Related papers: Counting points on hyperelliptic curves with expli…

To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups.…

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…

By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted…

We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability…

Let $k$ be a number field. We refine a construction of Mestre--Shioda to construct (infinite) families of hyperelliptic curves $X/{k}$ having a record number of rational points and record Mordell--Weil rank relative to the genus of $g$ of…

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the…

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-$5$ curves that are…

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

A connected Kuga-Sato variety $\mathbf{W}^r$ parameterizes tuples of $r$ points on elliptic curves (with level structure). A special point of $\mathbf{W}^r$ is a tuple of torsion points on a CM elliptic curve. A sequence of special points…

We present an efficient deterministic algorithm which outputs exact expressions in terms of $n$ for the number of monic degree $n$ irreducible polynomials over $\mathbb{F}_{q}$ of characteristic $p$ for which the first $l < p$ coefficients…

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the…

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental…

In this paper we describe a generalisation and adaptation of Kedlaya's algorithm for computing the zeta function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm…

Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a…

We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of…

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

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…

A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of…

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic…