i has a unique local antiderivative g i on J ( Z p ) R such that dg i = ω i and g i ∈ Q p [[ t , . . . , t g ]] with constant term . Let ˜ R ∈ J ( Z p ) R be the point where all t i vanish. We may then evaluate the power series at j using the local parameters at R by(9) log( j ) := ( g ( t ( j ) , . . . , t g ( j )) /p, . . . , g j ( t ( j ) , . . . , t g ( j )) /p ) + log( ˜ R ) Remark . For computational purposes, it is easier to exploit the isomorphism AJ ∗ b : H ( C, Ω C ) ≃ H ( J, Ω J ) . Then we may evaluate log( j ) using linearity of the logarithm andexpanding in a local parameter on C Z p at each point. As C is one-dimensional over Z ( p ) , weonly need one parameter, see [Bal15] for example.Consider the inclusion of M into J ( Z p ) . Let(10) V := { v ∈ H ( J Z p , Ω J Z p ) : 1 /p Z m v = 0 for all m ∈ M } . Since rank Z p H ( J Z p , Ω J Z p ) = g but rank Z p M = r ′ , we see V is a rank ( g − r ′ ) Z p -module.Let B be a basis for V and let v : Hom Z p ( H ( J Z p , Ω J Z p ) , (1 / Ann J ( F p )) · Z p ) → Z g − r ′ p denotethe map ψ ( ψ ( ν )) ν ∈ B . By construction, the map v vanishes on log( j ) for j ∈ M .Next consider the Abel–Jacobi embedding AJ b : C ( Z p ) → J ( Z p ) and the composition Λ CC := v ◦ log ◦ AJ b . We get the following diagram:(11) M J ( Z p ) Hom Z p ( H ( J Z p , Ω J Z p ) , J ( F p ) · Z p ) ( J ( F p ) ) · Z g − r ′ p C ( Z p )0 AJ b log v Λ CC Definition 3.15.
We define the Chabauty–Coleman set to be the following:(12) C ( Z p ) CC := Z (Λ CC ) = { P ∈ C ( Z p ) : 1 /p Z P − b ν = 0 for all ν ∈ B } . Then AJ b ( C ( Z p )) ∩ M is contained in C ( Z p ) CC . Lemma 3.16.
Fix R = AJ b ( Q ) ∈ J ( F p ) with Q ∈ C ( F p ) . Let e z : C ( Z p ) Q ∼ −→ Z p be given bya parameter z at Q , and denote Q µ for the element with image µ . Then log([ Q µ − b ]) can beexpressed as f + c where f ∈ Z p h µ i g and c ∈ / Ann( F p ) · Z gp Proof.
Write log( Q µ − b ) = log( Q µ − Q ) + log( Q − b ) , and t , . . . , t g for parameters of J at . Then by [Spe20, Remark 2.3] the function µ e t i ( Q µ − Q ) is given by a convergentpower series in µ . Also, log : J ( Z p ) → Z gp consists of g convergent power series in e t , . . . , e t g .The composition of convergent power series exists and is itself a convergent power series, so log( Q µ − Q ) ∈ Z p h µ i g .By Remark 3.13, the constant term log( Q − b ) lives in / Ann( F p ) · Z dp . (cid:3) In practice, to compute C ( Z p ) CC , we must truncate the power series by working modulo p n for some n ∈ Z > . The choice of n depends on the Newton polygon of the power series: must be large enough so that the truncated power series has the same number of zerosas the original power series, allowing us to Hensel lift the solutions of the truncated powerseries to Z p .To compare the geometric linear Chabauty and Chabauty–Coleman methods, we describehow the Chabauty–Coleman method works in a single residue disk. We fix an F p -point Q ∈ C ( F p ) , and assume Q passes the Mordell–Weil sieve, i.e. J ( Z ( p ) ) Q − b contains an element T . Since T ∈ M , we know /p R Tb v vanishes, so ker( v ◦ log) = ker( v ◦ log ◦ tr − T ) . Thendiagram (11), restricted to this residue disk, becomes(13) M Q − b J ( Z p ) Q − b H ( J Z p , Ω J Z p ) ∨ Z g − r ′ p C ( Z p ) Q AJ b log ◦ tr − T vλ Q CC where λ Q CC is now the composition v ◦ log ◦ tr − T ◦ AJ b . Note that log ◦ tr − T is a bijection J ( Z p ) Q − b → H ( J Z p , Ω J Z p ) ∨ .Unfortunately, the sequence → M Q − b log ◦ tr − T ◦ κ −−−−−−−→ H ( J Z p , Ω J Z p ) ∨ v −→ Z g − r ′ p → is not necessarily exact at the middle term. In fact, ker v ◦ log is the p -saturation N of M inside J ( Z p ) , by the following lemma. Lemma 3.17.
