NTS ABSTRACTSpring2020: Difference between revisions

The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.

The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.

Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.

The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program.

How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.

The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups). In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group. This is joint work with Sug Woo Shin.

By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.

Grothendieck's section conjecture predicts that over arithmetically interesting fields (e.g. number fields or [math]\displaystyle{ p }[/math]-adic fields), rational points on a smooth projective curve X of genus at least 2 can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter.

If [math]\displaystyle{ X }[/math] is a curve over a number field [math]\displaystyle{ K }[/math], then we are motivated to understand the action of the absolute Galois group [math]\displaystyle{ G_K }[/math] on the \'etale fundamental group [math]\displaystyle{ \pi_1(X) }[/math]. When [math]\displaystyle{ X }[/math] is the Fermat curve of degree [math]\displaystyle{ p }[/math] and [math]\displaystyle{ K }[/math] is the cyclotomic field containing a [math]\displaystyle{ p }[/math]th root of unity, Anderson proved theorems about this action on the homology of [math]\displaystyle{ X }[/math], with coefficients mod [math]\displaystyle{ p }[/math]. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of [math]\displaystyle{ G_K }[/math] on the lower central series of [math]\displaystyle{ \pi_1(X) }[/math], with coefficients mod [math]\displaystyle{ p }[/math]. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren.