Date

Speaker

Title

Sept 29

ChengChiang Tsai (MIT)

Computing padic orbital integrals
We sketch an inductive algorithm to compute padic orbital integral of a general orbit on nice test functions. Our emphasis will be on geometric structures, that is, varieties over the residue field on which we have to count rational points. We also give some formal results such as uniform bounds for orbital integrals and a local constancy result. If time permits, we can formulate an endoscopic conjecture regarding BorelMoore homology of the above varieties.
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Oct 6

Ana Caraiani (Princeton)

On vanishing of torsion in the cohomology of Shimura varieties
I will discuss joint work in progress with Peter Scholze showing that torsion in the cohomology of certain compact unitary Shimura varieties occurs in the middle degree, under a genericity assumption on the corresponding Galois representation.
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Oct 13

Christian Johansson (IAS)

Overconvergent modular forms using perfectoid modular curves
We give a new definition of overconvergent modular forms roughly analogous to the complex analytic definition of modular forms, and discuss an application to the overconvergent EichlerShimura isomorphism of AndreattaIovitaStevens. Joint with Przemyslaw Chojecki and David Hansen.
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Oct 20

Jessica Fintzen (Harvard)

Stable vectors in the MoyPrasad filtration
Reeder and Yu gave recently a new construction of certain supercuspidal representations of padic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first MoyPrasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes.
Some of the above work forms part of a joint research project with Beth Romano.
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Oct 27

Erick Knight (Harvard)

A padic JacquetLanglands correspondence
I will construct a padic JacquetLanglands correspondence, which is a correspondence between Banach space representations of GL_2(Q_p) and Banach space representations of the unit group of the quaternion algebra D over Q_p. The correspondence satisfies localglobal compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Langlands correspondence, in the sense that the representations of the unit group of D can often be shown to have the expected locally algebraic vectors.
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Nov 3

Samit Dasgupta (UC Santa Cruz)

On the higher rank GrossStark conjecture
In 1980, Gross stated a conjecture relating the leading term of the padic Lfunction of
a ray class character of a totally real field at s=0 to a padic regulator of punits in the field cut out by the character. In previous joint work with Darmon and Pollack, we proved this conjecture in the rank one case under certain assumptions; these assumptions were later removed by Ventullo. In this talk, we describe work in progress with Ventullo and Kakde on the higher rank case. In particular, we present a proof in the rank two setting under a certain assumption. As a corollary of our result, we obtain an unconditional proof of the conjecture when the ground field is real quadratic.
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Nov 10

Florian Herzig (UToronto)

On de Rham lifts of local Galois representations
It is an open problem to show that a given ndimensional mod p local Galois representation \rho has a (regular) de Rham lift. We discuss several results concerning the existence of de Rham lifts of \rho of prescribed weights and types, assuming that \rho admits a nice lift to start with (for example, a FontaineLaffaille lift). Our arguments combine local and global methods. This is joint work with T. Gee, T. Liu, and D. Savitt.
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Nov 17

Akshay Venkatesh
(Stanford)

Height of modular forms
I will define the notion of "height" of a (cohomological) automorphic form, for any group over a number field, and give some example of where this notion comes up naturally. I will then speculate (rather wildly) that it should be tightly related to the height of the associated motive (as recently defined by Kato). I have only a tiny bit of evidence for this speculation but if true it is rather surprising, because the two notions of height are quite different.
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Nov 24

No Seminar

No Seminar
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Dec 1

Bjorn Poonen
(MIT)

Most odd degree hyperelliptic curves have only one rational point
We prove that the probability that a curve of the form y^2 = f(x) over Q
with deg f = 2g+1 has no rational point other than the point at infinity
tends to 1 as g tends to infinity. This is joint work with Michael Stoll.
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