Let A be a free Z p -module of rank n , let B be a Z p -submodule of rank m , andlet v : A → Z n − mp be a linear map vanishing on B . Then ker v is the p -saturation of B .Proof. By linearity of v , ker v contains the p -saturation of B . Comparing dimensions, we seethat (ker v ) ⊗ Z p Q p must equal B ⊗ Z p Q p . Then ker v is contained in ( B ⊗ Z p Q p ) ∩ A , whichis exactly the p -saturation of B . (cid:3) Applying Lemma 3.17 to A = J ( Z p ) and B = M , we see ker( v ◦ log ◦ tr − T ) = T + N .That gives us the following corollary. Corollary 3.18.
Let Q be an F p -point of C that passes the Mordell–Weil sieve, with T ∈ J ( Z ( p ) ) Q − b . Then Z ( λ Q CC ) is exactly the intersection ( N + T ) ∩ AJ b ( C ( Z p ) Q − b ) pulled backalong AJ b . Explicit Geometric linear Chabauty mod p
We outline a practical method for doing explicit geometric linear Chabauty modulo p using Coleman integration. Previously, in [Spe20], this was done by using the birationalityof the map Sym g C → J given by subtracting a generic degree g divisor, and using theKhuri-Makdisi representation of elements of the Jacobian ([KM04]), where elements of theJacobian are represented as certain submodules of Riemann–Roch spaces. This approach ofusing the birationality of the map Sym g C → J is taken in [EL19] for geometric quadraticChabauty as well. he advantage to using Coleman integration is that the map J ( Z /p Z ) → F gp can bemade much more explicit, making the computations simple linear algebra. In what follows,we describe this map and give examples of the method.The logarithm is linear modulo p for p > [Spe20, Lemma 3.7] (when p = 2 the logarithmis not necessarily linear modulo p , hence we exclude this case). After choosing parameterswe have a map J ( Z /p Z ) ≃ F gp . After applying log : F gp → F gp , the resulting map is anisomorphism of vector spaces over F p , allowing us to carry out the methods in Section3.2. For example, the vector in F gp corresponding to j ∈ J ( Z /p Z ) is (1 /p R j ω i ) gi =1 . Thistranslates linear algebra in a vector space of dimension g where addition is difficult, to linearalgebra in F gp , the final step needed to perform geometric linear Chabauty modulo p .4.1. Examples.Example 4.1.
Let C/ Z (5) be (the smooth projective monic model of) the genus 2 curve(LMFDB label ) given by y = x + x + x + 1 / with Mordell–Weil rank 1. Then C has the known rational points C ( Z ( p ) ) known = {∞ = (1 : 0 : 0) , P = (0 : − , P = (0 : 1 : 1) } . The Mordell–Weil group of J is isomorphic to Z and generated by P − ∞ .Let p = 5 . Over F we have the points C ( F ) = { (1 : 0 : 0) , (0 : 1 : 1) , (0 : 4 : 1) } . At the finite non-Weierstrass residue disks corresponding to the points P = (0 : 4 : 1) and P = (0 : 1 : 1) , we have isomorphisms x : C ( Z / Z ) P i ∼ −→ Z / Z , giving C ( Z / Z ) P i = { x = 5 λ : λ ∈ F } . At the infinite point, the local parameter is t = x /y . We identify J ( Z /p Z ) with F p by choosing the basis of differentials ω = dx/y, ω = xdx/y , then applying log : j (1 /p R j ω , /p R j ω ) .Fix a residue disk C ( Z ) P : our goal is to show there is only one point in this disk (andeach other disk). We start by computing M . Since P − ∞ generates the Mordell–Weilgroup, computing M is equivalent to finding the smallest n such that n ( P − ∞ ) = 0 in J ( F p ) . We find n = 15 , that is, m = 15( P − ∞ ) generates M . A simple calculation withtiny Coleman integrals shows that log m = (3 , ∈ F p . That automatically means that themap M → J ( Z p ) is of good reduction, and M is the F p -vector space generated by (3 , .By specializing λ = 0 and λ = 1 we see D P ⊂ J ( Z /p Z ) is generated by d = (5 : − − (0 : − and log d = (4 , .Now the matrix A representing the map φ : D P ⊕ M → J ( Z /p Z ) is (cid:18) (cid:19) . As A is invertible, | φ − ( v ) | = 1 . Hence, by (5) the only rational point in the residue disk of P is P itself. Using the hyperelliptic involution, we see the same holds for P .For the point ∞ , we carry out a similar calculation. We change our basepoint to ∞ ,allowing us to again work in J ( Z /p Z ) = F p . Then D ∞ is generated by d ′ = (1 : 5 : 0) − and log d ′ = (0 , . So we again conclude that the only rational point in the residuedisk containing ∞ is ∞ .As we have now treated all three residue disks, we have proven C Q ( Q ) = { (1 : 0 : 0) , (0 : − , (0 : 1 : 1) } . Example 4.2.
