Abstract: I will discuss the problem of computing asymptotics of Betti numbers of congruence subgroups in unitary groups as a function of the level. In degrees below the middle, these dimensions of cohomology grow sub-linearly in the volume of the corresponding locally symmetric space. I will present a strategy to compute the exact growth using automorphic representations and the stable trace formula.

Abstract:We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O'Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the common phenomenon of ``square root cancellation'', while in another case we find more cancellation.

Abstract: By a "modular form" for a reductive group G we mean an automorphic form that has some sort of very nice Fourier expansion. Building upon work of Gan, Gross, Savin, and Wallach, I will explain how there is a notion of modular forms on certain real forms of the exceptional groups. Some examples of these modular forms include the minimal representation on E_{8,4}, singular and distinguished automorphic forms on E_{7,4} and E_{6,4}, and cusp forms on G_2 all of whose Fourier coefficients are integers.

Abstract: Bertolini and Darmon formulated in the mid 1990s conjectures of Birch and Swinnerton-Dyer type for certain Heegner distributions. In this talk, after recalling the statements, I will describe the ingredients in the proof of one of the inequalities predicted by the rank part of the conjectures of Bertolini-Darmon. Based on a joint work with A. Agboola.

Abstract: Szpiro's conjecture relates two important quantities associated to elliptic curves, namely its minimal discriminant and its conductor. Applied to the Frey curve, it implies the abc conjecture of Oesterle and Masser. We give an overview of the inequality proposed by Shinichi Mochizuki and present work done with Taylor Dupuy making explicit the nature and effect of the indeterminacies that play an important role in the approach as well as computations assuming Corollary 3.12 of Mochizuki's IUT III paper.

Abstract: We propose a general framework to inductively prove new results for counting number fields. By using this method, we prove the precise asymptotic distribution of $G$-extensions for a family of Galois groups $G$ that could be constructed via taking towers of extensions. The key ingredient is a uniform estimate on the number of relative extensions with dependency on the base field. This is a joint work with Robert J.Lemke Oliver and Melanie Matchett Wood.

Abstract: In the early 20th century, Hecke studied the diagonal restrictions of Hilbert Eisenstein series, motivated by the principle that the properties of its higher Fourier coefficients can be used to infer information about its constant term. The same principle was later employed by Serre as an approach towards p-adic L-functions of totally real fields, and was recently exploited to efficiently compute them in joint work with Alan Lauder. I will discuss how the same principle, curiously reversed, leads to a full proof of some of the conjectures in the p-adic approach towards singular moduli for real quadratic fields, in the special case of the so-called Dedekind-Rademacher cocycle. This leads to results in RM theory similar to the analytic investigations appearing in the work of Gross-Zagier on differences of singular moduli from CM theory. This is joint work with Henri Darmon and Alice Pozzi.

Abstract: In the representation theory of p-adic groups, second adjointness is an interesting and important adjointness between functors of parabolic induction and parabolic restriction (also known as the Jacquet functor). In the representation theory of real groups, the situation becomes more complicated. I will try to recall the situation in the p-adic case and then describe some parts of the situation in the real case. Based partially on works by Y., Chen-Y., Chen-Gaitsgory-Y., Gaitsgory-Y.

Abstract: Cherednik's theorem states that a Shimura curve attached to a quaternion algebra over a totally real field $F$ which is split at exactly one archimedean place and is ramified at some $p$-adic place admits $p$-adic uniformization by the Drinfeld $p$-adic upper half plane. This theorem is now almost 50 years old but is still not understood, one of the main difficulties being that outside the case $F=\BQ$ the Shimura curve does not represent a moduli problem of abelian varieties. I will show that when the multiplicative group of the quaternion algebra is replaced by the group of unitary similitudes of a binary hermitian space, things become drastically better. This is joint work with S. Kudla and Th. Zink, and improves on P. Scholze (yes, this can be done!).

Abstract: The Breuil-Mezard conjecture predicts the geometry of local deformation rings with p-adic Hodge theory condition in terms of modular representation theory. I will reformulate this conjecture in terms of the Emerton-Gee moduli stack of mod p Galois representations. I will then describe joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra where we prove the conjecture in generic situations for a class of potentially crystalline deformation rings. The key ingredient is the construction of a local model for the singularities of regular weight potentially crystalline deformation rings.

Abstract: Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero.

For example, we show that if $X$ is a normal, connected variety over $\mathbb{C}$, the (typically infinite) set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(\overline{\mathbb{Q}_\ell})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of $\mathbb{Q}_\ell$, then the set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.

Abstract: We will compare two different constructions of p-adic L-functions for modular forms and their relationship to Galois cohomology: one using Kato?s Euler system and the other using Emerton?s p-adically completed cohomology of modular curves. At a more technical level, we will prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato?s Euler system, and the other from the theory of modular symbols and p-adic local Langlands correspondence for GL_2(Qp). We will show that this equality holds even in the cases when the construction of p-adic L-functions is still unknown (i.e. when the modular form is supercuspidal at p).

