We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this elementary combinatorial object encodes the curve's Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants.
Arithmetic inflection formulae for linear series on hyperelliptic curves
Instituto de Matemática, UFF（UFF数学研究所）；Department of Mathematics, University of Chicago（芝加哥大学数学系）；Department of Mathematics, University of Georgia（佐治亚大学数学系）
We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and widely available. Here we address the general case, in which we do not assume the existence of a rational Weierstrass point, using a balanced divisor approach.
A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.
This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\math…
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 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.
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.
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.
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which sp…
We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. 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 $C(\mathbb{Q})$.
We establish a connection between torsion packets on curves of genus $2$ and pairs of elliptic curves realized as double covers of the projective line $\mathbb{P}_{x}^{1}$ that have many common torsion $x$-coordinates. This can be used to show that the set of common torsion $x$-coordinates has size at least $22$ infinitely often and has $34$ elements in some cases. We also explain how we obtained the current record example of a hyperelliptic torsion packet on a genus $2$ curve.
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 fo…
For integers $N\geq 3$ and $g\geq 1$, we study bounds on the cardinality of the set of points of order dividing $N$ lying on a hyperelliptic curve of genus $g$ embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.
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.
In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then determine the sets of rational points by using the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant.
For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.
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,…
Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton d…
We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus.
We give an efficient algorithm to compute equations of twists of hyperelliptic curves of arbitrary genus over any separable field (of characteristic different from 2), and we explicitly describe some interesting examples.
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.
Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $\mathbb{Q}$-rational points on the Atkin-Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.
Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture.
