**FGC-HRI-IPM** Joint Number Theory Seminar

**When:**Wednesdays

**Where:**Zoom

**Meeting ID:**85613860958

**Seminar mailing list:**google group

## Talks

** Speaker: ** Soumya Sankar, Utrecht University

** Title:** *Counting points on stacks and elliptic curves with a rational N-isogeny*

** Abstract:** The classical problem of counting elliptic curves with a rational N-isogeny can be phrased in terms
of counting rational points on certain moduli stacks of elliptic curves. Counting points on stacks, while posing various challenges,
has also opened up several new avenues of exploration in the last few years. In this talk, I will give an introduction to modular
curves from the stacky perspective, discuss some notions of height on them, and use some of these notions to answer the counting
question mentioned above. Time permitting, I will talk about this in the context of the stacky version
of the Batyrev-Manin conjecture. The talk assumes no prior knowledge of stacks and is based on joint work with Brandon Boggess.

**Date:** November 22, 2023

** Speaker: ** Farzad Aryan, IPM

** Title:** *Cancellations in Character Sums and the Vinogradov Conjecture*

** Abstract:** In this talk, we will explore conjectures related to cancellations in character sums.
Additionally, we will examine the potential impact on the distribution of zeros of Dirichlet L-functions if these conjectures prove to be false.

**Date:** November 8, 2023

** Speaker: ** Berkay Kebeci, Koc University

** Title:** *Mixed Tate Motives and Aomoto Polylogarithms*

** Abstract:** Grothendieck proposed the category of motives as a Tannakian category, offering a
universal framework for Weil cohomology theories. In this talk, we will consider motives in the sense
of Nori. One expects the Hopf algebra R of mixed Tate motives to be isomorphic to the bi-algebra A
of Aomoto polylogarithms. Our aim is to reconstruct A using Nori motives. This allows us to write a morphism from A to R.

**Date:** Octopber 25, 2023

** Speaker: ** Türkü Özlüm Çelik, Koç University

** Title:** * Algebraic Curves from Polygons*

** Abstract:**We study constructing an algebraic curve from a Riemann surface given via a
translation surface, which is a collection of finitely many polygons in the plane with sides
identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm
for approximating the Jacobian variety of a translation surface whose polygon can be decomposed
into squares. We first implement the algorithm in the case of L-shaped polygons where the algebraic
curve is already known. The algorithm is also implemented in any genus for specific examples
of Jenkins-Strebel representatives, a dense family of translation surfaces that, until now,
lived on the analytic side of the transcendental divide between Riemann surfaces
and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.

**Date:** Octopber 11, 2023