Abstract: The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing $p$-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the $U_p$ Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension $2$, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions. We will then discuss the ideas of the proof, which goes through the classification of automorphic representations that are principal series at $p$, and a geometric consequence.

Abstract: To any Hodge type Shimura datum, Caraiani and Scholze have attached an adic space called the Hodge-Tate period domain. This period domain admits natural stratifications, analogous to the Ekedahl-Oort and Newton stratifications from the theory of integral models. These are easy to define but difficult to describe. I'll give an explicit description of the Ekedahl-Oort stratification in the first nontrivial case: that of compact Shimura curves.

Abstract: Let $A$ denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. This is joint work with Davesh Maulik and Yunqing Tang.

Abstract: In the mid 90's Colliot-Thélène formulated a conjecture about zero-cycles on smooth projective varieties over p-adic fields. A weaker form of this conjecture was recently established, but the general conjecture is only known for very limited classes of varieties. In this talk I will present some recent joint work with Isabel Leal, where we prove this conjecture for products of elliptic curves, under some assumptions on their reduction type. Our methods often allow us to obtain very sharp results about the structure of the group of zero-cycles on such products and also give us some promising global-to-local information.

Abstract: We'll discuss a work in progress describing properties of p-adic L-functions of a modular form whose Galois representation is residually reducible. As an application, we prove cases of a conjecture of Greenberg about mu-invariants of Selmer groups of elliptic curves, paying particular attention to the case of X_0(11) with p=5. This is joint work with Rob Pollack.

Abstract: In 1966 Artin-Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. I will present a resolution to this conjecture, based on a new connection to Steenrod operations and other ideas originating in algebraic topology.

Abstract: In 2012, Maulik proved a conjecture of Oblomkov-Shende relating: (1) the Hilbert schemes of a plane curve (equivalently, its compactified Jacobian), (2) the HOMFLY polynomials of the links of its singularities. In this talk, we recast his theorem from the viewpoint of representation theory. We state a stronger conjecture relating two representations of a Weyl group W, constructed from: (1) fibers of a parabolic Hitchin map, (2) generalized Bott-Samelson spaces attached to elements in the braid group of W. For each W, we can establish an infinite family of cases of our conjecture. Time permitting, we will explain how the new conjecture bridges: (1) harmonic analysis on nonarchimedean groups, (2) representations of finite groups of Lie type.

The equidistribution conjecture is essentially known only for modular and Shimura curves following Duke's pioneering result in the 80's. I will discuss the relation of this problem to homogeneous dynamics and periodic torus orbits. I will then present two new theorems, for products of modular curves and for Kuga-Sato varieties, establishing partial results for the equidistribution conjecture by combining measure rigidity and a novel method to show that Galois/Torus orbits of special points do not concentrate on proper special subvarieties.

Abstract: Let K be a nonarchimedean local field of characteristic l, F be a finite field of characteristic p, and r be continuous mod p representation of G_K. When l = p, and K = Q_p, the Breuil-Mezard conjecture relates the geometry of the mod p reduction of the universal deformation ring of r to mod p reduction of representations of GL_n(O_K). In this talk, I will formulate an analog of the conjecture when K is local function field, and l different from p, which asks for compatibility between inertia local Langlands correspondence and a certain mod p inertia local Langlands correspondence. I will then sketch a proof using the Taylor--Wiles--Kisin patching method.

Abstract: Under the Langlands correspondence, cuspidal automorphic representations (of GL(n)) should have irreducible Galois representations. This was proven by Ribet for classical modular forms, and has been proven more generally for many classes of regular/cohomological automorphic representations.

In this talk, I will discuss this conjecture for low weight Siegel modular forms, a class of irregular automorphic representations, which are of particular interest due to their conjectural relationship with abelian surfaces.

In this talk we will present a precise formulation of the local and global conjectures for arbitrary connected reductive groups in characteristic zero. It is based on the construction of certain Galois gerbes defined over local and global fields and the study of their cohomology. These cohomological results place the conjectures for classical groups well within reach of the currently available methods.


SPECIAL TIME AND ROOM: April 23 (Thursday), 4:30 in Krieger 205
Speaker: Jim Owings (Maryland)
Title: Two clues are no better than one
Abstract: A bag contains a finite number of integers, no two the same. Players A and B have no knowledge of what numbers are in the bag or how many there are, only that they are all different. Each pulls a number from the bag. Let x be A's number, m be B's number. A must guess whether or not x is bigger than m. To aid in his guess A is allowed to examine t more numbers (the clues) from the bag. For a given t what strategy should A use to maximize his probability of being correct?

We will show that, in a precisely defined sense, 3 clues are better than 1, but that 2 clues are not better than 1. The proof of the latter utilizes Ramsey's Theorem for hypergraphs. We will also discuss certain related results.