Chabauty–Coleman experiments for genus 3 hyperelliptic curves
We describe a computation of rational points on genus hyperelliptic curves defined over whose Jacobians have Mordell–Weil rank . Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in .
Let be a non-singular curve over (or more generally, a number field ) of genus . In the case where or , has extra structure given by the fact that if is non-empty, then is rational (if ) or is an elliptic curve (if ). In these cases, computing the set of rational points is either trivial by the Hasse Principle, or highly non-trivial in the case of elliptic curves. In the latter case the rational points form a finitely generated abelian group, and methods specific to this case exist for computing upper bounds on the rank of , and the possibilities for the torsion subgroup of are completely understood by the work of Mazur [Maz77, Theorem 8].
On the other hand, if , then is of general type, and the Mordell conjecture, proved by Faltings in 1983 [Fal83], implies that is finite. Our main motivation is to compute explicitly in this case. We will focus our attention on hyperelliptic curves of genus 3 such that the group of rational points of the Jacobian of has Mordell–Weil rank . This falls into the special case where which was considered by Chabauty in 1941 [Cha41], and techniques developed by Coleman in the 1980s allow us to use -adic integration to bound, and often, in practice, explicitly compute, the set of rational points [Col85b, Col85a].
In addition to these methods, we will also use the algorithm of Balakrishnan, Bradshaw, and Kedlaya [BBK10] and its implementation in Sage [S17] to explicitly compute the relevant Coleman integrals by computing analytic continuation of Frobenius on curves. Nonetheless, we note that the algorithms presented in this article (see Section 3) have not been implemented previously by other authors or carried out on a large collection of curves. (See, however, [BS08] for related work in genus 2.) Our code is available at [BBCF].
We consider the case of genus hyperelliptic curves for two reasons:
When , we can impose the condition that , i.e. , which, by a dimension argument, makes the method more effective. Indeed, in this case, the set is contained in the intersection of the zero sets of the integrals of two linearly independent regular -forms on the base-change of to , where is any odd prime of good reduction.
When , the Jacobian of is a surface, and its geometry and arithmetic is better understood. In particular, methods developed by Cassels and Flynn have been implemented by Stoll in
Magmato make the computations needed much more efficient. More precisely, in this case, one can simplify the algorithm further by working with the quotient of the Jacobian by , which is a quartic surface in , known as the Kummer surface. In order to make the search of rational points more effective, the Chabauty method can also be combined with the Mordell–Weil sieve, which uses information at different primes (see also [BS10]).
We begin with an overview of the Chabauty–Coleman method and explicit Coleman integration in Section 2. In Section 3, we present an algorithm to find a finite set of -adic points containing the rational points of a hyperelliptic curve of genus , which admits an odd model, and whose Jacobian has rank . We fix a prime and work under the assumption that we know a -rational point whose image in the Jacobian has infinite order (here the embedding of into is via the base-point ). Besides -rational points, the output will include all points in which are in the pre-image of the -adic closure of in .
We then proceed to run our code on a list of relevant curves taken from the forthcoming database of genus 3 hyperelliptic curves [BPSS]. Our list consists of 16,977 curves, and we separately do a point search in
Magma to find all -rational points whose -coordinates with respect to a fixed integral affine model have naive height at most (cf. Section 4). Our Chabauty–Coleman computations then show that there are no -rational points of larger height on any of these curves.
In some cases, our algorithm outputs points in . Besides -rational (but non--rational) Weierstrass points, on curves we find that the local point is the localization of a point where is a quadratic field in which the prime splits. In all these cases, we are able to explain why these points appear in the zero locus that we are studying. The following three scenarios occur, and we discuss representative examples of each in Section 4:
It may happen that is a torsion point in the Jacobian (see Example 4.1). In this case, the integral of any -form would vanish between and .
As in Example 4.2, it may happen that some multiple of the image of in the Jacobian actually belongs to : the vanishing here follows by linearity in the endpoints of integration.
The Jacobian may decompose over as a product of an elliptic curve and an abelian surface. Then if the subgroup generated by and the point comes from the elliptic curve, the dimension of the -adic closure of in must be equal to , even if is a point of infinite order (see Example 4.3).