Let C/ Z (3) be (the smooth projective model of) the genus 2 curve (LMFDBlabel ) given by the equation y + ( x + x + 1) y = x − x + x with Mordell–Weil rank 1. Then C has known rational points C ( Z ( p ) ) known = {∞ = (1 : 0 : 0) , (0 : 0 : 1) , (0 : − } . The Mordell–Weil group is J ( Z ( p ) ) ≃ Z and is generated by d := (0 : − − ∞ .Let p = 3 , then C ( F ) = { (1 : 0 : 0) , (0 : 0 : 1) , (0 : 2 : 1) , (1 : 1 : 1) , (1 : 2 : 1) , (2 : 0 : 1) , (2 : 2 : 1) } . In this example we show how to rule out some of the residue disks that do not containrational points using geometric linear Chabauty.For each residue disk C ( Z p ) Q not containing a known rational point, using arithmetic in J ( F p ) we are able to find T := md such that T ∈ J ( Z (3) ) Q −∞ . The order of d is so m < .Since all Q ∈ C ( F ) pass the Mordell–Weil sieve, we proceed to compute the matrix A foreach Q .First we compute M , which does not depend on Q . To do this, we compute log(29 d ) =(2 , ∈ F .Then, for each residue disk C ( Z p ) Q without a known rational point, we will compute theone-dimensional subspace D Q . To do this, we lift Q in two different ways Q and Q to finiteprecision, and then take the tiny Coleman integral log( Q − Q ) . Q m Q Q log( Q − Q )(1 : 1 : 1) 20 (1 + O (3 ) : 4 + O (3 ) : 1) (4 + O (3 ) : 1 + O (3 ) : 1) (0 , O (3 ) : 20 + O (3 ) : 1) (4 + O (3 ) : 5 + O (3 ) : 1) (0 , O (3 ) : 6 + O (3 ) : 1) (5 + O (3 ) : 6 + O (3 ) : 1) (2 , O (3 ) : 14 + O (3 ) : 1) (5 + O (3 ) : 17 + O (3 ) : 1) (1 , Table 1.
Values for D Q For the F p -points, (2 : 0 : 1) and (2 : 2 : 1) , we see modulo that M and D Q give adeterminant zero matrices: A (2:0:1) = (cid:18) (cid:19) , A (2:2:1) = (cid:18) (cid:19) . We check whether for v = Q − ∞ − md , the vector log v is in the image of A Q . For (2 : 0 : 1) we have log v = ( − O (3 ) , − O (3 )) and for (2 : 2 : 1) we have log v = ( O (3 ) , O (3 )) .Therefore by (5) neither residue disk can contain a Z ( p ) -point.However, for (1 : 1 : 1) and (1 : 2 : 1) , we see that A Q is invertible: A (1:1:1) = (cid:18) (cid:19) , A (2:2:1) = (cid:18) (cid:19) . ence geometric linear Chabauty modulo p shows that there is at most one rational pointin each of the two corresponding residue disks.In the previous pair of examples, the geometric linear Chabauty method and Chabauty–Coleman method find the same set of p -adic points, i.e. C ( Z p ) CC = C ( Z p ) GLC . In thefollowing example we show this is not always the case. We will study the differences betweenthe two methods in the subsequent sections.
Example 4.3.
