March 20 (Fri) 2026, 10:30 ~ 11:30 (Asan science building 525)
Title: Markoff and Lagrange spectra for Hecke groups
Speaker: Byungchul Cha (Hongik University)
Abstract:
For an integer $q\ge 3$, \emph{the Hecke group $G_q$ of index $q$} is the subgroup generated by $z\mapsto -1/z$ and $z\mapsto z + 2\cos(\pi/q)$ of the orientation-preserving isometry group of the upper half plane. We study the structure of Markoff and Lagrange spectra of $G_q$. In particular, we characterize the initial discrete parts of these spectra. This mirrors classical results on the Markoff and Lagrange spectra of the real line. This is joint work with Dong Han Kim.
April 10 (Fri) 2026, 11:00 ~ 12:00 (Asan science building 525)
Title: Goldfeld conjecture for genus 2 curves
Speaker: Keunyoung Jeong (Chonnam National University)
Abstract:
Goldfeld's conjecture predicts that the average rank in the family of quadratic twists of an elliptic curve over the rationals is 1/2. Naive analogues of Goldfeld's conjecture have also been studied for elliptic curves over number fields and for genus 2 curves, together with the phenomenon of disparity. However, these analogues have mainly been considered in the setting of hyperelliptic twists. In this talk, we investigate a non-hyperelliptic analogue of Goldfeld's conjecture for a genus 2 curve.
May 29 (Fri) 2026, 10:30 ~ 11:30 (Asan science building 525)
Title: Serre weight conjectures and modularity lifting for GSp4
Speaker: Heejong Lee (KIAS)
Abstract:
Given a Galois representation attached to a regular algebraic cuspidal automorphic representation, the Hodge--Tate weight of the Galois representation is matched with the weight of the automorphic representation. Serre weight conjectures are mod p analogue of such a correspondence, relating ramification at p of a mod p Galois representation and Serre weights of mod p algebraic automorphic forms. In this talk, I will discuss how to understand Serre weight conjectures and modularity lifting as a relationship between representation theory of finite groups of Lie type (e.g. GSp4(Fp)) and the geometry of p-adic local Galois representations. Then I will explain the proof idea in the case of GSp4. This is based on a joint work with Daniel Le and Bao V. Le Hung.