# UCSD Number Theory Seminar (Math 209)

### Thursday 1-2pm, AP&M 7421

#### Fall Quarter 2013

 September 26 NO SEMINAR October 3 Kevin Ventullo (UCLA) The rank one abelian Gross-Stark conjecture 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. October 10 NO SEMINAR October 17 NO SEMINAR October 24 Christelle Vincent (Stanford) Weierstrass points on Drinfeld modular curves 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. October 31 NO SEMINAR November 7 Taylor Dupuy (UCLA) "Linear" Constructions with Nonlinear Fermat Quotient Operators 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. November 14 John Voight (Dartmouth College) Numerical calculation of three-point branched covers of the projective line 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. Luis Lomeli (University of Oklahoma) The Langlands-Shahidi method for the classical groups over function fields and the Ramanujan conjecture 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. November 21 NO SEMINAR Tuesday November 26 11am-12pm Elena Fuchs (Berkeley) Thin Monodromy Groups 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. December 5 Florian Herzig (University of Toronto); 3-4pm in 7421 On the classification of irreducible mod p representations of p-adic reductive groups 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. Sug Woo Shin (MIT); 1-2pm in 7421 Patching and the p-adic Langlands correspondence 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.