Arbeitsgruppe Algebra und Zahlentheorie, MI Köln

Seminare und Vorträge im WS 2025/2026

am Montag, 27. Oktober :

Oberseminar Zahlentheorie

Debmalya Basak
Title: Distributions Associated to Small Quadratic Non-Residues and their Partitions
Abstract: Assuming the Generalized Riemann Hypothesis, it is known that the least quadratic nonresidue modulo a prime \(p\) is less than or equal to \( (\mathrm{log} \, p)^2\). In this talk, we will discuss unconditional results on the distribution of quadratic nonresidues and their associated partitions in significantly smaller intervals of length \( (\mathrm{log} \, p)^A \) with 0 < A < 2. Along the way, we shall introduce an analogue of the Dedekind Eta function for Hecke Groups \(H(\sqrt{D})\), leading to some interesting conjectures. This is based on joint projects with Bruce Berndt, Dorian Goldfeld, Kunjakanan Nath, Nicolas Robles and Alexandru Zaharescu.

14:00

Seminarraum 3 des Mathematischen Instituts

am Montag, 17. November :

Oberseminar Zahlentheorie

Ozlem Imamoglu
Title: A Lyapunov exponent attached to modular functions
Abstract: Recently, in joint work with Paloma Bengoechea and Sebastian Herrero we defined a Lyapunov exponent attached to modular functions and proved its properties. Our work was motivated by the work of Spalding and Veselov as well as conjectures of Kaneko on the values of modular functions at real quadratic irrationalities. In this talk I will first explain the results of Spalding and Veselov and conjectures of Kaneko. I will then talk about the new results.

14:00

Seminarraum 3 des Mathematischen Instituts

am Montag, 01. Dezember :

Oberseminar Zahlentheorie

Likun Xie
Title: On a Conjecture of Erdős over Function Fields
Abstract: I will discuss a function-field analogue of a classical conjecture of Erdős. Given a squarefree polynomial f over F_q of degree n, one can ask whether every residue class modulo f can be written as a product of two irreducible polynomials of degree at most n. I will explain how Katz’s equidistribution framework leads to an affirmative answer when q is large in terms of n. Sawin previously proved this result with stronger square-root cancellation using a higher-dimensional geometric method. I will instead present a one-dimensional approach that produces a natural q^{-1/2} saving and is effective in the large-q regime.

14:00

Seminarraum 3 des Mathematischen Instituts