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April 23 |
Jishnu Ray (University of British Columbia)
The Iwasawa Theory of Selmer groups provides a natural way for p-adic approach to the celebrated Birch and Swinnerton Dyer conjecture. Over a non-commutative p-adic Lie extension, the (dual) Selmer group becomes a module over a non-commutative Iwasawa algebra and its structure can be understood by analyzing “(left) reflexive ideals” in the Iwasawa algebra. In this talk, we will start by recalling several existing conjectures in Iwasawa Theory and then we will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications in understanding the (two-sides) reflexive ideals. Generalizing Clozel’s work for SL(2), we will also show that such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. uniform pro-p groups and the pro-p Iwahori of GL(n,Z_p). Further, if time permits, I will also sketch some of my recent Iwasawa theoretic results joint with Sujatha Ramdorai. |
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April 30 |
Jize Yu (California Institute of Technology) (slides) (paper)
The geometric Satake equivalence establishes a link between two monoidal categories: the category of perverse sheaves on the local Hecke stack and the category of finitely generated representations of the Langlands dual group. It has many important applications in the study of the geometric Langlands program and number theory. In this talk, I will discuss the integral coefficient geometric Satake equivalence in the mixed characteristic setting. It generalizes the previous results of Lusztig, Ginzburg, Mirkovic-Vilonen, and Zhu. Time permitting, I will discuss an application of this result in constructing a Jacquet-Langlands transfer. |
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May 7 |
Carl Wang-Erickson (University of Pittsburgh) (slides) (paper)
In his landmark paper "Modular forms and the Eisenstein ideal," Mazur studied congruences modulo a prime p between the Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp forms, assuming these modular forms have weight 2 and prime level N. He asked about generalizations to squarefree levels N. I will present some work on such generalizations, which is joint with Preston Wake and Catherine Hsu. |
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May 14, 10:00 |
Federico Pellarin (U. Jean Monnet, Saint-Etienne, France)
In this talk we will describe some recent works on Drinfeld modular forms with values in Tate algebras (in 'equal positive characteristic'). In particular, we will discuss some remarkable identities (proved or conjectural) for Eisenstein and Poincaré series, and the problem of analytically interpolate families of Drinfeld modular forms for congruence subgroups at the infinity place. |
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May 14 |
Xin Tong (University of California, San Diego) (slides)
With the motivation of generalizing the corresponding geometrization of Tamagawa-Iwasawa theory after Kedlaya-Pottharst, and with motivation of establishing the corresponding equivariant version of the relative p-adic Hodge theory after Kedlaya-Liu aiming at the deformation of representations of profinite fundamental groups and the family of étale local systems, we initiate the corresponding Hodge-Iwasawa theory with deep point of view and philosophy in mind from early work of Kato and Fukaya-Kato. In this talk, we are going to discuss some foundational results on the Hodge-Iwasawa modules and Hodge-Iwasawa sheaves, as well as some interesting investigation towards the goal in our mind, which are taken from our first paper in this series project. |
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May 21 |
Jack Thorne (Cambridge University)
Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2), thought of as the L-group of the reductive group GL(2) over Q. I will discuss the proof, joint with James Newton, of the existence of the corresponding functorial liftings for a broad class of holomorphic modular forms, including Ramanujan’s Delta function. |
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May 28 |
Elena Fuchs (University of California, Davis)
Circle packings in which all circles have integer curvature, particularly Apollonian circle packings, have in the last decade become objects of great interest in number theory. In this talk, we explore some of their most fascinating arithmetic features, from local to global properties to prime components in the packings, going from theorems, to widely believed conjectures, to wild guesses as to what might be true. |
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June 4 |
Niccolo Ronchetti (University of California, Los Angeles)
When studying the cohomology of Shimura varieties and arithmetic manifolds, one encounters the following phenomenon: the same Hecke eigensystem shows up in multiple degrees around the middle dimension, and its multiplicities in these degrees resembles that of an exterior algebra. In a series of recent papers, Venkatesh and his collaborators provide an explanation: they construct graded objects having a graded action on the cohomology, and show that under good circumstances this action factors through that of an explicit exterior algebra, which in turn acts faithfully and generate the entire Hecke eigenspace. In this talk, we discuss joint work with Khare where we investigate the $p=p$ situation (as opposed to the $l \neq p$ situation, which is the main object of study of Venkatesh’s Derived Hecke Algebra paper): we construct a degree-raising action on the cohomology of the ordinary Hida tower and show that (under some technical assumptions), this action generates the full Hecke eigenspace under its lowest nonzero degree. Then, we bring Galois representations into the picture, and show that the derived Hecke action constructed before is in fact related to the action of a certain dual Selmer group. |