PKU/BICMR Number Theory Seminar - 2023

March 15

15:00-16:00

Yupeng Wang

(Chinese Academy of Sciences)

Integral \(p\)-adic non-abelian Hodge theory for small representations

April 5

15:30-16:30

Li Lai

(Tshinghua University)

On the irrationality of certain \(2\)-adic zeta values

Let \( \zeta_2(\cdot) \) be the Kubota-Leopoldt \(2\)-adic zeta function. We prove that, for every nonnegative integer \(s\), there exists an odd integer \(j\) in the interval \( [s+3,3s+5] \) such that \( \zeta_2(j) \) is irrational. In particular, at least one of \( \zeta_2(7),\zeta_2(9),\zeta_2(11),\zeta_2(13)\) is irrational.

Our approach is inspired by the recent work of Sprang. We construct explicit rational functions. The Volkenborn integrals of the (higher order) derivatives of these rational functions produce good linear combinations of \(1\) and \(2\)-adic Hurwitz zeta values. The most difficult step is to prove that certain Volkenborn integrals are nonzero, which is resolved by careful manipulation of the binomial coefficients.

April 12

15:30-16:30

Fan Gao

(Zhejiang University)

Some satisfactory and unsatisfactory aspects of the dual groups for central covers

We consider finite degree central covers of a linear reductive group in the local setting. Using some examples as the highlights, we will explain the dual group of such a central cover, and illustrate how much it captures the representation-theoretic information of the central cover, and also how much it fails for the same purpose. We concentrate on two aspects of a representation: formal degree and wavefront set.

April 26

15:30-16:30

King Fai Lai

同调代数的一个注记

谈一谈关于非交换环的同调代数的几方面。

May 17

15:00-16:00

Juan Esteban Rodríguez Camargo

(MPIM)

Solid locally analytic representations (Joint with Joaquín Rodrigues Jacinto)

In this talk I will introduce different categories of \(p\)-adic representations in the framework of condensed mathematics. We give different geometric interpretations to them, construct explicit adjunctions that serve to compare cohomology theories, and see an application to \(p\)-adic categorical local Langlands for \(\mathrm{GL}_1\).

May 31

15:30-16:30

Koji Shimizu

(YMSC)

Robba site and Robba cohomology

We will discuss a \(p\)-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic \(p\). After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.

June 7

09:00-10:00

Alexander Bett

(Harvard University)

p-adic obstructions and Selmer sections

In 1983, shortly after Faltings' resolution of the Mordell Conjecture, Grothendieck formulated his famous Section Conjecture, positing that the set of rational points on a projective curve Y of genus at least two should be equal to a certain section set defined in terms of the etale fundamental group of Y. To this day, this conjecture remains wide open, with only a small handful of very special examples known. In this talk, I will discuss recent work with Jakob Stix, in which we proved a Mordell-like finiteness theorem for the "Selmer" part of the section set for any smooth projective curve Y of genus at least 2 over the rationals. This generalises the Faltings-Mordell Theorem, and implies strong constraints on the finite descent locus from obstruction theory. The key new idea in our proof is an adaptation of the recent proof of Mordell by Lawrence and Venkatesh to the study of the Selmer section set. Time permitting, I will also briefly describe recent work with Theresa Kumpitsch and Martin Lüdtke in which we compute the Selmer section set in one example using the Chabauty-Kim method.

[Livrestream]

June 21

Rui Chen

(Zhejiang University)

TBA

July 12

Lie Qian

(Stanford University)

TBA