# 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 1:15pm.
Don't forget to register for Math 209 if you are a graduate student. As of this quarter, we are eligible for department funding as long as we maintain sufficient enrollment.

#### Fall Quarter 2016

 September 22 NO SEMINAR (organizational meeting) September 29 NO SEMINAR October 6 NO SEMINAR (preempted by RTG colloquium) October 13 Benedict Gross (UCSD) On Hecke's decomposition of the regular differentials on the modular curve of level p (part II) This is the continuation of last week's RTG colloquium lecture. October 20 NO SEMINAR October 27 (plus prep talk) Jinhyun Park (KAIST) Algebraic cycles and crystalline cohomology In the theory of "motives", algebraic cycles are central objects. For instance, the so-called "motivic cohomology", that give the universal bigraded ordinary cohomology on smooth varieties, are obtained from a complex of abelian groups consisting of certain algebraic cycles. In this talk, we discuss how one can go beyond it, and we show that an infinitesimal version of the above complex of abelian groups of algebraic cycles can be identified with the big de Rham-Witt complexes after a suitable Zariski sheafification. This in a sense implies that the crystalline cohomology theory admits a description in terms of algebraic cycles, going back to a result of S. Bloch and L. Illusie in the 1970s. This is based on a joint work with Amalendu Krishna. November 3 NO SEMINAR November 10 NO SEMINAR November 17 (plus prep talk) Rufei Ren (UC Irvine) Slopes for higher rank Artin-Schreier-Witt towers We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower. This extends the main result of Davis--Wan--Xiao from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$. November 24 NO SEMINAR (Thanksgiving) December 1 (plus prep talk) Vlad Matei (Wisconsin) Counting low degree covers of the projective line over finite fields In joint work with Daniel Hast and Joseph we count degree 3 and 4 covers of the projective line over finite fields. This is a geometric analogue of the number field side of counting cubic and quartic fields. We take a geometric approach, by using a vector bundle parametrization of these curves which is different from the recent work of Manjul Bhargava, Arul Shankar, Xiaoheng Wang "Geometry of numbers methods over global fields: Prehomogeneous vector spaces" in which the authors extend the geometry of numbers methods to global fields. Our count is just for $S_3$ and $S_4$ covers, and we put the rest of the curves in our error term. (There will also be a colloquium lecture by David Hansen at 4pm in APM 6402.) Friday, December 2 (2-3PM, AP&M 7421) (no prep talk) David Hansen (Columbia) Critical p-adic L-functions for Hilbert modular forms I will describe a construction which associates a canonical p-adic L-function with a refined cohomological Hilbert modular form (pi, alpha) under some mild and natural assumptions. The main novelty is that we do not impose any hypothesis of "small slope" or "noncriticality" on the allowable refinements. Over Q, this result is due to Bellaiche. Our strategy for dealing with critical refinements is roughly parallel to his, and in particular relies on a careful study of the local geometry of eigenvarieties at classical (but possibly critical) points. This is joint work with John Bergdall.