# 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 2019

 April 4 NO MEETING April 11 Nolan Wallach (UC San Diego) + pre-talk 1:20-1:50 The meromorphic continuation of smooth Eisenstein series In his monumental book "On the Functional Equations satisfied by Eisenstein Series" Langlands proved that the K-finite Eisenstein series, initially defined, convergent and holomorphic in appropriate open tube of the parameter space can be meromorphically continued to the entire parameter space. The K-finiteness was critical to his proof of the theorem. In this lecture I will show how to use Langlands' theorem to prove the meromorphic continuation for smooth Eisenstein series. These results are valid in the full context of Langlands' theorem but I will only talk about arithmetic groups for which the definitions are easier. (Indeed, Langlands' definition of the groups that that would be studied was only completed at the end of the induction in his notorious chapter 7). Note: There will be a pretalk aimed at graduate students and post-docs, 1:20-1:50. April 18 NO MEETING April 25 Rudy Perkins (Cuyamaca College) The Many Obstructions to Formulating a Maeda Conjecture for Drinfeld-Goss Modular Forms TBA May 2 NO MEETING May 9 NO MEETING May 16 Thomas Grubb (UC San Diego) Smooth semiample complete intersections over finite fields May 21 (TUESDAY) Wayne Raskind (Wayne State University) 4-5 PM in the Halkin Room (6402); pre-talk 3:20-3:50pm Etale cohomology of algebraic varieties over the maximal cyclotomic extension of a global field Let k be a global field, that is, a number field of finite degree over Q or the function field of a smooth projective curve C over a finite field F. Let X be a smooth projective variety over k, and let K be the maximal cyclotomic extension of k, obtained by adjoining all roots of unity. If X is an abelian variety, a famous theorem, due to Ribet in the number field case and Lang-Neron in the function field case when X has trace zero over the constant subfield of K, asserts that the torsion subgroup of the Mordell-Weil group of X over K is finite. Denoting by k_{sep} a separable closure of k, this result is equivalent to finiteness of the fixed part by G=Gal(k_{sep}/K) of the etale cohomology group H^1(X_{k_{sep}},Q/Z), where we ignore the p-part in positive characteristic p. In a recent paper, Roessler-Szamuely generalize this result to all odd cohomology groups. The trace zero assumption in the function field case is replaced by a "large variation" assumption on the characteristic polynomials of Frobenius acting on the cohomology of the fibres of a morphism f: \cal{X}\to C from a smooth projective variety \cal{X} over a finite field to C with generic fibre X. In this talk, I will discuss the case of even degree, proving some positive results in the number field case and negative results in the function field case. Note: There will be a preparatory talk for graduate students and postdocs 3:20-3:50pm in the seminar room. May 23 Piper H (University of Hawaii) pre-talk 1:20-1:50pm Joint Shapes of Quartic Fields and Their Cubic Resolvents The shape of a degree $n$ number field is a $n-1$-variable real quadratic form (up to equivalence and scaling) which keeps track of the lattice shape of its ring of integers relative to $\mathbb{Z}$. For number fields of small degree, in previous joint work with Bhargava, we showed that shapes of $S_n$-number fields are equidistributed, when ordered by absolute discriminant. The proof relies heavily on Bhargava's parametrizations which introduces but ultimately ignores the notion of resolvent rings. This talk discusses work in progress, joint with Christelle Vincent, in which we define the joint shape of a ring and its resolvent ring in order to prove equidistribution of joint shapes of quartic fields and their cubic resolvent fields. Note: The speaker will give a preparatory lecture for graduate students and postdocs in the seminar room from 1:20-1:50pm. May 30 Naser Sardari (University of Wisconsin-Madison) pre-talk 1:20-1:50pm Bounds on the multiplicity of the Hecke eigenvalues Fix an integer N and a prime (p,N)=1 where p>3. We show that the number of newforms f (up to a scalar multiple) of level N and even weight k such that T_p(f) = 0 is bounded independently of k, where T_p is the Hecke operator. June 6 Francois Thilmany (UC San Diego) TBA TBA