In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish.
We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.
Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(α_{1}, α_{2}, ... , α_{2g + 1})$, where the $α_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - α_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $Γ(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generator…
We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled using `cluster pictures', similarly to the case of odd residue characteristic.
Let p be an odd prime number and g $\ge$ 2 be an integer. We present an algorithm for computing explicit rational representations of isogenies between Jacobians of hyperelliptic curves of genus g over an extension K of the field of p-adic numbers Qp. It relies on an efficient resolution, with a logarithmic loss of p-adic precision, of a first order system of differential equations.
Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision. Consequently, after having possibly lifted the problem in the $p$-adics, we derive fast algorithms for computing explicitly Cantor's division polynomials of hyperelliptic curves defined over finite fields.
We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.
Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding hyperelliptic curve and $J$ its jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(J)$ for the $Q_{\ell}$-Tate module of $J$ and $e_{\ell}$ for the Riemann form on $V_{\ell}(J)$ attached to the theta divisor. (Here $Q_{\ell}$ is the field of $\ell$-adic numbers.) We write $sp(V_{\ell}(J))$ for the $Q_{\ell}$-Lie algebra of the symplectic group o…
Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.
We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.
This paper explores two techniques on a family of hyperelliptic curves that have been proposed to accelerate computation of scalar multiplication for hyperelliptic curve cryptosystems. In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, the $τ$-adic non-adjacent form ($τ$-NAF). It is shown that the $τ$-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. These properties are not only of intrinsic mathematical interest, but also desirable in some cryptographic applications. On the other hand…
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
Vologodsky's theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry. In contrast with the theory developed by Coleman, it has the advantage of being insensitive to the reduction type at $p$. Building on recent work of Besser and Zerbes, we describe an algorithm for computing Vologodsky integrals on bad reduction hyperelliptic curves. This extends previous joint work with Katz to all meromorphic differential forms. We illustrate our algorithm with numerical examples computed in Sage.
We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.
Lame curves are a particular class of elliptic curves (with a torsion point attached to them) which naturally arise when studying Lame operators with finite monodromy. They can be realized as covers of the projective line unramified outside three points and can be defined over number fields. This paper investigates their p-adic properties. The main ingredient is formal/rigid geometry and in particular the use of p-adic theta functions. As a consequence, we can completely enumerate Lamé curves with bad reduction (by giving, among the others, the p-adic valuation of their j-invariant) and descri…
We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the characteristic polynomial of Frobenius. For fixed p, the asymptotic running time for a curve of genus g over the field of p^n elements is O(g^{4+ε} n^{3+ε}).
Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - λ)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of discriminant $Δ$ and some element $λ\in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(λ))$ but not $(2Δ)$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_{2}(J))$ depending on the maximal power…
We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli …
In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over p-adic fields emerge. We present a novel strategy to solve this kind of equations in a memory efficient way. The main application is an algorithm requiring quasi-cubic time and only quadratic memory in the parameter n, that solves the following problem: for E a hyperelliptic curve of genus g over a finite field of extension degree n and small characteristic, compute its zeta function. This improves substantially upon Kedlaya's resu…
In this work we generalise the main result of arXiv:1812.05651 to the family of hyperelliptic curves with potentially good reduction over a $p$-adic field which have degree $p$ and the largest possible image of inertia under the $\ell$-adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$-extension of it. In this second case, t…
We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the $p$-adic approximation techniques introduced in previous work with the Mordell-Weil sieve
In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve i…
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian,…
We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators…
We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the $p$-adic de Rham period map $j^{dr}_n$ on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $\mathcal{U}$. Several algorithms forming part of the computation of finite level versions $j^{dr}_n$ of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $\mathcal{U}$ in general requires a description in terms of an open covering, and can be regarded as a simple ex…
Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the o…
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…
In this paper we provide an explicit construction of a $distinctive$ multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker function on the double cover of a Kac-Moody group. To do so, we first impose a set of axioms, independent of any group of functional equations, which the aforementioned object should satisfy. As a consequence, we deduce that the coefficients of the $p$-parts of the multiple Dirichlet series satisfy certain recurrence relations. These relations lead to a family of…
Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of $C$, along with the leading coefficient of $f$ and the action of $\mathrm{Gal}(\bar{K}/K)$ on the roots of $f$, completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of $C$. In particular, we give an explicit description of the special fibre in terms of this data.
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their…
Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification…
We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how thi…
We find equations for the higher dimensional analogue of the modular curve X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime 3. We prove the existence of a quasi-quadratic time algorithm for computing a canonical lift in characteristic 3 based on these equations, with a detailed description of our method in genus 1 and 2.
We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito t…
Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of t…
We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.
Let $X$ be an elliptic curve or a ramifying hyperelliptic curve over $\mathbb F_q$. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain $v$-adic convergence results for such exponential and logarithm series, for $v$ a 'finite' prime. As an application, we can show that the $v$-adic Goss $L$-value $L_v(1, Ψ)$ is log-algebraic for suitable characters $Ψ$.
We describe a more efficient algorithm to compute p-adic Coleman integrals on odd degree hyperelliptic curves for large primes p. The improvements come from using fast linear recurrence techniques when reducing differentials in Monsky-Washnitzer cohomology, a technique introduced by Harvey arXiv:math/0610973 when computing zeta functions. The complexity of our algorithm is quasilinear in $\sqrt p$ and is polynomial in the genus and precision. We provide timings comparing our implementation with existing approaches.
Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…
We develop a variant of Coleman and Perrin Riou's methods giving, for a de Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the $p$-adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato's Euler system, this gives $p$-adic $L$-functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at $p$. In the case of dimension $2$, we prove a functional equat…
In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space compl…
We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek's extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.
Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and P…
For a monic polynomial $D(X)$ of even degree, express $\sqrt D$ as a Laurent series in $X^{-1}$; this yields a continued fraction expansion (similar to continued fractions of real numbers): \[\sqrt D=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\ddots}}},\quad a_i\text{ polynomials in }X.\] Such continued fractions were first considered by Abel in 1826, and later by Chebyshev. It turns out they are rarely periodic unless $D$ is defined over a finite field. Around 2001 van der Poorten studied non-periodic continued fractions of $\sqrt D$, with $D$ defined over the rationals, and simultaneously the…