The first author is supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant #550023. The second author is supported by EPSRC and by Balliol College through a Balliol Dervorguilla scholarship. The third author was supported by a Conacyt fellowship. The fourth author is supported by NSF grant DMS-1352598.
This project began at “WIN4: Women in Numbers 4,” and we are grateful to the conference organizers for facilitating this collaboration. We further acknowledge the hospitality and support provided by the Banff International Research Station. We thank the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation for providing computational resources, and we are grateful to Bjorn Poonen, Andrew Sutherland, and Raymond van Bommel for helpful conversations.
2. The Chabauty–Coleman method and Coleman integration
In this section, we review the Chabauty–Coleman method, used to compute rational points in our main algorithm. For further details, see Section 3. We also give a brief overview of explicit Coleman integration on hyperelliptic curves.
2.1. Chabauty–Coleman method
Let be a smooth, projective curve over the rationals of genus at least 2. By the work of Faltings [Fal83], we know to be finite, but Faltings’ proof does not explicitly yield the set . However, before the work of Faltings, Chabauty considered the following set-up. Let be a prime and . Consider the embedding
Then let denote the -adic closure of and define
Chabauty proved the following case of Mordell’s conjecture:
Theorem 2.1 ([Cha41]).
Let be a curve of genus such that the Mordell–Weil rank of the Jacobian of over is less than , and let be a prime. Then is finite.
Chabauty’s result was later re-interpreted and made effective by Coleman, who showed the following:
Theorem 2.2 ([Col85a]).
Let be as above and suppose that is a prime of good reduction for . If , then
To obtain an explicit upper bound on the size of , and hence , Coleman used his theory of -adic integration on curves to construct -adic integrals of 1-forms on that vanish on and restrict them to . Here, we follow the exposition in [Wet97] in defining the Coleman integral.
Let , and be the unique homomorphism such that . Consider the map induced by
Observe that is an isomorphism of vector spaces which is independent of the choice of [Mil86, Proposition 2.2].
Define to be the corresponding differential on . On the Jacobian we have the natural pairing
Note that since is a homomorphism, it vanishes on . Now given we define
hence for a fixed point and we get a function with
We now restrict to the case where and , in which case . The exposition below can be generalized whenever . Let
This is a 2-dimensional -vector space; hence there exist two linearly independent differentials such that
Let be a -rational divisor on of degree , and consider the map such that . Define
Consider the set
While a priori we have defined in terms of , it is actually independent of the choice of , and [Wet97, §1.6].
The above discussion indicates how we would handle the case when our hyperelliptic curve has an even degree model. However, since we restrict our attention to hyperelliptic curves with an odd degree model, we are guaranteed a rational point and we use . Hence, we have two -valued functions on whose common zeros capture the rational points of .
2.2. Computing Coleman integrals
In order to compute , we need a way to evaluate for an arbitrary and arbitrary . Suppose that is a prime of good reduction for and let be the reduction of modulo , i.e. the special fiber of a minimal regular proper model of over . Then there exists a natural reduction map . Define a residue disk to be a fiber of the reduction map. To compute , we now consider two cases: either and lie in the same residue disk, or they do not.
Coleman integral within a residue disk
Let . By a local coordinate for we mean a rational function such that
is a uniformizer at ; and
the reduction of to a rational function for is a uniformizer at .
Hence, a local coordinate at establishes a bijection
where are Laurent series and .
Suppose now that is not identically zero modulo . We will often use the fact that the expansion of in terms of has the form , for some converging on the entire residue disk. Hence, if , we can compute by formally integrating a power series in the local coordinate ([Wet97, Lemma 7.2]). Such definite integrals are referred to as tiny integrals.
A local coordinate at a given point can be found using [Bal15, Algorithms 2-4]. In particular,
If , then and is the unique solution to such that .
If and , then and is the unique solution to such that .
If , one first finds by solving . Then .
In practice, in all three cases one can explicitly compute up to arbitrary -adic and -adic precision by Newton’s method.
