Seminars Sorted by Series

We consider a generalization of Serre's conjecture for irreducible, conjugate self-dual Galois representations rho : G_F --> GL_3(\bar F_p), where F is an imaginary quadratic field in which p splits. We previously gave a conjecture for the possible...

This talk is based on some results for real groups used in Arthur's classification of global packets for classical groups. The setting is twisted endoscopy for a connected reductive algebraic group over the reals. There is a geometric transfer which...

In this talk we consider the problem of giving explicit inequalities which relate heights, discriminants and conductors of a curve defined over a number field. We present such inequalities for some curves, including all curves of genus one or two...

In this talk we will show that many arithmetic sequences have asymmetries in their distribution amongst the progressions mod q. The general phenomenon is that if we fix an integer a having some arithmetic properties (these properties depend on the...

We will discuss the value distribution of the Epstein zeta function E_n(L,cn) for real c and a random lattice L of covolume 1 and large dimension n. Important ingredients in our study will be the distribution of vector lengths in a random lattice as...

We begin with a survey of recent results on the problem of bounding the sup-norm of automorphic forms. If f is a cuspidal automorphic forms on a reductive group G it is classical to study its value distribution and in the particular the maximum of...

A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David...

Modularity lifting theorems were introduced by Taylor and Wiles and formed a key part of the proof of Fermat's Last Theorem. Their method has been generalized successfully by a number of authors but always with the restriction that the Galois...

Let X a curve over F_q and G a semi-simple simply-connected group. The initial observation is that the conjecture of Weil's which says that the volume of the adelic quotient of G with respect to the Tamagawa measure equals 1, is equivalent to the...

A conjecture of Langlands-Rapoport predicts the structure of the mod p points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions. I will...

The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide a fairly wide class...

This talk will be about some new tools based on model theory that could be useful in the study of harmonic analysis on reductive p-adic groups. In 2008, R. Cluckers and F. Loeser defined the class of the so-called "constructible motivic exponential...

I will report some recent progress in the study of local models of Shimura varieties, including the proof of the coherence conjecture of Pappas-Rapoport and the Kottwitz conjecture

I will report on some recent work on multiple zeta values. I will sketch the definition of motivic multiple zeta values, which can be viewed as a prototype of a Galois theory for certain transcendental numbers, and then explain how they were used to...

1. A (very) special case of Deligne-Lusztig theory yields a construction of cuspidal irreducible representations of the finite group $GL_n(\mathbb F_q)$ in the cohomology of an algebraic variety equipped with an action of $GL_n(\mathbb F_q)$. There...

I will describe the appearance of special values of Eisenstein series on E6, E7, and E8 that arose in studying the low energy expansion of the 4-graviton scattering amplitude in string theory (see arxiv:1004.0163 and 1111.2983). Through methods to...

For GL(2) over Q_p, the p-adic Langlands correspondence is available in its full glory, and has had astounding applications to Fontaine-Mazur, for instance. In higher rank, not much is known. Breuil and Schneider put forward a conjecture, which is a...

We'll prove a converse theorem for forms forms on SL_2. While the theorem is easy to prove once it has been formulated, the number-theoretic considerations leading to its formulation nevertheless pose some interesting and apparently unsolved...

Recently Nicolas and Serre have determined the structure of the Hecke algebra acting on modular forms of level 1 modulo 2, and Serre has conjectured the existence of a universal Galois representation over this algebra. I'll explain the proof of this...

In his work of the early 1980s, Shimura observed that expressions of special values of automorphic L-functions in terms of period invariants could be used to obtain relations among the latter. This observation has since been applied in numerous...

The classical Kuga-Satake construction, over the complex numbers, uses Hodge theory to attach to each polarized K3 surface an abelian variety in a natural way. Deligne and Andre extended this to fields of characteristic zero, and their results can...

To any essentially self-dual, regular algebraic (ie cohomological) automorphic representation of GL(n) over a CM field one knows how to associate a compatible system of l-adic representations. These l-adic representations occur (perhaps slightly...

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. Under the generalized Sato-Tate conjecture, this is equal to the distribution of characteristic polynomials of...

Monodromy groups arise naturally in algebraic geometry and in differential equations, and often preserve an integral lattice. It is of interest to know whether the monodromy groups are arithmetic or thin. In this talk we review the Deligne-Mostow...

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for H^n(F, Z/mZ(n-1)) for all n>1...

Abstract: Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in...

Application of Plancherel's theorem to integral kernels approximating compact period functionals yields estimates on (global) automorphic Levi-Sobolev norms of the functionals. The utility of this viewpoint can be illustrated in reconsideration of...

Let $X$ be a locally symmetric variety, $\bar{X}$ its Baily-Borel compactification, $\bar{X}^{rbs}$ its reductive Borel-Serre compactification and $p:\bar{X}^{rbs} \to \bar{X}$ the canonical map. We prove that the derived direct image sheaf $Rp_*...

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on GL(n+1) x GL(n) along the diagonal subgroup GL(n) and express it in terms of the Rankin-Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This...

I will report on the construction of p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing ramification (including deep levels) at p, with good behaviors over the loci where certain...

The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. Its generalization, the relative trace formula, has also been used to prove...

In this talk, I will report on an on-going joint project with David Helm and Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort strata are defined by the vanishing locus of the partial Hasse-invariants; it is an...

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic...

Let x_1, x_2,... be a sequence of n-tuples of roots of unity and suppose X is a subvariety of the algebraic torus. For a prime number p , Tate and Voloch proved that if the p-adic distance between x_k and X tends to 0 then all but finitely many...

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite...

We will discuss some new automorphy lifting theorems for residually reducible Galois representations, and their application to proving new cases of symmetric power functoriality for elliptic modular forms. This is joint work with Laurent Clozel.

I will describe work with Yingkun Li on some arithmetic properties of the Fourier coefficients of harmonic modular forms of weight one. These are Maass forms of weight one whose eigenvalue under the Laplacian is zero and that are allowed to have...

A continuous representation of a profinite group induces a continuous pseudorepresentation, where a pseudorepresentation is the data of the characteristic polynomial coefficients. We discuss the geometry of the resulting map from the moduli formal...

Let \(\chi\) be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of \(L(s,\chi)\) and the distribution of zeros of the Dirichlet \(L\)-function \(L(s,\psi)\), with \(\psi\) belonging to a...

Iwasawa Theory for elliptic curves/modular forms has been traditionally in better shape at ordinary primes than at supersingular ones. After sketching the ordinary theory, we will indicate what makes the supersingular case more complicated, and then...

I will explain how to use Deligne's theory of nearby cycles over general bases to prove Thom-Sebastiani type theorems.

I will begin with a brief introduction to the deformation theory of Galois representations and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss...

Let \(E/F\) be a quadratic extension of \(p\)-adic fields. Let \(V_n\subset V_{n+1}\) be hermitian spaces of dimension \(n\) and \(n+1\) respectively. For \(\sigma\) and \(\pi\) smooth irreducible representations of \(U(V_n)\) and \(U(V_{n+1})\) set...