Speaker: 王六权 (武汉大学)
Abstract: Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's problem is to describe all such $A,B$ and $C$ with rational entries for which a specific $r$-fold $q$-hypergeometric series (denoted by $f_{A,B,C}(q)$) involving the parameters $A,B,C$ is a modular form. When the rank $r=2$, Zagier provided eleven sets of examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ is likely to be a modular form. We present a number of Rogers-Ramanujan type identities involving double sums, which give modular form representations for Zagier's rank two examples. Together with several known cases in the literature, we verified all of Zagier's examples. In particular, we give the first $q$-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011. Part of this talk is based on a joint work with Zhineng Cao and Hjalmar Rosengren.
Room: 216, East 32 Building.
Time: 15:30-16:30, Sep 21, 2023
Speaker: 林明辉 (华中师范大学)
Abstract: Around the 1960s, Iwasawa established a regularity in the growth of the Sylow p-subgroup of the class groups of the intermediate subfields of a $\mathbb{Z}_p$-extension of a number field F. Later works of Mazur, Greenberg and Lee showed that the Tate-Shafarevich group of an elliptic curve E also possesses such a regular growth when the elliptic curve E in question has good ordinary reduction at p. When E has good supersingular reduction at p, it was only thanks to the vision of Kobayashi, and a subsequent follow-up work of Iovita and Pollack, that we have an asymptotic formula for the growth in $\mathbb{Z}_p$-extension at which the Mordell-Weil rank of E is bounded. In this talk, we will study the growth of the Tate-Shafarevich groups of an elliptic curve with good supersingular reduction at p over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field under the generalized Heegner condition, where the latter condition forces the Mordell-Weil rank to be unbounded. This is a joint work with Antonio Lei and Katharina Mueller.
Room: 216, East 32 Building.
Time: 10:40-11:40, Oct 13, 2023
Speaker: 汪春晖 （武汉大学）
Abstract: Consider a local field F and a symplectic vector space W over F. Let Mp(W) be the Metaplectic covering group over Sp(W) and note that Weil representations of Mp(W) exist. In this talk, we will review models that allow realization of these representations, and then generalize some results of Pierre Cartier and Jae-Hyun Yang from the real field to the p-adic field.
Room: 216, East 32 Building.
Time: 15:30-16:30, Oct 26, 2023
Speaker: 严盼 （University of Arizona）
Abstract: We construct a family of integrals which represent the product of Rankin-Selberg L-functions of $GL(\ell)\times GL(m)$ and of $GL(\ell)\times GL(n)$ when $m+n<\ell$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. We study basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals and Cogdell--Shahidi--Tsai's theory on partial Bessel functions. This is joint work with Qing Zhang.
Online, Zoom 935 6255 8812.
Time: 14:30-15:30, Nov 2, 2023
Speaker: 许宾
Abstract: In this talk, we will introduce a local descent construction from an irreducible unitary supercuspidal representation GL(2n) of symplectic type, to the pure inner forms of the odd special orthogonal group SO(2n+1). We will give precise formulations of the construction, and talk about its basic properties, as well as some related problems. In particular, we will explain that this construction is capable of recovering the local Vogan L-packet for SO(2n+1) parametrized by simple L-parameters.Room: 216, East 32 Building.
Time: 16:00-17:00, Nov 23, 2023
Speaker: 齐治，浙江大学
Abstract: 首先我会回顾几个经典的Bessel积分公式及其在诸如Waldspurger公式， Beyond Endoscopy，Motohashi公式，Kuznetsov公式等数论问题与公式中的应用，然后我会介绍最近证明的 复数域上的Bessel积分公式以及Gauss数域上的Bruggeman-Motohashi和Kuznetsov-Motohashi公式。
Time and Room: 15:30-16:30 Dec 15, 2023, Room 216, East 32.