Date

Speaker

Title

January 6

Chao Li (Harvard)

Level raising mod 2 and arbitrary 2Selmer ranks
We prove a level raising mod p=2 theorem for elliptic curves over Q, generalizing theorems of Ribet and DiamondTaylor. As an application, we show that the 2Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the ppart of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd p. This is joint work with Bao V. Le Hung.
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January 13

David Zywina (Cornell)

Elliptic surfaces and the Inverse Galois Problem
By studying the Galois action on etale cohomology groups arising from
elliptic surfaces, we will prove several new cases of the Inverse
Galois Problem. In particular, we will explain why each of the simple
groups PSp_4(F_p) occur as the Galois group of some Galois extension
of Q. The key ingredients will be a big monodromy result along with
some known cases of the Birch and SwinnertonDyer conjecture.
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January 20

Takashi Suzuki (UChicago)

Grothendieck's pairing on Neron component group
The component group of the special fiber of the Neron model of an
abelian variety is called the Neron component group. In SGA7,
Grothendieck constructed a canonical pairing between the Neron
component groups of an abelian variety and its dual over a local field
with perfect residue field. He conjectured that it is perfect. In this
talk, we prove his conjecture. A key tool is the category of fields
viewed as a Grothendieck site. The cohomology of an abelian variety
can be regarded as a sheaf on this new site. This allows us to treat
Neron models completely functorial in the derived category of sheaves.
From the known case of semistable abelian varieties, we deduce the
perfectness in full generality. Time permitting, we also discuss a
global function field version of this duality, generalizing the
CasselsTate pairing to the case that the base field is perfect.
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January 27

Matthew Emerton (UChicago)

Moduli spaces of local Galois representations
I will describe work in progress, joint with Toby Gee. Our goal is to construct, for a finite extension K of Q_p and a natural number n, to construct a formal Artin stack over the formal spectrum of Z_p which parameterizes families of ndimensional representation of the absolute Galois group G_K. The underlying reduced closed subscheme of our formal scheme will be an equidimensional Artin stack over F_p whose F_pbar points naturally correspond to representations G_K > GL_n(F_pbar). My aim in the talk is to motivate our construction, to sketch some of the ideas underlying the construction, and also to explain what points remain to be worked out in order to complete the construction. If time permits, I will also explain some applications of the construction that we have in mind.
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February 3

Simion Filip (UChicago)

Hodge theory and arithmetic in Teichmuller dynamics
The dynamics of a billiard ball in a polygon is a classical dynamical system for which many questions remain open. These questions are intimately related to a natural action of the group SL(2,R) on the tangent bundle to the moduli space of Riemann surfaces. It can be viewed as a "complexified" geodesic flow.
By recent results of Eskin and Mirzakhani, this action of SL(2,R) enjoys rigidity properties akin to Ratner's theorems  in particular, orbit closures are submanifolds. In this talk, I will explain that these orbit closures are in fact algebraic varieties with interesting arithmetic properties. For instance, they parametrize algebraic curves with real multiplication and torsion conditions on (factors of) their Jacobian. These results depend on extending results about variations of Hodge structures to this special setting (in particular, giving a different approach to some results of Schmid).
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February 10

Brandon Levin (UChicago)

Moduli of finite flat groups schemes with descent
In Kisinâ€™s work on modularity lifting, he resolves flat deformation rings by moduli spaces of finite flat group schemes. The geometry of this resolution can be related to local models of Shimura varieties. I will discuss a generalization of this story to potentially flat Galois deformation rings. Time permitting I will also discuss how this relates to the moduli spaces of local Galois representations constructed by Matt Emerton and Toby Gee. This is joint work with Ana Caraiani.
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February 17

Keerthi Madapusi Pera (UChicago)

Heights of special divisors on orthogonal Shimura varieties
The GrossZagier formula relates two complex numbers obtained in seemingly very disparate ways: The NeronTate height pairing between Heegner points on elliptic curves, and the central derivative of a certain automorphic Lfunction of Rankin type. I will explain a variant of this in higher dimensions. On the geometric side, the intersection theory will now take place on Shimura varieties associated with orthogonal groups. On the analytic side, we will find RankinSelberg Lfunctions involving modular forms of halfintegral weight. This is joint work with Fabrizio Andreatta, Eyal Goren and Ben Howard.
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February 24

KaiWen Lan
(UMN)

Compactifications of PELtype Shimura varieties in ramified characteristics
I will report on what we know about the compactifications of PELtype Shimura varieties in ramified characteristics, allowing arbitrary levels, when they are simply defined by taking normalizations in certain auxiliary good reduction integral models. While we cannot expect them to be smooth, I will explain that they still enjoy many nice properties comparable to the good reduction cases. If time permits, I will also talk about a construction for the splitting models introduced by Pappas and Rapoport.
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March 3

Mohammad Hadi Hedayatzadeh
( Purdue University )

Exterior powers of LubinTate groups
After defining exterior powers of pdivisible groups, we prove that the exterior powers of pdivisible groups of dimension at most one over any base scheme exist and their construction commute with arbitrary base change. If time permits, we will prove the similar results for LubinTate groups in mixed and equal characteristic cases.
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March 10

Vaidehee Thatte
(UChicago)

Ramification Theory for Arbitrary Valuation Rings: Equal Characteristic Case
In the classical ramification theory we consider complete discrete valued field extensions with perfect residue fields and separable residue extensions. It is interesting to see if we can have ramification theory for arbitrary valuations, that is compatible with the classical theory. A lot of work has been done for complete discrete valued fields with arbitrary residue fields. In this talk, we will discuss ArtinSchreier extensions of arbitrary valued fields (with arbitrary residue fields, possibly with nondiscrete value groups). Special difficulties arise in the defect case (i.e., a ramified extension with no extension of the residue field and no extension of
the value group). Complete discrete valued fields are defectless, but the defect occurs often when we study arbitrary valuations.We will also introduce a generalization and further refinement of Kato's refined Swan conductor.
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March 17

Michael Rapoport
(Univ. of Bonn)

On an Arithmetic Transfer conjecture for formal moduli spaces
In the relative trace formula approach to the arithmetic GanGrossPrasad conjecture, there is a Arithmetic Transfer analogue of the Arithmetic Fundamental Lemma conjecture of Wei Zhang. In joint work with B. Smithling and W. Zhang we prove this conjecture in the case of a unitary group in three variables.
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