Abstracts for Conference for Young Number Theorists in Bonn

There have been a number of papers on statistical questions concerning the sign changes of Fourier coefficients of newforms. In one such paper, Linowitz and Thompson gave a conjecture describing when, on average, the first negative sign of the Fourier coefficients of an Eisenstein series newform occurs. In this talk, we correct their conjecture and prove the corrected version.

In this talk, we will study the problem of understanding if the property of admitting an isogeny of fixed degree satisfies the local-global principle. Specifically, we seek to address the following question. Let $A$ be an abelian variety and $K$ be a number field. Assume that, for all but finitely many primes $p$ in $K$, the abelian variety $A$ reduced modulo $p$ is isogenous to an abelian variety via an isogeny of fixed degree $N$. Is it true that $A$ is isogenous to an abelian variety via an isogeny of degree $N$? We will present the known results on the topic and some new results.

Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$.

Arakelov theory is a tool to transfer techniques from function fields to number fields. The Arakelov intersection products are usually difficult to determine or even estimate, and usually people only prove equalities which are rare. Purpose of the talk is to change this.

Video-Recording