Date

Speaker

Title

March 29

Henri Darmon (McGill)

Generalised Kato classes and arithmetic applications
I will discuss some of the arithmetic applications of the eponymous cohomology classes of the title, notably, their relations with the socalled ``StarkHeegner points" that are conjecturally defined over ring class fields of real quadratic fields.
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April 5

Matthew Emerton (UChicago)

Moduli stacks of local padic Galois representations
I will describe work in progress with Toby Gee, in which we construct moduli stacks parameterizing padic representations of the absolute Galois groups of padic local fields.
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April 14 (Thur) Note special day!

Toby Gee (Imperial)

Potentially automorphic components of local deformation rings
We show that there is a wellbehaved notion of a "potentially automorphic component" of a potentially semi stable universal deformation ring of an ndimensional mod p representation of the absolute Galois group of a finite extension of Q_p, whenever n is odd (or n=2) and p>2(n+1). This is joint work with Frank Calegari and Matthew Emerton.
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April 19

Kartik Prasanna (UMichigan)

Hodge classes on products of quaternionic Shimura varieties
I will discuss the relation between Langlands functoriality and the theory of algebraic cycles in one of the simplest instances of functoriality, namely the JacquetLanglands correspondence for Hilbert modular forms. In this case, functoriality gives rise to a family of Tate classes on products of quaternionic Shimura varieties. The Tate conjecture predicts that these classes come from an algebraic cycle, which in turn should give rise to a Hodge class that is compatible with the Tate classes. While we cannot yet prove the Tate conjecture in this context, I will outline an unconditional proof of the existence of such a Hodge class. This is joint work (in progress) with A. Ichino.
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April 26

Preston Wake (UCLA)

Level structures beyond the Drinfeld case
Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional pdivisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case:
i) full level structures on the group \mu_p x \mu_p;
ii) Gamma_1(p^r)type level structures on an arbitrary pdivisible group (joint work with R. Kottwitz).
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May 3

Jared Weinstein (BU)

The absolute Galois group of Q_p as a geometric fundamental group
We construct an object defined over an algebraically closed field, whose fundamental group equals the absolute Galois group of Q_p. Formally, this object is a quotient of a perfectoid space, and is closely related to the "fundamental curve of padic Hodge theory" of Fargues and Fontaine. Time permitting, we will explain the connect between this object and the moduli spaces of RapoportZink, which are linked to local Langlands.
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May 10

Hansheng Diao (Princeton)

Log adic spaces and overconvergent modular forms
The main objects of the talk are adic spaces with logarithmic structures. We study their Kummer etale and Kummer proetale topologies. In particular, we show that log adic spaces are locally "perfectoid". As an application, we establish an oveconvergent EichlerShimura morphism connecting modular symbols and overconvergent modular forms.
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May 17

Ellen Eischen
(Oregon)

padic families of Eisenstein series and applications
I will discuss a construction of a padic family of Eisenstein series. I will also describe how it feeds into a program to construct padic Lfunctions associated to automorphic forms (in particular, on unitary groups). The latter part is joint with Michael Harris, JianShu Li, and Chris Skinner.
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May 24

Gil Moss
(Oklahoma St.)

A local converse theorem and the local Langlands correspondence in families
In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for GL(n) of a padic field should interpolate in \elladic families, where \ell is a prime different from p. Recently, Helm showed that the conjecture follows from the existence of an appropriate map from the integral Bernstein center to a Galois deformation ring. Such a map would connect moduli spaces of representations on either side of the correspondence. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image.
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May 31

Brandon Levin
(UChicago)

Weight elimination in Serretype conjectures
I will discuss recent results towards the weight part of Serre's conjecture for GL_n as formulated by Herzig. The conjecture predicts the set of weights where an odd ndimensional mod p Galois representation should appear in cohomology (modular weights) in terms of the restriction to decomposition group at p. We show that the set of modular weights is always contained in the predicted set in generic situations. This is joint work with Daniel Le and Bao V. Le Hung.
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