2022 봄학기 정수론 세미나
8월 8일 (월) 11:00 AM(*), 27동 116호(*)
Speaker : Serin Hong (University of Michigan)
Title : Geometry of the $B_{dR}^+$-Grassmannian
Abstract : The classical affine Grassmannian plays a pivotal role in geometric representation theory and the geometric Langlands program. Its p-adic analogue, known as the $B_{dR}^+$-Grassmannian, similarly plays a crucial role in the geometrization of the local Langlands correspondence, and also provides a way to study the cohomology of Shimura varieties. In this talk, we give a brief overview on this object and present a recent progress towards understanding its geometry.
6월 21일 (화) 11:00 AM(*), 27동 116호(*)
Speaker : Gyujin Oh (Columbia University)
Title : Arithmetic local systems over the moduli space of curves
Abstract : It is well-known that the moduli space of genus g curves with n marked points, denoted M_g,n, can be defined over the rational numbers Q. In this talk, we will discuss arithmetic local systems over M_g,n (i.e. representations of the arithmetic etale fundamental group of M_g,n) and their relation to Galois representations. While there are many arithmetic local systems of more obvious origins, there is also an exotic family of local systems that can be constructed using the ideas of physics (conformal field theory).
6월 3일 (금) 4:00 PM, 27동 325호
Speaker : Soogil Seo (Yonsei University)
Title : The Coleman conjecture on circular distributions and the equivariant Tamagawa Number conjecture
Abstract : We introduce the Coleman conjecture on circular distributions made around 1980's and its implications in Euler systems over Q. In our joint work with Burns, Bullach and Daoud, we will provide a proof of the Coleman conjecture and introduce "Coleman's conjecture for higher ranks" with its application to the equivariant Tamagawa Number conjecture for Dirichlet L-functions at s=0 (eTNC (G_m), in short) of Bloch, Kato, Burns and Flach.
Based on the method used in the proof of Coleman's conjecture, we provide a proof of the minus part of eTNC(G_m) for CM extensions of totally real fields unconditionally and imaginary quadratic fields under mu vanishing condition, which accordingly recovers or implies refinements and variants of ‘Stark-type conjectures’ (in particular, conjectures of Rubin-Stark, Brumer, Chinberg, and the lifted root number conjecture of Gruenberg, Rittel and Weiss, among others).
5월 20일 (금) 4:00 PM, 27동 325호
Speaker : Dohoon Choi (Korea University)
Title : Distribution of Hecke eigenvalues of GL_2 automorphic representations
Abstract : Let A be the ring of adeles in a number field F. Let \pi be a automorphic representation of
GL(2,A) such that a local representation of \pi at each infinite plase is a principal series. In this talk,
I will talk about the distribution of Hecke eigenvalues of \pi.
5월 13일 (금) 4:00 PM, 27동 325호
Speaker : DoYong Kwon (Chonnam National University)
Title : A singular function containing all Lagrange numbers less than three
Abstract : Given a real number $\alpha$, the Lagrange number of $\alpha$ is the supremum of all real numbers $L>0$ for which the inequality $|\alpha -p/q|<(L q^2)^{-1}$ holds for infinitely many rational numbers $p/q$. If Lagrange numbers are less than $3$, then they characterize some badly approximable real numbers in the context of Diophantine approximations. Moreover, they can be arranged as a set $\{l_{p/q}: p/q\in \mathbb{Q}\cap [0,1] \}$ using the Farey index. The present talk considers a singular function devised from Sturmian words. After investigating its regularity and singularity, we demonstrate that this function contains all information on Lagrange numbers less than $3$.
5월 6일 (금) 4:00 PM, 27동 325호
Speaker : Keunyoung Jeong (Chonnam National University)
Title : On an upper bound of the average analytic rank of a family of elliptic curves
Abstract : The average of the rank of elliptic curves over rational numbers in a ``natural'' family is expected to be 1/2. For example, Goldfeld conjectured that the average of analytic ranks of the quadratic twist family of an elliptic curve over rational numbers is 1/2. In this talk, we will introduce machinery which gives an upper bound of the average of analytic ranks of a family of elliptic curves. To run the machinery, we need to know the probability that an elliptic curve in the family has good/multiplicative/additive reduction (actually we need something more) and use trace formulas. Using the machinery on the set of all elliptic curves over rationals and the set of elliptic curves with a given torsion subgroup respectively, we can compute an upper bound of the n-th moment of the average. This is the first result on an upper bound of the average of the family of elliptic curves with a fixed torsion group, as far as we know. This is joint work with Peter J. Cho.
4월 15일 (금) 4:00 PM, 27동 325호
Speaker : Seokho Jin (Chung-Ang University)
Title : A note on non-ordinary primes for some genus-zero arithmetic groups
Abstract : Suppose that O_L is the ring of integers of a number field L, and suppose that f(z) is a normalized Hecke eigenform for Γ0(N)+. We say that f is non-ordinary at p if there is a prime ideal p ⊂ OL above p for which a_f(p) ≡ 0 (mod p). In authors’ previous paper it was proved that there are infinitely many Hecke eigenforms for SL2(Z) such that are non-ordinary at any given finite set of primes. In this talk, we extend this result to some genus 0 subgroups of SL2(R), namely, the normalizers Γ0(N)+ of the congruence subgroups Γ0(N). This is joint with Wenjun Ma.
4월 1일 (금) 10:30 AM, Zoom meeting: 910 7203 8314 (passcode: 176444)
Speaker : Victor Wang (Princeton University)
Title : Randomness and structure for sums of cubes
Abstract : I will discuss some perspectives, from my thesis work under Peter Sarnak, on counting integral solutions to x_1^3+...+x_6^3 = 0, and on the related problem of "producing" sums of three cubes on average. Some aspects I may discuss are randomness-structure dichotomies, modularity/automorphy, geometry of low-degree varieties, statistics of L-function families, and/or the delta method.
3월 21일, 22일, 24일 10:00PM, Zoom meeting: 220 305 8101 (passcode: 196884)
Speaker : Preston Wake (Michigan State University)
Title : Three lectures on the Eisenstein ideal
Abstract : In his landmark 1977 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and then Eisenstein series of weight 2 and prime level N. He proved many deep results describing the structure of these congruences, and noted, based on computational evidence, that there is usually only one such cusp form, but sometimes there are several. He asked if there any arithmetic significance to this number of cusp forms. In this series of three lectures, we will address this question and see that this number is significant both algebraically (in terms of Galois cohomology) and analytically (in terms of L-functions). The first lecture will be an introduction to the subject with many examples. The second lecture will focus on the analytic aspects (Eisenstein series and L-functions) and will include generalizations to higher weight. The third lecture will focus on algebraic aspects (Galois representations and Galois cohomology) and will include generalizations to non-prime level. This is joint work with Carl Wang-Erickson.
3월 18일 (금) 4:00 PM, 27동 325호
Speaker : Donghyeok Lim (Ehwa Womans University)
Title : On p-rationality of $\mathbb{Q}(\zeta_{2l+1})^{+}$ for Sophie Germain primes $l$
Abstract : A number field F is called p-rational for an odd prime p, if the Galois group of the maximal pro-p extension of F that is unramified outside p is pro-p free. It was introduced by Abbas Movahhedi to find non-abelian number fields for which the Leopoldt conjecture at p is true. In this talk, we briefly explain the theory of p-rationality. We also show that if l is a Sophie Germain prime such that p is a primitive root modulo l, then Q(\zeta_{2l+1})^{+} is p-rational if p is less than 4l. We also give a heuristic evidence for the recent conjecture of Georges Gras that a number field is p-rational for all but finitely many primes p.