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Tuesday,
September 22
(MIT) |
Yiannis Sakellaridis (University of Toronto)
Title : A "relative" Langlands program and periods of automorphic forms
Abstract: TBA
&
Matthew Emerton (Northwestern University)
Title: p-adically completed cohomology and the p-adic Langlands program
Abstract: Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic global Galois representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e. that come from geometry, are expected to correspond to classical automorphic eigenforms.)
In certain contexts (namely, when one has Shimura varieties at hand), one can begin with a geometric definition of automorphic forms, and generalize it to obtain a geometric definition of p-adic automorphic forms. However, in the non-Shimura variety context, such an approach is not available. Furthermore, this approach is somewhat remote from the representation-theoretic point of view on automorphic forms, which plays such an important role in the classical Langlands program.
In this talk I will explain a different,, and very general, approach to the problem of p-adic interpolation, via the theory of P-adically completed cohomology. This approach has close ties to the p-adic and mod p representation theory of p-adic groups, and to non-commutative Iwasawa theory.
After introducing the basic objects (namely, the p-adically completed cohomology spaces attached to a given reductive group), I will explain several key conjectures that we expect to hold, including the conjectural relationship to Galois deformation spaces. Although these conjectures seem out of reach at present in general, some progress has been made towards them in particular cases. I will describe some of this progress, and along the way will introduce some of the tools that we have developed for studying p-adically completed cohomology, the most important of these being the Poincare duality spectral sequence.
This is joint work with Frank Calegari. |
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Tuesday,
October 20
(BC) |
Ze'ev Rudnick (Tel-Aviv University and IAS)
Title: Statistics of the zeros of zeta functions over a function field
Abstract: Much of the Diophantine information concerning curves over a finite field is encoded in the zeta function of the curve. We investigate the statistics of such zeta functions as the curve varies in a moduli space of hyperelliptic curves over a finite field. When the genus g is fixed and the finite field grows, Katz and Sarnak showed that the statistics are those of random unitary symplectic 2g by 2g matrices. I wil discuss what is known for the opposite limit, when the finite field is fixed and the genus grows.
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Haruzo Hida (UCLA)
Title: Characterization of abelian components of the "big" Hecke algebra
Abstract: In the first 20 minutes, we try to explain why such a characterization is important. Then, we state a conjectural example of such characterization, and at the end, we explain a (partial) proof of the conjecture. |
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Tuesday,
November 17
(MIT) |
Akshay Venkatesh (Stanford University)
Title: Torsion in the homology of arithmetic groups
Abstract: TBA
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Ken Ono (University of Wisconsin)
Title: p-adic coupling of harmonic Maass forms
Abstract: Recently there have been many applications of the theory of harmonic Maass forms. Such forms have two components, a holomorphic part and a non-holomorphic part. One of the central basic problems about these functions is that of "directly" computing one part from the other. It has been generally quite difficult to determine how to link such functions together to obtain a Maass form. In this lecture I will present a p-adic solution in the case of integer weight forms.
This is joint work with Pavel Guerzhoy and Zachary Kent. |