Let C/ Z (3) be (the smooth projective model of) the genus 2 curve (LMFDBlabel ) given by the equation y + xy = x + 2 x + 4 x + 4 x + 3 x + 1 with Mordell–Weil rank 1. Then C has the known rational points C ( Z ( p ) ) known = { (1 : 0 : 0) , (0 : − , (0 : 1 : 1) } . But C ( F ) = { (1 : 0 : 0) , (0 : 1 : 1) , (0 : 2 : 1) , (1 : 0 : 1) , (1 : 2 : 1) , (2 : 2 : 1) } . The Mordell–Weil group of J is isomorphic to Z , generated by d := (0 : − − (1 : 0 : 0) .In J ( F p ) , d has order . Sieving, we find the only residues c ∈ C ( F p ) such that there exists m ∈ Z such that c − ∞ = md are the images of rational points under the reduction map.In their corresponding residue disks, the geometric linear Chabauty method only finds onesolution, so C ( Z p ) GLC = C ( Z ( p ) ) known = C ( Z ( p ) ) .However, the Chabauty–Coleman method finds the rational points along with the p -adicpoints { (2 + 3 + 3 + 2 · + 2 · + 3 + 3 + O (3 ) : 2 + 2 · + 3 + 3 + 2 · + O (3 ) : 1) , (1 + 2 · + 2 · + 2 · + 3 + O (3 ) : 2 · + 2 · + O (3 ) : 1) , (1 + 2 · + 2 · + 2 · + 3 + O (3 ) : 2 + 2 · + 2 · + 2 · + O (3 ) : 1) } . One of the p -adic points lies in a Weierstrass disk, and so is -torsion, while the other twopoints do not readily have explanations for being in the Chabauty–Coleman set (in particular,they are not torsion in J ( Z p ) and not recognizably algebraic).5. Comparison
Throughout this section, Q still denotes a point in C ( F p ) and T still denotes a point in J ( Z ( p ) ) Q − b (i.e., we assume Q passes the Mordell–Weil sieve at p , see Section 3.4).To compare the geometric linear Chabauty and Chabauty–Coleman methods, we first re-call some notation in the following commutative diagram, which is the union of the diagrams
2) and (13):(14) Z g − p M Q − b J ( Z p ) Q − b H ( J Z p , Ω J Z p ) ∨ Z g − r ′ p C ( Z p ) Q κ f AJ b log ◦ tr − T vλ Q GLC λ Q CC where we recall • the maps κ and f are defined as in diagram (2); • AJ b is the Abel–Jacobi embedding at b ∈ C ( Z ( p ) ) from C ( Z p ) Q → J ( Z p ) Q − b ; • the map v is given by g − r ′ nonzero Coleman integrals that vanish on M ; • and log : J ( Z p ) ∼ −→ H ( J Z p , Ω J Z p ) ∨ is given by the (normalized) Coleman integral log : x ( ω /p R x ω ) .Now we can give a comparison theorem for the geometric linear Chabauty and Chabauty–Coleman methods. Theorem 5.1.
