Number Theory Seminar at Johns Hopkins University

Fall 2017

Except where (frequently) noted to the contrary, the seminar meets from 4:30-5:30 every other Tuesday in Krieger 306.

September 12
Speaker: Michael Lipnowski (IAS)
Title: Growth of the smallest 1-form eigenvalue on hyperbolic manifolds and applications to torsion homology growth
Abstract: Joint work with Mark Stern. We relate small 1-form eigenvalues to relative cycle complexity on hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth in families of congruence, arithmetic hyperbolic 3-manifolds.

October 10
Speaker: Jessica Fintzen (IAS/Michigan)
Title: On the Moy-Prasad filtration and supercuspidal representations
Abstract: Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group.

If time permits, we will present the underlying general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields new descriptions of the Moy-Prasad filtration representations.

SPECIAL TIME AND ROOM: October 26 (Thursday), 3:00 in Maryland 201
Speaker: Lucia Mocz (Princeton)
Title: A New Northcott Property for Faltings Height
Abstract: The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.

SPECIAL TIME AND ROOM: November 2 (Thursday), 3:00 in Maryland 201
Speaker: Eric Katz (Ohio State)
Title: Buium-Manin theory and periods of p-adic Abelian varieties
Abstract: In the 1960's Manin gave a proof of Mordell's conjecture for function fields in characteristic 0 that made use of "Manin maps" which are differential functions on Abelian varieties. These functions vanished on torsion. He defined these functions by differentiating period integrals. Manin maps were introduced by Buium in the p-adic setting using the very different language of differential algebra. In this talk, we discuss a construction of p-adic Manin maps through Coleman integration. This will involve an investigation of the notion of period in a very general context, finding a connection between Buium's notion of p-adic period and the Colmez--Fontaine notion in p-adic Hodge theory. This answers a question of Manin. The ultimate hope of this project is understanding functions that vanish on points of number theoretic interest, putting Manin's techniques on the same footing as the Chabauty--Coleman method.

SPECIAL TIME AND ROOM: November 14, 3:00 in Shaffer 301
Speaker: Xuhua He (Maryland)
Title: Some results on affine Deligne-Lusztig varieties
Abstract: In Linear Algebra 101, we encounter two important features of the group of invertible matrices: Gauss elimination method, or the LU decomposition of almost all matrices, which is an important special case of the Bruhat decomposition; the Jordan normal form, which gives a classification of the conjugacy classes of invertible matrices.

The study of the interaction between the Bruhat decomposition and the conjugation action is an important and very active area. In this talk, we focus on the affine Deligne-Lusztig variety, which describes the interaction between the Bruhat decomposition and the Frobenius-twisted conjugation action of loop groups. The affine Deligne-Lusztig variety was introduced by Rapoport around 20 years ago and it has found many applications in arithmetic geometry and number theory.

In this talk, we will discuss some recent progress on the study of affine Deligne-Lusztig varieties, and some applications to Shimura varieties.

November 28
Speaker: Yiannis Sakellaridis (IAS/Rutgers Newark)
Title: Transfer operators between relative trace formulas in rank one
Abstract: I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms.

More precisely, for a spherical variety X=H\G of rank one, I will prove that there is an explicit "transfer operator" which transforms the orbital integrals of the relative trace formula for X x X/G to the orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped with suitable non-standard test functions. The operator is determined by the L-value associated to the square of the H-period integral, and the proof uses a deep theory of Friedrich Knop on the cotangent bundles of spherical varieties. This is part of an ongoing joint project with Daniel Johnstone and Rahul Krishna, who are proving instances of the fundamental lemma. Globally, this transfer will induce an identity of relative trace formulas and global relative characters, translating to an Ichino-Ikeda type formula that relates the square of the H-period to the said L-value.

This can be viewed as part of the program of relative functoriality, a generalization of the Langlands functoriality conjecture, predicting relations between the automorphic spectra of two spherical varieties when there is a map between their dual groups. The case under consideration here is the simplest non-abelian case of this, when the dual groups are equal and of rank one. If time permits, I will discuss how the transfer operator here and in a few examples of higher rank where it is known is a "deformation" of an abelian transfer operator obtained by replacing the spherical variety by its asymptotic cone (or boundary degeneration).

There will also be following special series of three lectures:
November 16, November 30, and December 7, 2:00 in Krieger 413
Speaker: Michael Rapoport (University of Bonn/University of Maryland)
Title: Reductions of Shimura varieties
Past semesters

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