Coleman integral between different residue disks
In our intended application of computing rational points, we will fix a basepoint as one endpoint of integration and consider the various Coleman integrals given by varying the other endpoint of integration over all residue disks. This makes it essential that the tiny integrals constructed in the previous section are consistent across the set of residue disks: in other words, we need a notion of analytic continuation between different residue disks.
Coleman solved this problem by using Frobenius to write down a unique “path” between different residue disks and presented a theory of -adic line integration on curves [Col85b] satisfying a number of natural properties, among them linearity in the integrand, additivity in endpoints, change of variables via rigid analytic maps (e.g., Frobenius), and the fundamental theorem of calculus. This was made algorithmic in [BBK10] for hyperelliptic curves by solving a linear system induced by the action of Frobenius on Monsky-Washnitzer cohomology, with an implementation available in Sage.
The upshot is that given two points , one can compute the definite Coleman integral from to as directly via [BBK10], as well as the Coleman integral from to the residue disk of , by further computing a local coordinate at (such that ), which gives:
where now plays the role of the constant of integration between different residue disks.
3. The algorithm
We now specialize to our case of interest, where is a genus hyperelliptic curve given by an odd degree model, i.e.,
where is monic of degree . We will further assume that the Jacobian of has Mordell–Weil rank over . Finally, we will assume that we have computed a point with the property that is of infinite order in . (This last assumption is straightforward to remove.)
Fix an odd prime of good reduction for , denote by the base change of to and let denote a list of known points in . Given this input, the algorithm in this section returns the set of common zeros of and , as defined in Section 2.1, excluding the known rational points .
3.1. Upper Bounds in Residue Disks
Define for . These differentials form a basis for . Let and be -forms in such that and form a basis for and such that and are not identically zero modulo . We may assume that we are in one of the following two situations:
and is a -linear combination of and , or
is a -linear combination of and and is a -linear combination of and .
Let be the local expansion of or in the residue disk of a point . Ultimately we want to compute the zeros of a particular antiderivative lying in up to a desired -adic precision. In certain cases, we will be able to avoid this calculation by instead obtaining an upper bound for the number of zeros of in which we know to be sharp. To do this, we use the theory of Newton polygons for -adic power series (see, e.g., [Kob84, IV.4]).
Given such that , let and define
Let such that . If , then the number of roots of in is less than or equal to .
The following lemmas give a refinement of this result for our particular choice of . We refer to a point of or as a Weierstrass point if it is fixed by the hyperelliptic involution.
Let be the local expansion of or in the residue disk of a point . If is non-Weierstrass, then
Moreover, the minimum of the orders of vanishing of and at is less than or equal to for all non-Weierstrass .
By construction, the differential is a linear combination of two of the differentials , , and is non-trivial modulo . The assumption that is non-Weierstrass implies that is a local coordinate, where is any lift of to characteristic zero. Write , where , , . Then in local coordinates we have
where has no zeros or poles in the residue disk. Since is a polynomial of degree less than or equal to in , the first part of the first claim is proved. Furthermore, the polynomial has a double root modulo at if and only if , , , and ; i.e., if and only if (up to rescaling) and . The last statement also follows, since by construction is a linear combination of and (see the beginning of §3.1). ∎
Let be an odd prime greater than or equal to of good reduction for . Let be a non-Weierstrass point. Then the set
has size less than or equal to .
Let be the local expansion of or in the residue disk of the point . Then . In particular, the minimum of the orders of vanishing of and at is less than or equal to .
We may take , (cf. [Bal15, Algorithm 4]). Then has a zero of order at . ∎
Let be an odd prime greater than or equal to of good reduction for . Then the set
has size less than or equal to . In particular, there are at most two points different from the point at infinity and reducing to it modulo in the above set.
Let be the local expansion of or in the residue disk of a point (with the notation of the algorithm). If is Weierstrass, then
Moreover, the minimum of the orders of vanishing of and at is less than or equal to .