Let C ( Z p ) GLC and C ( Z p ) CC be the finite subsets of C ( Z p ) defined in Defini-tions 3.5 and 3.15. We have the inclusions C ( Z ( p ) ) ⊆ C ( Z p ) GLC ⊆ C ( Z p ) CC . Furthermore, for any point R ∈ C ( Z p ) CC \ C ( Z p ) GLC , one of the following two conditionshold:(1) the point R fails the Mordell–Weil sieve at p , i.e. the image of R − b in J ( F p ) is notcontained in the image of M in J ( F p ) ;(2) or for T ∈ J ( Z ( p ) ) R − b , the element log( R − b − T ) is not in the Z p -submodule log M of H ( J Z p , Ω J Z p ) ∨ , only in its p -saturation log N .Proof. We may prove this disk-by-disk. Let Q ∈ C ( F p ) fail the Mordell–Weil sieve. Then M Q − b = Z ( λ Q GLC ) = ∅ , and so (1) holds.Otherwise, we can find T ∈ J ( Z ( p ) ) Q − b . Then, by Proposition 3.4, we know Z ( λ Q GLC ) =( M + T ) ∩ AJ b ( C ( Z p ) Q − b ) and by Corollary 3.18 we know AJ b ( Z ( λ Q CC )) = ( N + T ) ∩ AJ b ( C ( Z p ) Q − b ) . So we see that R − b belongs to AJ b ( Z ( λ Q CC )) − κ ( Z ( λ Q GLC )) if and only if log ◦ tr − T ( R − b ) = log( R − b − T ) is in log N \ log M . (As any two choices of T differ byan element of M , this statement is choice-independent.) (cid:3) Remark . In the case of good reduction of the Mordell–Weil group, the obstruction (2)cannot happen, as then by definition M is its own p -saturation. orollary 5.3. If p ∤ | J ( F p ) | , then (2) is equivalent to log( R − b ) not lying in the submodule log M of H ( J Z p , Ω J Z p ) ∨ .Proof. Recall from Remark 3.13 that the isomorphism log : J ( Z p ) → H ( J Z p , Ω J Z p ) ∨ ex-tends to a map log : J ( Z p ) → m − H ( J Z p , Ω J Z p ) ∨ where m = Ann J ( F p ) , by sending x to log( mx ) /m . Under the condition p ∤ | J ( F p ) | , we see that m − is a p -adic unit, so the loga-rithm extends to a map log : J ( Z p ) → H ( J Z p , Ω J Z p ) ∨ . Hence log M is equal to log M , assubmodules of H ( J Z p , Ω J Z p ) ∨ . As log T is an element of log M , we conclude that log( R − b − T ) not lying in log M is equivalent to log( R − b ) not lying in log M . (cid:3) So there are two theoretical obstructions to the Chabauty–Coleman method calculating AJ b ( C ( Z ( p ) )) ∩ M exactly. We give two examples that show these both occur, and wheregeometric linear Chabauty outperforms Chabauty–Coleman. Example 5.4.
Let C/ Z (7) be (the smooth projective model of) the genus 2 curve withMordell–Weil rank 1 (LMFDB label ) given by the equation y = x − x + x. Computing the Chabauty–Coleman set with p = 7 we find C ( Z ) CC = {∞ , (0 : 0 : 1) , (1 : − , (1 : 1 : 1) ,R := ( α : √ , R := ( α : −√ ,R := ( − α + 1 : √ , R := ( − α + 1 : −√ } where α is a root of x − x +1 in Z . Similar to [BBCF + , Example 4.4], J has endomorphismsinduced by ι the hyperelliptic involution and φ : ( x, y, z ) ( z, y, x ) . So J decomposes up toisogeny as a product of elliptic curves J ≃ E × F , where E := C/ h φ i and F := C/ h ι ◦ φ i .We compute that E ( Z ( p ) ) is rank and F ( Z ( p ) ) is rank .The points R i map to -torsion points on F . We show that this forces R i to belongto C ( Z ) CC . Let ν be a basis for the -dimensional space of differential forms (10) whoseColeman integrals vanish on J ( Z ( p ) ) . Then under pullback by E × F → J , we obtaindifferentials ν E and ν F on E and F whose Coleman integrals vanish on E ( Z ( p ) ) and F ( Z ( p ) ) ,respectively. As E ( Z ( p ) ) has rank , the differential ν E is . The Coleman integral of ν F vanishes on the images of the R i because they are torsion. In total, we conclude the Colemanintegral of ν vanishes on the R i , explaining their presence in C ( Z ) CC .The Mordell–Weil group of the Jacobian is isomorphic to Z × Z / Z with the torsionpart generated by (0 : 0 : 1) − ∞ and the free part generated by (1 : − − ∞ . Let P = (1 : − . We claim n ( R i − ∞ ) cannot be in J ( Z ( p ) ) for n ∈ Z > and all i . Assumefor contradiction there is some n . Then n ( R i − ∞ ) is linearly equivalent to m ( P − ∞ ) forsome m > . For any σ an automorphism of L , we have that nR σi − ∞ σ is linearly equivalentto mP σ − m ∞ σ . Subtracting, we have nR σi − nR i is linearly equivalent to . However, usingColeman integration, we can show that /p R R R dx/y = 0 , and /p R R R dx/y = 0 , hence thiscannot happen.Running the geometric linear Chabauty method on this curve, we first compute C ( F ) = { (1 : 0 : 0) , (0 : 0 : 1) , (1 : 1 : 1) , (1 : 6 : 1) , (3 : 4 : 1) , (3 : 3 : 1) , (5 : 4 : 1) , (5 : 3 : 1) } . he order of P − ∞ ∈ J ( F p ) is 12. We construct the subgroup of J ( F p ) S := { n ( P − ∞ ) + m ((0 : 0 : 1) − ∞ ) : n ∈ { , . . . , } , m ∈ { , }} . This allows us to sieve at p by intersecting with the image of C ( F p ) S ′ := { c : c ∈ C ( F p ) and ( c − ∞ ) ∈ S } = { (1 : 0 : 0) , (0 : 0 : 1) , (1 : 1 : 1) , (1 : 6 : 1) } , showing that only the reductions of the Z ( p ) -points modulo p do not fail the Mordell–Weilsieve. The Chabauty–Coleman method finds points in residue disks corresponding to the F p -points { (3 : 4 : 1) , (3 : 3 : 1) , (5 : 4 : 1) , (5 : 3 : 1) } , which are ruled out by this test.The extra points R i found by the Chabauty–Coleman method but not the geometric linearChabauty method are torsion in J ( Z p ) /M but not in J ( Z p ) /J ( Z ( p ) ) . Example 5.5.
