Moduli spaces of shtukas over function fields
Jared Weinstein
Boston University
February 13, 2020
Joint IAS/PU Number Theory
Eisenstein series and the cubic moment for PGL(2)
Paul Nelson
ETH Zürich
January 30, 2020
Canonical integral models of Shimura varieties
George Pappas
Member, School of Mathematics
November 21, 2019
We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.
A slice or two of a diagonal cubic: arithmetic stratification via the circle method
Trevor Wooley
Purdue University
October 24, 2019
Extremal cases of Rapoport-Zink spaces
Michael Rapoport
Universität Bonn, University of Maryland
October 10, 2019
This talk is about qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.
A stacky approach to crystalline (and prismatic) cohomology.
Vladimir Drinfeld
The University of Chicago; Visiting Professor, School of Mathematics
October 3, 2019
The stacky approach was originated by Bhatt and Lurie. (But the possible mistakes in my talk are mine.)
Let X be a scheme over F_p. Many years ago Grothendieck and Berthelot defined the notion of crystal on X; moreover, they defined the notion of crystalline cohomology of a crystal.
Golden Gates in PU(n) and the Density Hypothesis.
Shai Evra
Member, School of Mathematics
September 26, 2019
In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS to higher rank compact Lie groups.
Taking the Hecke algebra to its limits
Raphael Steiner
Member, School of Mathematics
September 12, 2019
We parametrise elements in the full Hecke algebra in a way such that the parametrisation represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.
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
How does the rank of an elliptic curve grow in towers of number fields?
Florian Sprung
Arizona State University
April 18, 2019
On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates $(x,y)$ in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field.