Swiss Number Theory Days 2024

• David Loeffler (david.loeffler@unidistance.ch)

Sarah Peluse (Stanford)

Integer distance sets

Abstract: I'll speak about joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance set if the distance between any pair of points in the set is an integer. Our main result is that any integer distance set in the plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size n and a strong upper bound on the size of any integer distance set in [-N,N]^2 with no three points on a line and no four points on a circle.
 

Victor Rotger (Barcelona)

Stark-Heegner points associated to Hilbert modular forms

Abstract: In this talk I will describe ongoing joint work with H. Darmon and A. Lauder, in which we explore a conjectural relationship between the diagonal restriction of p-adic families of Hilbert modular forms over a real quadratic field and logarithms of points on elliptic curves, rational over number fields cut out by Asai Galois representations.

Michael Stoll (Bayreuth)

Formalizing Mordell's Conjecture?

Abstract: I will give a short introduction to interactive theorem proving (using Lean) and will then discuss what would be needed to a) state and b) give a proof of Mordell's Conjecture in Lean, building on its mathematical library Mathlib.
 

Jan Vonk (Leiden)

Points and geodesics on non-split Cartan modular curves

Abstract: The study of CM points on non-split Cartan modular curves has a long history, notably including the work of Heegner on the class number one problem. This talk will explore RM geodesics on non-split Cartan curves, and their p-adic height pairing. I will discuss joint works with Darmon and Balakrishnan-Dogra-Müller-Tuitman, as well as work of Braat and Daas.

Jessica Fintzen (Bonn)

Representations of p-adic groups and Hecke algebras

Abstract: An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will give an overview of what we know about the structure of the Bernstein blocks. In particular, I will discuss a joint project with Adler, Mishra and Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

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