KIAS-LFANT Winter School on Number Theory
January 7-10, 2025 KIAS 1503
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Jaesung Kwon (Seoul National University)
Title: TBA
Abstract: TBA
Jungwon Lee (MPIM-Bonn)
Title: Bowen--Series Theory: applications
Abstract: The seminal work of Bowen and Series confirmed the existence of an expanding Markov map which encodes the dynamics of Fuchsian group actions on hyperbolic surfaces and has proved useful in a number of different applications.
We outline the theory and discuss new arithmetic application in the value distribution of general quantum modular forms for Fuchsian groups. Examples include classical modular symbols and central values of additive twists L-functions of Maass forms. This is based on the variants of the Bowen--Series map and spectral analysis of a family of transfer operators (joint with Sandro Bettin and Sary Drappeau).
Min Lee (University of Bristol)
Title: Murmurations
Abstract: A murmuration, originally referring to the wave-like patterns in the movement of flocks of starlings, has taken on a new meaning in analytic number theory: a correlation between the Dirichlet coefficients of a family of L-functions and their root numbers. Murmurations were first observed in elliptic curves by He, Lee, Oliver and Pozdnyakov in 2022, using machine learning algorithms. This phenomenon has since attracted considerable attention from the analytic number theory community. In this three-lecture mini-course, we will explore the murmurations of modular forms.
Youngmin Lee (KIAS)
Title: TBA
Abstract: TBA
Jun-Yong Park (University of Sydney)
Title: Totality of Rational points on Moduli stacks - Counting Families of Varieties
Abstract: The study of fibrations of curves and abelian varieties over a smooth algebraic curve lies at the heart of the classification theory of algebraic surfaces and rational points on varieties. For the case of elliptic curves, it is natural to want to count elliptic curves over global fields such as the field Q of rational numbers or the field Fq(t) of rational functions over the finite field Fq. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}. In these lectures, I will explain the exact counting formula for all elliptic curves over Fq(t) along with an explanation for the geometric origin of lower order main terms, as well as basic context, relevant ideas and methods.
Junyeong Park (Chonnam National University)
Title: TBA
Abstract: TBA