Finally, we give an example where M does not have good reduction, and theChabauty–Coleman set contains extra points R such that log( R − b ) is not in log M , onlyin its p -saturation. These extra points pass the Mordell–Weil sieve but are ruled out by thegeometric linear Chabauty method.Let C/ Z (3) be (the smooth projective model of) the genus 2 curve with Mordell–Weil rank1 (LMFDB label ) given by the equation y = x − / x + 19 / x − / x + 10 x − . The Jacobian of C has Mordell–Weil group J ( Z ( p ) ) ≃ Z , generated by d := (1 : − − (1 : 0 : 0) . We compute log d = (3 + 3 + 2 · + 3 + 3 + 3 + O (3 ) , · + 3 + O (3 )) so in particular M does not have good reduction.The Chabauty–Coleman method produces the set C ( Z ) CC = { (1 : 0 : 0) , (1 : − , (1 : 1 : 1) , (3 + 3 + 3 + 2 · + 2 · + O (3 ) : 1 + 3 + 2 · + 3 + 2 · + O (3 ) : 1) , (3 + 3 + 3 + 2 · + 2 · + O (3 ) : − (1 + 3 + 2 · + 3 + 2 · + O (3 )) : 1) } . Then log( R − ∞ ) = (1 + 2 · + O (3 ) , + 3 + 3 + O (3 ))log( R − ∞ ) = (2 + 2 · · + 2 · + O (3 ) , + 3 + 3 + O (3 )) . Hence R and R are not in AJ ∞ ( C ( Z )) ∩ M and the geometric linear Chabauty algorithmsuccessfully rules out these p -adic points from consideration as possible rational points.Hence the geometric method is theoretically strictly better. However, depending on thecurve and the level of precision needed, the geometric linear Chabauty method can be trickyto execute; the best known method for expressing λ Q GLC as polynomials modulo some powerof p uses interpolation, then one has to solve multiple power series in r variables. Sometimesone can use the implicit function theorem for power series [Haz12, Proposition A.4.5] toreduce to fewer variables, but in general this can be an arduous task. Hence in practice, weadvise the following adjustment of the Chabauty–Coleman method. Algorithm 5.6.
1) Calculate S := C ( Z p ) CC using the Chabauty–Coleman method.(2) Let ( D i ) r ′ i =1 ∈ J ( Z ( p ) ) be a set of topological generators of M and ( ω j ) gj =1 a basis of H ( J Z p , Ω J Z p ) .(3) Calculate H the image of M inside J ( F p ) and ( E i ) r ′ i =1 a set of topological generatorsof M .(4) Calculate log E i = (1 /p R E i ω j ) gj =1 ∈ Z gp for i = 1 , . . . , r ′ .(5) For R ∈ S , remove R from S if R − b := R − b mod p is not an element of H .(6) For R ∈ S , let T ∈ J ( Z ( p ) ) R − b . If log( R − b − T ) is not a Z p -linear combination of (log E i ) r ′ i =1 , remove R from S .(7) Return S .From the previous discussion, we have the following theorem. Theorem 5.7.
Algorithm 5.6 computes AJ b ( C ( Z p )) ∩ M .Proof. This is immediate from Theorem 5.1 and Proposition 3.4. (cid:3)
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