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September 26 |
NO SEMINAR
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October 3 |
Kevin Ventullo (UCLA) Let $\chi$ be a totally odd character of a totally real number
field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of
Stark which expresses the leading term at s=0 of the p-adic L-function
attached to $\chi\omega$ as a product of a regulator and an algebraic
number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the
rank one case under two assumptions: that Leopoldt's conjecture holds for F
and p, and a certain technical condition when there is a unique prime above
p in F. After giving some background and outlining their proof, I will
explain how to remove both conditions, thus giving an unconditional proof
of the conjecture. If there is extra time I will explain an application to
the Iwasawa Main Conjecture which comes out of the proof, and make a few
remarks on the higher rank case. |
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October 10 |
NO SEMINAR
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October 17 |
NO SEMINAR
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October 24 |
Christelle Vincent (Stanford) We consider the so-called Drinfeld setting, a function field analogue of some aspects of the theory of modular forms, modular curves and elliptic curves. In this setting Drinfeld constructed families of modular curves defined over a complete, algebraically closed field of characteristic p. We are interested in studying their Weierstrass points, a finite set of points of geometric interest. In this talk we will present some tools from the theory of Drinfeld modular forms that were developed to further this study, some geometric and analytic considerations, and some partial results towards computing the image of these points modulo a prime ideal of the base ring. |
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October 31 |
NO SEMINAR
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November 7 |
Taylor Dupuy (UCLA) The integers do not admit any nontrivial derivations. We will explain
how the operation x ↦ (x-xp)/p can be thought of as a replacement for
the derivative operator on the integers. After the introduction we hope
to explain some recent work on the meaning of "linearity" in this theory.
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November 14 |
John Voight (Dartmouth College) We exhibit a numerical method to compute a three-point branched covers
of the complex projective line. We develop algorithms for working
explicitly with Fuchsian triangle groups and their finite index
subgroups, and we use these algorithms to compute power series
expansions of modular forms on these groups. As one application, we
find an explicit rational function of degree 50 which regularly
realizes the group PSU3(5) as a Galois group over the rationals.
This is joint work with Michael Klug, Michael Musty, and Sam
Schiavone.
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Luis Lomeli (University of Oklahoma) The Langlands-Shahidi method provides us with a constructive
way of studying automorphic L-functions. For the classical groups over
function fields we will present recent results that allow us to obtain
applications towards global Langlands functoriality. This is done via
the Converse Theorem of Piatetski-Shapiro, which we can apply since our
automorphic L-functions have meromorphic continuation to rational
functions and satisfy a functional equation. We lift globally generic
cuspidal automorphic representations of a classical group to an
appropriate general linear group. Then, we express the image of
functoriality as an isobaric sum of cuspidal automorphic representations
of general linear groups, where the symmetric and exterior square
automorphic L-functions play a technical role. As a consequence, we can
use the exact Ramanujan bounds of Laurent Lafforgue for GL(N) to prove
the Ramanujan conjecture for the classical groups. Our results are
currently complete for the split classical groups under the assumption
that characteristic p is different than two. |
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November 21 |
NO SEMINAR
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Tuesday |
Elena Fuchs (Berkeley) In recent years, it has become interesting from a
number-theoretic point of view to be able to determine whether a
finitely generated subgroup of GLn(Z) is a so-called thin
group. In general, little is known as to how to approach this question.
In this talk we discuss this question in the case of hypergeometric
monodromy groups, which were studied in detail by Beukers and Heckman in
1989. We will convey what is known, explain some of the difficulties in
answering the thinness question, and show how one can successfully
answer it in many cases where the group in question acts on hyperbolic
space. This work is joint with Meiri and Sarnak. |
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December 5 |
Florian Herzig (University of Toronto); 3-4pm in 7421 Suppose that G is a connected reductive p-adic group. We will
describe the classification of irreducible admissible smooth mod p
representations of G in terms of supercuspidal representations. This is
joint work with N. Abe, G. Henniart, and M.-F. Vigneras. |
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Sug Woo Shin (MIT); 1-2pm in 7421 This is a report on joint work with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, and Vytautas Paskunas. We will explain some ideas around the global construction of new representation-theoretic objects called "patched modules" by a variant of the Taylor-Wiles-Kisin method. They are in many ways better suited than p-adically completed cohomology for a global attempt to understand the p-adic local Langlands correspondence. As an application, we obtain new cases of the Breuil-Schneider conjecture. |