In this case we may take and solve for using . In particular, then (cf. [Bal15, Algorithm 3]). Therefore, has no zero or pole at , has either no zero or pole or a zero of order if . Now consider
where is a unit power series and is non-zero modulo . For any choice of , , it can be verified that by distinguishing between the cases or . If when or and then equals if . However, by construction, and cannot both be of this form. ∎
Let be an odd prime greater than or equal to of good reduction for . Let be a finite Weierstrass point. Then the set
has size less than or equal to .
3.2. Roots of -adic power series
Let be the local expansion of (resp., ) in a residue disk, and let be an antiderivative of whose constant term is either zero or the Coleman integral of (resp., ) between and a -rational point on . To provably determine the roots of lying in a residue disk up to a desired -adic precision, we need to do the following:
make sure that we truncate at a -adic precision that is able to detect all the roots (up to where , see Proposition 3.11);
determine such that to compute a root up to , we only need to consider the power series up to where the coefficient of is in for all if the roots are simple and is suitably normalized (i.e. ).
Write where is an integer greater than or equal to and is a polynomial of degree less than or equal to . Then and have the same number of roots in of -adic valuation greater than or equal to , as can be deduced from the same considerations on the Newton polygon of which imply Lemma 3.1 (for more details, see the proof of [MP12, Lemma 5.1]). We are interested in the zeros of in . Note that
Hence, and as polynomials in have exactly the same zeros (including multiplicities). Furthermore, if a zero of (and ) modulo is simple, then it lifts to a root of in by an inductive application of Hensel’s lemma.
To compute a suitable choice of , we require two more lemmas.
Let for some , and . If has order prime to , then . In particular, this holds if is a prime of non-anomalous reduction for .
Let be the order of the reduction of modulo . Then , the kernel of reduction at , and we have , where . Now can be computed by writing as a power series in where is a local coordinate system for around , formally integrating and evaluating at . ∎
We have , where , for all . Therefore for all , . Furthermore if then .
The first assertion is clear. For the latter, recall that is either or of the form , for some and . The proof is then similar to Lemma 3.9. ∎
By Lemma 3.10, we know that has coefficients in , except possibly when . Let be the minimum of the valuations of the coefficients of . Note that, since has order of vanishing equal to , if , it follows that the valuation of the coefficient of in is precisely . Therefore . Furthermore, for , we have .
Let be an antiderivative of or , let , and let be the minimal valuation of the coefficients of . Fix an integer such that . Let be the smallest integer greater than or equal to with and , and set
Then each simple root of in equals the approximation modulo of a root of . Furthermore, if all such roots are simple, then these are all the roots of in .
It suffices to show that for , . Since , the statement is clear for by Lemma 3.10. Now suppose for some . Hence, where and , and . Then by the definition of , we know that
We now have two cases to consider:
Case 1: Assume that . It follows that which in turn implies that . Then since , we have that
Case 2: Assume that . It follows that . Thus
So if then and hence , contradicting our assumption on . ∎
3.3. Outline of the algorithm
We retain the notation of the beginning of Section 3. The algorithm will always work if and may or may not work if or (see Remark 3.2 and the comments in the main steps of the algorithm below). We now list the input and output of our algorithm followed by its main steps.
: a hyperelliptic curve of genus over given by a model where is monic of degree , such that its Jacobian has rank ;
: an odd prime of good reduction for not dividing the leading coefficient of and ;
: a point in such that has infinite order;
: a list of all known rational points on ;
the -adic precision (by Proposition 3.11, is sufficiently large);
the -adic precision (if by Proposition 3.11, we can set ).
Output: The set defined in (2.1) modulo the action of the hyperelliptic involution. In our code, this set is split into the following:
a list of points of which can be recognized as points in up to the hyperelliptic involution;
a list of points such that , up to the hyperelliptic involution (here, if is not -torsion and is the localization of a point defined over a quadratic extension of then the coordinates in are given as the corresponding minimal polynomials over );
a list of all remaining points (as above, if is the localization of a point defined over a quadratic extension of then the coordinates in are given as the corresponding minimal polynomials over ).
Main steps of the algorithm:
A basis for the annihilator.
