# UCSD Number Theory Seminar (Math 209)

### Thursday 2-3pm, AP&M 7421

This quarter, most talks will be preceded by a 30-minute "prep talk" for graduate students and postdocs. These will be in AP&M 7421 starting at 1:20pm.
Don't forget to register for Math 209 if you are a graduate student. We are eligible for department funding as long as we maintain sufficient enrollment.

#### Spring Quarter 2018

 April 6 (FRIDAY), 2-3pm in APM 7218 Karol Koziol (Toronto) + pre-talk Some calculations with higher pro-p-Iwahori cohomology Let G denote a p-adic reductive group, and I_1 a pro-p-Iwahori subgroup. A classical result of Borel and Bernstein shows that the category of complex G-representations generated by their I_1-invariant vectors is equivalent to the category of modules over the (pro-p-)Iwahori-Hecke algebra H. This makes the algebra H an extremely useful tool in the study of complex representations of G, and thus in the Local Langlands Program. When the field of complex numbers is replaced by a field of characteristic p, the equivalence above no longer holds. However, Schneider has shown that one can recover an equivalence if one passes to derived categories, and upgrades H to a certain differential graded Hecke algebra. We will attempt to understand this equivalence by examining the H-module structure of certain higher I_1-cohomology spaces, with coefficients in mod-p representations of G. If time permits, we'll discuss how these results are compatible with Serre weight conjectures of Herzig and Gee--Herzig--Savitt. April 12 Claus Sorensen (UCSD) + pre-talk Local Langlands in rigid families Motivated by local-global compatibility in the $p$-adic Langlands program, Emerton and Helm (and others) studied how the local Langlands correspondence for $\GL(n)$ can be interpolated in Zariski families. In this talk I will report on joint work with C. Johansson and J. Newton on the interpolation in rigid families. We take our rigid space to be an eigenvariety $Y$ for some definite unitary group $U(n)$ which parametrizes Hecke eigensystems appearing in certain spaces of $p$-adic modular forms. The space $Y$ comes endowed with a natural coherent sheaf $\mathcal{M}$. Our main result is that the dual fibers $\mathcal{M}_y'$ essentially interpolate the local Langlands correspondence at all points $y \in Y$. This make use of certain Bernstein center elements which appear in Scholze's proof of the local Langlands correspondence (and also in work of Chenevier). In the pre-talk I will talk about the local Langlands correspondence, primarily for $\GL(2)$. April 19 NO MEETING April 26 Watson Ladd (UC Berkeley) + pre-talk Lattice methods for paramodular forms Using Ibukiyama's conjecture on transfers from inner forms of GSp(4) we compute paramodular forms with prime levels up to 400. This is joint work with Jeffery Hein and Gonzalo Tornaria. May 3 Jesse Elliott (CSU Channel Islands) + pre-talk Extensions of the prime number theorem and the Riemann hypothesis We provide two asymptotic continued fraction expansions of the prime counting function. We also develop a "degree" calculus that enables us to strengthen the connections between various reformulations and extensions of the Riemann hypothesis. May 10 Marc-Hubert Nicole (Aix-Marseille) + pre-talk Families of Drinfeld modular forms on Drinfeld modular varieties Classical modular curves associated to GL(2) are moduli spaces of elliptic curves with additional structure. Taking advantage of the analogy between number fields and function fields, Drinfeld modules (of rank 2) were introduced as a good analogue of elliptic curves. While there are no Shimura varieties associated to the general linear group GL(N) for N>2, the situation is sharply different over function fields. The Drinfeld modular variety for GL(N) is a moduli space of Drinfeld modules of rank N (with auxiliary level structure). It is an affine scheme of dimension N-1. In this talk, I will explain how analogues of well-established theories due to Hida and Coleman in the classical p-adic context extend to Drinfeld modular varieties and their associated modular forms. Joint with G. Rosso (Montréal). May 17 Preston Wake (UCLA) + pretalk Eisenstein ideal with squarefree level In his influential paper "Modular curves and the Eisenstein ideal", Barry Mazur studied congruences modulo p between cusp forms and the Eisenstein series of weight 2 and prime level N. In particular, he defined the Eisenstein ideal in the relevant Hecke algebra, and showed that it is locally principal. We'll discuss the analogous situation for certain squarefree levels N, and show that, while the Eisenstein ideal may not be locally principal, we can count the minimal number of generators and explain the arithmetic significance of this number. This is joint work with Carl Wang-Erickson. May 24 Jukka Keranen (UCLA) + pre-talk L-Functions of Unitary Group Shimura Varieties We will discuss two different approaches to computing the L-functions of Shimura varieties associated with GU(2,1). Both approaches employ the comparison of the Grothendieck-Lefschetz formula with the Arthur-Selberg trace formula. The first approach, carried out by the author, takes as its starting point the recent work of Laumon and Morel. The second approach is due to Flicker. In both approaches, the principal challenge is that the Shimura varieties in question are non-compact, and one must use cohomology with compact supports. Time permitting, we will discuss the prospects for extending these approaches to the non-compact Shimura varieties associated with higher-rank unitary groups. May 31 NO MEETING (preempted by Jacob Lurie colloquium) June 7 Shishir Agrawal (UC Berkeley) + pre-talk Rigid local systems and rigid isocrystals A local system on the Riemann sphere minus finitely many points is defined to be "rigid" if it is determined by the conjugacy classes of its monodromy operators along the missing points. Katz proves a convenient cohomological criterion characterizing irreducible rigid local systems, which is based on an analysis of the moduli of local systems on the punctured Riemann sphere. In this talk, we will discuss this story, and then proceed towards an analogous story in the arithmetic setting, where, in place of local systems on the punctured Riemann sphere, we consider overconvergent isocrystals on the punctured projective line over a field of positive characteristic. June 14, APM B402A Samuele Anni (MPIM Bonn) + pre-talk Congruence, modular forms and graphs The theory of congruences of modular forms is a central topic in contemporary number theory. Congruences between modular forms play a crucial role in understanding links between geometry and arithmetic: cornerstone example of this is the proof of Serre's modularity conjecture by Khare and Wintenberger. Congruences of Galois representations govern many kinds of representations of the absolute Galois group of number fields. Even though our understanding is improving, many aspects remain very mysterious, some are theoretically approachable, many are not; and amongst the latter, some allow numerical studies to reveal first insights. In this talk I will introduce congruence graphs, which are graphs encoding congruence relations between classical newforms. Then I will explain first how to construct analogous graphs for congruences of Galois representations, and then how to use these graphs to study questions regarding Hecke algebras and Atkin-Lehner operators.