Christopher Skinner (Princeton University)
Families of automorphic forms and applications

Families of automorphic forms, particularly p-adic families, have been a key ingredient in efforts to understand the arithmetic significance of L-values. I will discuss some recent results about the existence of such families and their connections to values of L-functions and order of Selmer groups.

Biography

David Helm (University of Texas)
On l-adic families of admissible representations of GL₂(Q_p)

The “mod l” local Langlands correspondence of Marie-France Vigneras establishes a bijection between admissible representations of GL_n(Q_p) and n-dimensional Frobenius-semisimple representations of G_{Qp} over an algebraic closure of F. For n = 2, we compare the deformation theory of a given admissible representation with that of the corresponding galois representation; in almost all cases there is a canonical isomorphism between the deformation rings. The failure of this to occur in all cases is closely connected to the existence of congruences between modular forms; when such congruences exist the deformation theory of the Galois representation is “richer” than that of the corresponding admissible representation.

A result of Matthew Emerton provides a way of remedying this deficiency. His ideas give rise to a notion of "family of admissible representations'' that is weaker than the deformation theoretic notion.

We show that using this weaker notion one can associate a family of admissible representations to a given l-adic family of Galois representations in a way that gives a "local Langlands correspondence in families.''

Ben Howard (Boston College)
Intersection theory on Shimura surfaces

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla-Rapoport-Yang. I will describe results in a higher dimensional setting: on the integral model of a Shimura surface one can consider the intersection of an embedded Shimura curve with a family of codimension two cycles of complex multiplication points. The intersection numbers of these cycles are related to Fourier coefficients of a Hilbert modular form of half-integral weight.

Ben Howard was an undergraduate at the University of Chicago, and went on to complete his Ph.D. at Stanford University under the supervision of Karl Rubin. He is currently an associate professor at Boston College.

Philippe Michel (Ecole Polytechnique Federale de Lausanne)
The subconvexity problem for GL₂

In this talk we will describe the general subconvexity problem for central value of L-functions.
We will also explain the resolution of this problem for GL₁ and GL₂ automorphic L-functions over a general number field, and this uniformly in all parameters (the spectral, level and s-aspects). The main ingredient of the proof are -suitable representations of the central values in terms of automorphic periods which factor over local period integrals of matrix coefficients, -the spectral decomposition of such periods -the spectral gap property for GL₂ matrix coefficients -and the amplification method of Iwaniec If time permits we will also describe an application -explained to us by Andre Reznikov- of the uniformity of our bounds to the study of the restriction of Maass forms of large Laplace eigenvalue along a fixed closed geodesic. This is joint work with Akshay Venkatesh.