Date

Speaker

Title

September 27

Sean Howe (Chicago)

Overconvergent modular forms and the padic JacquetLanglands correspondence.
We explain an explicit transfer of
Hecke eigensystems from the space of overconvergent modular forms to the space of continuous
padic automorphic functions on the units of the definite quaternion algebra of invariant p,
giving a partial answer to an old question of Serre. Conjecturally, these padic automorphic
functions should satisfy a localglobal compatibility with the local padic JacquetLanglands
of Knight and Scholze; if this holds, then our construction can be used to obtain new
information about the quaternion algebra representations arising in this correspondence. The
construction proceeds by evaluating overconvergent modular forms at special points in the
infinite level modular curve. To make sense of this evaluation we employ a construction of
overconvergent modular forms using the infinite level modular curve and the HodgeTate period
map. Control over the quaternion algebra representation and the field of coefficients is
obtained from a reciprocity law intertwining the Galois, GL_2, and quaternion algebra
actions. ( Hide Abstract)

October 4

David Hansen (Columbia)

The geometry of padic period
domains.
The rigid generic fiber of a
RapoportZink space admits a remarkable etale map to a
rigid analytic flag variety; this is the socalled
GrothendieckMessing period map. Studying the geometry of
this map and its image  the "admissible locus"  is a
difficult problem, and our knowledge is poor outside of a
few special cases. I'll describe some recent progress on
understanding these maps; in particular, I'll explain some
ideas for getting to grips with the *complement* of the
admissible locus. We'll look in detail at the
RapoportZink space associated with the isoclinic
pdivisible group over F_pbar of dimension 3 and height 7;
here the relevant flag variety is 12dimensional, and I'll
try to convince you that the inadmissible locus is
5dimensional and naturally stratified into six disjoint
strata. This is joint work with Jared Weinstein.
(Hide Abstract)

October 11

Jonathan Wang (Chicago)

An invariant bilinear form on the space of automorphic forms.
Let F be a function field and G a reductive group over F. We define a bilinear form B on the space of Kfinite smooth compactly supported functions on G(A)/G(F). For G = SL(2), the definition of B generalizes to the case where F is a number field (and this is expected to be true for any G). The definition of B relies on the constant term operator and the standard intertwining operator. This form is natural from the viewpoint of the geometric Langlands program via the functionssheaves dictionary. To see this, we show the relation between B and S. Schieder's geometric Bernstein asymptotics, and we highlight the connection between the classical nonArchimedean Gindikinâ€“Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series.
(Hide Abstract)

October 18

Tasho Kaletha (Michigan)

Regular supercuspidal representations
HarishChandra has given a simple and explicit classification of the discrete series representations of real reductive groups. We will describe a very similar classification that holds for a large proportion of the supercuspidal representations of padic reductive groups (which we may call regular). The analogy runs deeper: there is a remarkable parallel between the characters of regular supercuspidal representations and the characters of discrete series representations of real reductive groups. Guided by this parallel we will give an explicit construction of the local Langlands correspondence for regular supercuspidal representations and discuss some of its properties.
(Hide Abstract)

October 25 Special room: Eckhart 312

Joel Specter (Northwestern)

Commuting Endomorphisms of the padic Formal Disk
Any one dimensional formal group law
over Z_p is uniquely determined by the series expansion of
its multiplication by p map. This talk addresses the
converse question: when does an endomorphism f of the
padic formal disk arise as the multiplication by
pmap of a formal group? Lubin, who first studied this
question, observed that if such a formal group were to
exist, then f would commute with an automorphism of
infinite order. He formulated a conjecture under which a
commuting pair of series should arise from a formal
group. Using methods from padic Hodge theory, we prove
the height one case of this conjecture.
(Hide Abstract)

November 1

No seminar


November 8

Claus Sorensen (UCSD)

Deformations and parabolic induction
One of the characteristics of the padic local Langlands correspondence is that it relates deformations of Galois representations to those of mod p representations of GL(2) over Q_p. For a general padic reductive group G the deformations of its mod p representations are not very well understood. Often the universal deformation ring exists as a pseudocompact ring, but it is not known to be Noetherian in general. In this talk we will present some modest steps towards a better understanding of these rings. For instance, that they are insensitive to parabolic induction. In view of the recent classification of Abe, Herzig, Henniart, and Vigneras, this reduces many questions (such as Noetheriannity) to the case of supersingulars. Our main result is an application of Hauseux's computation of Emerton's higher ordinary parts for parabolically induced representations. This is joint work with Julien Hauseux and Tobias Schmidt.
(Hide Abstract)

November 15

Jennifer Park
(Michigan)

A heuristic for boundedness of
elliptic curves
I will discuss a heuristic that predicts that the ranks of all but finitely
many elliptic curves defined over Q are bounded above by 21. This is joint work with Bjorn Poonen,
John Voight, and Melanie Matchett Wood. (Hide
Abstract)

November 22

No seminar


November 29

Daniel Litt
(Columbia)

Arithmetic restrictions on geometric monodromy
Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X.
As a sample application of our techniques, we show that if X is a smooth variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible padic representation of the fundamental group of X which arises from geometry is nontrivial mod p^N.
(Hide Abstract)
