# 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. We are eligible for department funding as long as we maintain sufficient enrollment.

#### Winter Quarter 2017

 January 12 (plus prep talk) Ozlem Ejder (USC) Torsion subgroups of elliptic curves in elementary abelian 2-extensions Let $E$ be an elliptic curve defined over $\Q$. The torsion subgroup of $E$ over the compositum of all quadratic extensions of $\Q$ was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of $31$ possible groups and Fujita proved that the list of $20$ different groups is complete. In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. $\Q(i)$ and $\Q(\sqrt{-3})$. January 19 (plus prep talk) Brian Hwang (Cornell) An application of (harmonic (families of)) automorphic forms to Galois theory A number of questions in Galois theory can be phrased in the following way: how large (in various senses) can the Galois group G of an extension of the rational numbers be, if the extension is only allowed to ramify at a small set of primes? If we assume that G is abelian, class field theory provides a complete answer, but the question is open is almost every nonabelian case, since there is no known way to systematically and explicitly construct such extensions in full generality. However, there have been some programs that are gaining ground on this front. While the problem above is natural and the objects are classical, we will see that to answer certain questions about of this Galois group, it seems necessary to use techniques involving automorphic forms and their representation-theoretic avatars. In particular, it will turn out that some recent results on "harmonic" families of automorphic forms translate to the fact that such number fields, despite not being explicitly constructible by known methods, turn out to "exist in abundance" and allow us to find bounds on the sizes of such Galois groups. January 26 (plus prep talk) Peter Stevenhagen (Leiden) Artin's conjecture: multiplicative and elliptic Artin's conjecture on primitive roots, which was originally formulated for multiplicative groups, has a natural analogue for elliptic curves. In this survey talk, I will discuss the analogy and focus on "new" phenomena such as the existence of "never-primitive" points. February 2 Michiel Kosters (UC Irvine) Slopes of L-functions of Z_p-covers of the projective line (2-3) Let P: ... -> C_2 -> C_1 -> P^1 be a Z_p-cover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the p-adic Newton slopes of L-functions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu. Niccolò Ronchetti (Stanford) A Satake homomorphism for the mod p derived Hecke algebra (3-4) Recently, Venkatesh introduced the derived Hecke algebra to explain extra endomorphisms on the cohomology of arithmetic manifolds: the crucial local construction is a derived version of the spherical Hecke algebra of a reductive p-adic group. Working with p-torsion coefficients, we will describe a Satake homomorphism for the derived spherical Hecke algebra of a p-adic group. This will allow us to understand its structure well enough to attack some global questions, which are work in progress. February 9 (plus prep talk) Jennifer Balakrishnan (Boston University) Databases of elliptic curves ordered by height Elliptic curves defined over the rational numbers are of great interest in modern number theory. The rank of an elliptic curve is a crucial invariant, with many open questions about its behavior. In particular, there is great interest in the "average" rank of an elliptic curve. The minimalist conjecture is that the average rank should be 1/2. In 2007, Bektemirov, Mazur, Stein, and Watkins [BMSW], using well-known databases of elliptic curves, set out to numerically compute the average rank of elliptic curves, ordered by conductor. They found that "there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either." In joint work with Ho, Kaplan, Spicer, Stein, and Weigandt, we have assembled a new database of elliptic curves ordered by height. I will describe the database and examine some of the questions posed by [BMSW]. I will also discuss ongoing work by a team of undergraduates at Oxford on similar questions about families of elliptic curves. February 16 (plus prep talk) James Maynard (Oxford) Polynomials representing primes It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed $X^2+Y^4$ is prime infinitely often, and Heath-Brown showed the same for $X^3+2Y^3$. We will demonstrate a family of multivariate sparse polynomials all of which take infinitely many prime values. February 23 (plus prep talk) Serin Hong (Caltech) Harris's conjecture for Rapoport-Zink spaces of Hodge type The l-adic cohomology of Rapoport-Zink spaces is expected to realize local Langlands correspondences in many cases. Along this line is a conjecture by Harris, which roughly says that when the underlying Rapoport-Zink space is not basic, the l-adic cohomology of the space is parabolically induced. In this talk, we will discuss a result on this conjecture when the Rapoport-Zink space is of Hodge type and "Hodge-Newton reducible". The main strategy is to embed our Rapoport-Zink space to an appropriate space of EL type, for which the conjecture is already known to hold. If time permits, we will also discuss other applications of this strategy. March 2 (plus prep talk) Nathan Kaplan (UC Irvine) Rational point count distributions for del Pezzo surfaces over finite f5Dields A del Pezzo surface of degree d over a finite field of size q has at most q^2+(10-d)q+1 F_q-rational points. A surface attaining this maximum is called 'split', and if all of these rational points lie on the exceptional curves of the surface, then it is called 'full'. Can we count and classify these extremal surfaces? We focus on del Pezzo surfaces of degree 3, cubic surfaces, and of degree 2, double covers of the projective plane branched over a quartic curve. We will see connections to the geometry of bitangents of plane quartics, counting formulas for points in general position, and error-correcting codes. March 9 Amir Mohammadi (UCSD) Effective equidistribution of certain adelic periods We will present a quantitative equidistribution result for adelic homogeneous subsets whose stabilizer is maximal and semisimple. Some number theoretic applications will also be discussed. This is based on a joint work with Einsiedler, Margulis and Venkatesh. March 16 (plus prep talk) Lucia Mocz (Princeton) A new Northcott property for Faltings height The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.