For each basis differential (), compute
where is the given -adic precision. Set and
In either case, and are reductions modulo of the pullback of a basis for the annihilator of , where
By Lemma 3.9, if is non-anomalous. If we are guaranteed to be able to carry out all computations in the next steps when .
Ruling out residue disks. Observe that we only need to consider residue disks up to the hyperelliptic involution.
Reduce and modulo . For each , expand and in a local coordinate around , calculate the orders of vanishing of and at , and let denote their minimum. Note that by Lemmas 3.3, 3.5 and 3.7, and hence it suffices to compute and up to to find .
If equals the number of -rational points in reducing to modulo and , then by Lemma 3.1 the set contains all -rational points in the residue disk of . Otherwise, proceed to the next step.
Searching for the remaining disks.
If, for a given point , the number of -rational points in reducing to modulo is strictly smaller that , then we need to compute the set of -rational points reducing to such that for a (any) rational point . For computational convenience we distinguish between two cases:
If there exists reducing to , let be a uniformizer at . Then expand and in and formally integrate to obtain two power series , , which parametrize the integrals of and between and any other point in the residue disk.
If we do not know any -rational point in the residue disk of , then we may assume that and hence write . If , let where is the Hensel lift of to a root of . Otherwise, if is not a Weierstrass point, we take where is any lift to of (the Teichmüller lift of would be a particularly convenient choice for ) and is obtained from using Hensel’s Lemma on . Let and be the integrals between and any other point reducing to in terms of a local parameter at . Then write and
Recall that in (1), we have computed the coefficients of the in and modulo . To provably compute the set of common zeros to a desired precision, we require that one of or have only simple roots, except possibly at (in practice, this has been the case for every curve that we have considered). To check this requirement, we compute their discriminants, which are correct up to the -adic precision of the coefficients.
Upon normalizing so that is not a root of either or , assume without loss of generality that has only simple roots. The -adic precision we should compute to in order to find provably correct approximations of its simple zeros is determined by Proposition 3.11. In practice, we truncate at where , unless the smallest multiple of greater than or equal to satisfies , in which case take . For the -adic precision, the coefficients are computed modulo . Then the simple roots are correct up to where is the minimal valuation of the coefficients of (cf. the discussion preceding Proposition 3.11). To compute the roots we use the function
PARI/GP. Finally, we take the list of roots which lie in and check whether they are also roots of .
If or and the order of vanishing of or modulo is greater than or equal to , then we cannot provably find the zeros of and . Currently the algorithm assumes that to avoid these pitfalls.
Identifying the remaining classes.
Once we have found the common zeros of and , we recover the corresponding -rational points that do not come from points in . We now have the output set that we will now break into sublists.
If we fail to recognize a point as -rational, we can check whether the integral between and of any non-zero differential not in the span of and also vanishes: if this is the case, the point is torsion (cf. [Col85b, Proposition 3.1]) and if we know explicitly (which in general is computable) we can verify whether is -rational or not. Furthermore, by increasing the degree in algdep, we may even try to identify the number field over which the coordinates of are defined
1. This may require high -adic precision; however, it was possible for every curve we considered.
If the integral of the differential is non-zero, and we have not recognized as a -rational point, we can still check whether the point is defined over some number field . For instance could equal a point in plus some torsion element in , with (see Example 4.1).
3.4. Generalizations of the algorithm
What if we do not know such that ?
The hyperelliptic curve we input in the algorithm is assumed to have rank . Calculation of the rank is attempted by
Magma [BCP97] by working out both an upper bound and a lower bound, the former coming from computation of the -Selmer group and the latter from an explicit search for linearly independent points on the Jacobian. The success of the rank computation relies on the two bounds being equal. In particular, if we suppose that we know provably that the rank of the Jacobian is one, we may as well assume that we know a point of infinite order and a divisor on representing it. Then we may proceed as follows. The first task is to write in the form , where is an effective -rational divisor. In order to achieve this, we follow step by step the proof of [Sto14, Corollary 4.14]. That is, we compute the dimension of for (here