Reduction and isogenies of elliptic curves

Let $R$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $k$ of characteristic $p>0$. Let $E/K$ be an elliptic curve with a $K$-rational isogeny of prime degree $\ell$. In this article, we study the possible Kodaira types of reduction that $E/K$ can have. We also prove some related results for elliptic curves over $\mathbb{Q}$.Comment: Preprint. Submitted for publicatio

Universal quadratic forms and Dedekind zeta functions

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption that the codifferent of $K$ is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields.Comment: 12 pages. Preprin

An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields

Let $L$ be a number field and let $\ell$ be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the $\ell$-power torsion is both a pro-$\ell$ extension of $L(\mu_\ell)$ and unramified away from $\ell$ are quite rare. In this paper, we formulate an analogue of the Rasmussen--Tamagawa conjecture for non-isotrivial abelian varieties defined over function fields. We provide a proof of our analogue in the case of elliptic curves. In higher dimensions, when the base field is a subfield of the complex numbers, we show that our conjecture is a consequence of the uniform geometric torsion conjecture. Finally, using a theorem of Bakker and Tsimerman we also prove our conjecture unconditionally for abelian varieties with real multiplication

Universal quadratic forms and Dedekind zeta functions

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption that the codifferent of $K$ is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields