Related papers: On Mordell-Weil groups of Jacobians over function …

Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the…

{Let $K$ be a number field, and $A_1,A_2$ abelian varieties over $K$. Let $P$ (resp. $Q$) be a non-torsion point in $ A_1(K)$ (resp. $A_2(K)$) such that for almost all places $v$ of $K$, the order of $Q$ mod $v$ divides the order of $P$ mod…

In this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We achieve this by using a generalization of the notion of Prym variety to higher dimensions and a structure theorem for the Mordell-Weil group of…

Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(\tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of…

We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to…

Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$…

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share…

Let $k$ be a number field and $\mathcal{E}$ an elliptic curve defined over the function field $k(T)$ given by an equation of the form $y^2 = a_3x^3 + a_2x^2 + a_1x + a_0$, where $a_i \in k[T]$ and $deg(a_i) \leq 2$. We explore the conic…

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is…

This is mainly a review of my results related to the title. We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically closed…

We construct a Taylor-Wiles system using a family of $p$-adically completed modular Jacobians over suitable Hecke algebras and prove that certain $p$-adically completed Mordell-Weil groups of these Jacobians is free of finite rank over a…

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group…

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a…

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is…

We construct three families of pairs of genus 2 curves over a field K, whose Jacobians are isomorphic as unpolarized abelian varieties. Each family is parameterized by an open subset of the Projective line over K. Our construction is based…

These are notes related to a 12-hour course of lectures given at the Centre de Recerca Mathem\`atica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with…

In this paper the fields of multiply periodic, or Kleinian $\wp$-functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses…

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

Let $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(\tau,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/\tau B(k)$ varies when we take a finite cover…