Alternatively have a look at the program.

## What does Spec ${\mathbb Z}$ look like?

Mazur observed that the etale cohomology groups of Spec ${\mathbb Z}$ indicate that it looks like a 3-manifold, and for each prime p the closed subset Spec ${\mathbb F}_p$ of Spec ${\mathbb Z}$ looks like a circle. Deninger suggested that moreover, there should be an action of the reals on Spec ${\mathbb Z}$, with the periodic orbits being precisely those Spec ${\mathbb F}_p$'s, each one becoming an orbit of length $\log p$.

## An application of number theory over function fields to combinatorics

In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set S of positive integers that is both a Sidon set and an asymptotic basis of order 3. This means that the sums of two elements of S are all distinct, while the sums of three elements of S cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.

## The first negative Fourier coefficient of an Eisenstein series newform

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.

## On divisor bounded multiplicative functions in short intervals

In 2016, the celebrated Matomaki-Radziwill theorem showed that there are cancellations for 1-bounded multiplicative functions in almost all short intervals. Our recent work proved that Matomaki-Radziwill theorem can be extended to divisor bounded multiplicative functions.

## A local-global principle for isogenies of abelian surfaces

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.

## Local-global divisibility on algebraic tori

In 2001, Dvornicich and Zannier introduced the so-called Local-Global Divisibility Problem for commutative algebraic groups, as a generalization of a particular case of the Hasse principle for quadratic forms. During the last twenty years, several results have been produced for different algebraic groups. In this talk, after an introduction to the problem, we will present some results in the case of algebraic tori.

## Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions

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 $S$.

## On ordinary isogeny graphs with a level structure

I will briefly introduce the basic concepts of Iwasawa theory of finite connected graphs and then introduce ordinary isogeny graphs with and without level structures. I will then explain how these graphs form a $\mathbb{Z}_p$-tower and what can be said about their structure at finite level. This is joint work with Antonio Lei.

## Estimates in Arakelov geometry

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.

## Square-free integers represented by binary quadratic forms of a fixed discriminant (online broadcast in lecture hall)

In this Work, we prove a result concerning the infinitude of square-free integers represented by a class of polynomials in two variables. More precisely, we prove that infinitely many square-free positive integers are represented by a primitive integral positive-definite binary quadratic form of a given discriminant $D$. We obtain our result by deriving an asymptotic formula for the summatory function associated to it using some known analytic properties of $L$-functions. This is a joint work with Manish Kumar Pandey.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |