## University of Chicago Number Theory Seminar## Autumn 2017: Tuesday 2:00-3:20pm, Eckhart 308 |

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Date

Speaker

Title

September 26th

**Yihang Zhu**

(*Columbia*)**The Hasse-Weil zeta functions of orthogonal Shimura varieties**

Initiated by Langlands, the problem of comparing the Hasse-Weil zeta functions of Shimura varieties with automorphic L-functions has received continual study. The strategy proposed by Langlands, later made more precise by Kottwitz, is to compare the Grothendieck-Lefschetz trace formula for Shimura varieties with the trace formula for automorphic forms. Recently the program has been extended to some Shimura varieties not treated before. In the particular case of (non-compact) orthogonal Shimura varieties, we discuss the proof of Kottwitz's conjectural comparison, between the intersection cohomology of their minimal compactifications and the stable trace formulas. Key ingredients include point counting on these Shimura varieties, Morel's theorem on intersection cohomology, and explicit computation in representation theory mostly for real Lie groups. ( Hide Abstract)October 3rd

**Hang Xue**

(*Univerity of Arizona*)**Fourier-Jacobi periods on symplectic groups.**

I will explain what the Fourier--Jacobi periods on sympletic groups are and its conjectural relation with certain central L-values. I will also explain the relation with the conjecture of Ichino--Ikeda. As a byproduct, I will correct a minor inaccuracy in the Ichino--Ikeda conjecture. ( Hide Abstract)October 10th

**Ruochuan Liu**

(*(Peking University)***Logarithmic p-adic Riemann Hilbert correspondence and periods on Shimura varieties.**

In the previous work with Xinwen Zhu we construct a p-adic analogue of the classical Riemann-Hilbert correspondence. As a by-product the de Rham periods of a general Shimura variety are obtained. In a recent joint work in progress with Hansheng Diao, Kai-Wen Lan and Xinwen Zhu, we further establish a logarithmic version of the p-adic Riemann-Hilbert correspondence which enables us to compare the de Rham periods and complex periods for a general Shimura variety. ( Hide Abstract)October 17th

**Chen Wan**

(*IAS*)**The local trace formula for the Ginzburg-Rallis model and the generalized Shalika model.**

We will first discuss a local trace formula for the Ginzburg-Rallis model. This trace formula allows us to prove a multiplicity formula for the Ginzburg-Rallis model, which implies that the summation of the multiplicities on every tempered Vogan L-packet is always equal to 1. Then we will talk about an analogy of this trace formula for the generalized Shalika model, which implies that the multiplicity for the generalized Shalika model is a constant on every discrete Vogan L-packet. ( Hide Abstract)October 24th

**François Loeser**

(*Jussieu*)**A non-archimedean Ax-Lindemann theorem**

The Ax-Lindemann theorem is a functional algebraic independence statement, which is a geometric version of the classical Lindemann-Weierstrass theorem. Its generalizations to uniformizing maps of arithmetic varieties played a key role in recent progress on the Andr\'e-Oort conjecture. In this talk I will present a non-archimedean analogue for the uniformization of products of Mumford curves. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization. This is joint work with Antoine Chambert-Loir. ( Hide Abstract)October 31st

**Carl Wang-Erickson**

(*Imperial*)**The Eisenstein ideal at square-free level**

In the previous term of this seminar, Preston Wake discussed our joint work on the congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We answer questions of Mazur, describing the rank and Newton polygon of the associated Hecke algebra in terms of cup products and Massey products in Galois cohomology. In this talk, we discuss continuing joint work toward partial generalizations to square-free level. This involves an adaptation of the work of Calegari and Specter on Galois pseudorepresentations. ( Hide Abstract)November 7th

**Koji Shimizu**

(*Harvard*)**Existence of a compatible system of a local system**

Fontaine and Mazur conjectured that an l-adic Galois representation of Q comes from algebraic geometry if it is unratified almost everywhere and de Rham at l. The conjecture implies that such a representation should be embedded into a compatible system of l’-adic Galois representations with various l’. My talk is about the relative version of the Fontaine-Mazur conjecture replacing Galois representations by etale local systems on an algebraic variety. I will give supporting evidence of the conjecture by discussing existence of a compatible system of a local system. ( Hide Abstract)November 14th

**No seminar**

November 21st

**Jan Vonk**

(McGill) **Singular moduli for real quadratic fields**

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. Hilbert's 12th problem asks for a satisfactory analogue of this theory for arbitrary number fields. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. ( Hide Abstract)November 28th

**George Boxer**

(University of Chicago) **Potential modularity for abelian surfaces**

I will describe some joint work with Frank Calegari, Toby Gee, and Vincent Pilloni where we prove that abelian surfaces (or genus 2 curves) over totally real fields are potentially modular. The main ingredient is a new modularity lifting theorem for certain non-regular 4-dimensional symplectic representations. This is proved by a combination of Calegari-Geraghty's modification of the Taylor-Wiles method and some recent advances in the theory of p-adic modular forms due to Pilloni. This talk will aim to be accessible to non-specialists. ( Hide Abstract)

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This page is maintained by Jack Shotton; it was shamelessly copied from Brandon Levin's page, which in turn was shamelessly copied from Davide Reduzzi page, which was shamelessly copied from Liang Xiao's page, which was shamelessly copied from Kiran Kedlaya's page, which in turn was shamelessly copied from Jason Starr's page, which in turn was shamelessly copied from Ravi Vakil's page, which in turn was shamelessly copied from Pasha Belorousski's page at the University of Michigan. For more sites with a similar pedigree, see Michael Thaddeus's list or Jim Bryan's list.