Related papers: The arithmetic of Prym varieties in genus 3
We compute the rational cohomology of the moduli space of non-singular complex projective curves of genus 3 with two marked points.
In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in \cite{fer}, we specialize it to double structures on…
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm…
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…
A non-square-tiled Veech surface has finitely many periodic points, i.e., points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic…
We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general \'etale double covers of curves of genus g>=2. We then specialize to curves on Nikulin surfaces and…
We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank…
We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic…
We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
The purpose of this paper is to list the refined Humbert invariants for a given automorphism group of a curve $C/K$ of genus 2 over an algebraically closed field $K$ with characteristic $0$. This invariant is an algebraic generalization of…
Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the…
We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and…
Let $C_k$ be a smooth projective curve over a global field $k$, which is neither rational nor elliptic. Harris-Silverman, when $p=0$, and Schweizer, when $p>0$ together with an extra condition on the Jacobian variety…
We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational…
We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the…
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by P\'epin in a series of articles on unsolvable quartic diophantine equations, and for constructing…
We prove the generic injectivity of the Prym map, sending a double covering of an elliptic curve ramified at r>4 points to its polarized Prym variety. For r=6 the map is birational.
Let E be the supersingular elliptic curve defined over k, the algebraic closure of the finite field with two elements, which is unique up to k-isomorphism. Denote by 0 its identity element and let C be the quotient of E-{0} under the action…
In this paper, we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p. Our techniques involve related computations in the case of three branch…