KIAS-LFANT Winter School on Number Theory
January 7-10, 2025 KIAS 1503
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Jaesung Kwon (Seoul National University)
Title: Modular curves and modular symbols
Abstract: This lecture provides an introduction to modular curves and modular symbols. We begin by defining modular curves as Riemann surfaces associated with congruence subgroups of $mathrm{SL}_2(mathbb{Z})$. Modular symbols are defined in terms of integrals over geodesics on modular curves. They play an important role in studying modular $L$-values. This talk aims to equip participants with a clear understanding of basic notions.
(Introductory lecture for Jungwon Lee's lecture)
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: Introduction to murmurations of Elliptic curves
Abstract: Elliptic curves play a crucial role in number theory and cryptography. While they are widely studied, many arithmetic properties of elliptic curves remain mysterious, such as the BSD conjecture. In 2022, He, Lee, Oliver, and Pozdnyakov made a surprising discovery regarding the average Frobenius traces of elliptic curves using machine learning algorithms: 'murmurations of elliptic curves’.
In this introductory lecture, I will explain the fundamental concepts of elliptic curves and the BSD conjecture. Following this, I will introduce the phenomenon of 'murmurations of elliptic curves’.
(Introductory lecture for Min Lee's lecture)
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: Spaces as functor of points
Abstract: In this introductory talk, I try to explain the functor of points approach to algebraic geometry. Using this, I will cover some terminologies required to understand the main lecture series.
(Introductory lecture for Jun-Yong Park's lecture)