Joint IAS/PU Number Theory

Singularities in reductions of Shimura varieties

Thomas Haines
University of Maryland
May 2, 2019

The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a fundamental
question to understand the singularities which arise in the reductions
modulo $p$ of integral models of Shimura varieties. For PEL type
Shimura varieties with parahoric level structure at $p$, this question
has been studied since the 1990's. Due to the recent construction of

Singular moduli for real quadratic fields

Jan Vonk
Oxford University
April 4, 2019

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. 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. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and analytic families of Eisenstein series.

Local aspects of Venkatesh's thesis.

Yiannis Sakellaridis
Rutgers University
March 14, 2019

The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy'' proof of stable functorial transfer from tori to ${\rm SL}(2)$, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.

An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

Emmanuel Lecouturier
Tsinghua University
February 14, 2019

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by
Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined $\mathscr{L}$
invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's
Eisenstein ideal to study in which power of the Eisenstein ideal $\mathscr{L}$ belongs. One corollary of our

Upper bounds for constant slope p-adic families of modular forms

John Bergdall
Bryn Mawr College
January 31, 2019
This talk is concerned with the radius of convergence of p-adic families of modular forms --- q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the nineties predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa--Mazur prediction was false. It has since remained open question how to salvage it.

Slopes in eigenvarieties for definite unitary groups

Lynnelle Ye
Harvard University
December 6, 2018
The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing $p$-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the $U_p$ Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension $2$, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions.

The Lucky Logarithmic Derivative

Will Sawin
Columbia University
November 29, 2018
We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials.

Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

Rong Zhou
Member, School of Mathematics
October 25, 2018
Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group.

Explicit formulae for Stark Units and Hilbert's 12th problem

Samit Dasgupta
Duke University
October 11, 2018
Hilbert's 12th problem is to provide explicit analytic formulae for elements generating the maximal abelian extension of a given number field. In this talk I will describe an approach to Hilbert’s 12th that involves proving exact p-adic formulae for Gross-Stark units. This builds on prior joint work with Kakde and Ventullo in which we proved Gross’s conjectural leading term formula for Deligne-Ribet p-adic L-functions at s=0. This is joint work with Mahesh Kakde.