Date

Speaker

Title

March 31

Xin Wan (Columbia)

Iwasawa main conjecture for supersingular elliptic curves
We prove the + main conjecture formulated by Kobayashi, for supersingular elliptic curves with a_p=0.
(Show Abstract)

April 7

John Bergdall (BU)

Arithmetic properties of Fredholm series
The slopes of modular forms are encoded in the slopes of the Newton polygon of the Fredholm determinant of Up acting on spaces of overconvergent padic modular forms. In this talk I will discuss an old technique, due to Koike, of computing this characteristic power series, and give new results regarding the mod p reduction of this series. This is joint work with Rob Pollack.
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April 14

Francesc Castella (UCLA)

padic heights of Heegner points and BeilinsonFlach elements
About 10 years ago, Ben Howard proved a Lambdaadic GrossZagier formula relating the padic heights of Heegner points over ring class fields of ppower conductor to the derivative of a twovariable padic Lfunction. In this talk, we will explain a strategy for extending Howard’s theorem to higher weights. Rather than on calculations inspired by the original work of Gross and Zagier, our approach is via Iwasawa theory, based on the connection between Heegner points and BeilinsonFlach elements, and their variation in padic families.
(Show Abstract)

April 21

Martin Luu (UIUC)

Numerical local Langlands duality and Weil’s Rosetta Stone
The analogy between number fields, curves over finite fields, and Riemann surfaces has a long and fruitful history, in particular with respect to the various Langlands dualities. More recently, through the work of Kapustin and Witten, quantum physics has been added as a fourth pillar to the story. In this talk I will describe some local aspects of these analogies. In particular, I will explain how the numerical local Langlands duality and the Tduality of 2D quantum gravity can be derived from the same symmetry principle of local Langlands parameters.
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April 28

Junecue Suh (UCSanta Cruz)

New vanishing theorems for mixed Hodge modules and applications
We'll review various vanishing theorems (of Kodaira, Nakano, Kawamata, Viehweg, Esnault, Illusie, ...) and then present new vanishing theorems, with coefficients in mixed Hodge modules. If time permits, we'll mention applications to the cohomology of Shimura varieties.
(Show Abstract)

April 30 (Thurs) Note special day!
(1:303PM in E203) 
Nike Vatsal
(UBC)

Congruences for modular forms of halfinteger weight
Suppose F and G are holomorphic cuspidal newsforms of even weight and trivial characters of levels M and N respectively, such that F and G are congruent modulo a prime P in the algebraic closure of Q. We can then pose the question of whether or not the modular forms associated to F and G by the ShimuraWaldspurger correspondence are also congruent modulo P. In considering this question, one quickly realizes that the in the most naive form the answer to this question is negative, but the reason for the failure turns out to be quite subtle. One is faced with the obvious fact that there’s no evident way to single out a specific form on the metaplectic group that corresponds to F or G, but a more subtle issue is that the usual ShimuraWaldspurger correspondence does not even yield a canonical bijection on the level of automorphic representations. In attempting to formulate a statement that might conceviably be true, one has to consider in some detail the structure of the Waldspurger packets on the metaplectic group, and the existence of a congruence on the metaplectic side is related to a hypothetical multiplicity one theorem for metaplectic modular forms in positive characteristic. Informal speculations along these lines were first made some years ago by K. Prasanna, and we will attempt to make some of his speculations more precise and state an actual conjecture.
(Show Abstract)

May 5

Bianca Viray (Washington)

Obstructions to the Hasse principle on degree 4 del Pezzo surfaces
In 1970, Manin showed that the Brauer group can obstruct the existence of rational points. ColliotThélène and Sansuc have conjectured that this obstruction completely explains the failure of rational points on del Pezzo surfaces. We show that on degree 4 del Pezzo surfaces, this BrauerManin obstruction manifests itself through linear projections. As a consequence of the proof, we obtain a simple and efficient for computing the Brauer classes of a degree 4 del Pezzo surface. This is joint work with Anthony VárillyAlvarado.
(Show Abstract)

May 12

Rebecca Bellovin (UCBerkeley)

Local epsilonisomorphisms in families
Given a representation of Gal_{Q_p} with coefficients in a
padically complete local ring R, Fukaya and Kato have conjectured the
existence of a canonical trivialization of the determinant of a certain
cohomology complex. When R=Z_p and the representation is a lattice
in a de Rham representation, this trivialization should be related to
the \varepsilonfactor of the corresponding WeilDeligne
representation. Such a trivialization has been constructed for certain
crystalline Galois representations, by the work of a number of authors.
I will explain how to extend these trivializations to certain families
of crystalline Galois representations. This is joint work with Otmar
Venjakob.
(Show Abstract)

May 19

Liang Xiao
(UConn)

Eigencurve over the boundary of the weight space
Eigencurve was introduced by Coleman and Mazur to parametrize modular forms
varying padically. It is a rigid analytic curve such that each point
corresponds to an overconvegent eigenform. In this talk, we discuss a
conjecture on the geometry of the eigencurve: over the boundary annuli of
the weight space, the eigencurve breaks up into infinite disjoint union of
connected components and the weight map is finite and flat on each
component. This was first verified by Buzzard and Kilford by an explicit
computation in the case of p = 2 and tame level 1. We will explain a
generalization to the definite quaternion case with no restriction on p
(except p > 2) or the tame level. This is a joint work with Ruochuan Liu
and Daqing Wan, based on an idea of Robert Coleman.
(Show Abstract)

May 26

No Seminar

No Seminar
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June 2

Ariane Mézard
(Jussieu)

Genetic of local padic Galois representations
In this talk, we define a combinatorial data, said the gene, associated to a modulo $p$ Galois representation $\bar{\rho}$ and a Galois type. We prove that this gene encodes explicit information on geometric deformations of $\bar{\rho}$. First, the gene provides an explicit easy instant description of Kisin variety parametrizing BreuilKisin modules associated to potentially BarsottiTate deformations of modulo $p$ Galois representations of dimension 2. Then we explain how we may deduce the associated deformation ring in non generic cases. This is joint work with Xavier Caruso and Agnes David.
(Show Abstract)
