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Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $nrightarrow infty$. This method has its origins in work of A.Cuoco, who studied $mathbb{Z}_p^2$-extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.
We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example,
Let $X$ be a curve of genus $ggeq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Neron-Tate height of generators of its Mordell-Weil group. The bounds are implie
In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.