University of Chicago Number Theory Seminar

Fall 2016: Tuesday 1:30-2:50pm, Eckhart 308


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NOTE: the room for this quarter is E-308.

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Click here to see the schedule of previous quarters: Spring 2013 / Fall 2013 / Spring 2014 / Fall 2014 / Winter 2015 / Spring 2015 / Fall 2015 / Winter 2016 / Spring 2016

Fall 2016 Schedule

Date

Speaker

Title

September 27

Sean Howe
(Chicago)

Overconvergent modular forms and the p-adic Jacquet-Langlands correspondence.
We explain an explicit transfer of Hecke eigensystems from the space of overconvergent modular forms to the space of continuous p-adic 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 p-adic automorphic functions should satisfy a local-global compatibility with the local p-adic Jacquet-Langlands 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 Hodge-Tate 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 p-adic period domains.
The rigid generic fiber of a Rapoport-Zink space admits a remarkable etale map to a rigid analytic flag variety; this is the so-called Grothendieck-Messing 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 Rapoport-Zink space associated with the isoclinic p-divisible group over F_pbar of dimension 3 and height 7; here the relevant flag variety is 12-dimensional, and I'll try to convince you that the inadmissible locus is 5-dimensional 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 K-finite 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 functions-sheaves 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 non-Archimedean Gindikin–Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. (Hide Abstract)

October 18

Tasho Kaletha
(Michigan)

Regular supercuspidal representations
Harish-Chandra 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 p-adic 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 p-adic 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 p-adic formal disk arise as the multiplication by p-map 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 p-adic 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 p-adic 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 p-adic 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 p-adic representation of the fundamental group of X which arises from geometry is non-trivial mod p^N. (Hide Abstract)

One may also want to check out:

  • Geometric Langlands Seminar Monday and Thursday 4:30pm;
  • Algebraic Geometry Seminar biweekly on Wednesday 4:30pm-6pm;
  • Northwestern University Number Theory Seminar Monday 4pm;
  • UIC Number Theory Seminar Tuesday 1pm.

    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.