# Number Theory Seminar 2017-2018

This seminar is held on Tuesdays at 10am in 104 Deady Hall unless otherwise stated.

### Spring Quarter, 2018

- April 10,
**Asif Zaman** (Stanford)

Primes represented by positive definite binary quadratic forms
**Abstract**: The distribution of primes represented by positive definite integral binary quadratic forms is a classical topic within number theory and has been intensely studied over many centuries. We will briefly describe some of its history and discuss results giving new upper bounds for the number of primes up to $x$ represented by a given form. These estimates will come in several different flavours: unconditional, conditional on the Generalized Riemann Hypothesis, and on average over discriminants. The key feature will be that non-trivial bounds are obtained when the size of $x$ is a small power of the discriminant and, in many cases, this size will be essentially optimal.

- April 10,
**Florian Sprung** (Arizona State University)

**Special Time/Location:** 11am in Deady 301

The Iwasawa Main Conjecture for Elliptic Curves at Supersingular Primes
**Abstract**: Iwasawa theory is a connection between algebra and analysis. In the context of elliptic curves, the main conjecture of Iwasawa theory states that an analytically constructed p-adic L-function matches its algebraic counterpart. This conjecture is known when p is an ordinary prime. In this talk, we will explain our results in the case where the prime is a general supersingular prime.

- May 17,
**Kevin McGown** (California State University, Chico)

**Special Time/Location:** 3 pm in Anstett 192

Grosswald’s conjecture on the least primitive root
- May 19,
**Rachel Pries** (Colorado State University)

Speaking at Oregon Number Theory Days
- May 22,
**Aaron Pollack** (Duke University)

**Special Time/Location : **11am in Deady 301

The Fourier expansion of modular forms on exceptional groups
**Abstract**: The classical group Sp(2n) has a nice notion of classical modular forms, namely, the holomorphic Siegel modular forms. However, exceptional groups such as G_2, F_4, and E_8 do not obviously have any classical “modular forms,” because their associated symmetric spaces don’t have complex structure. Nevertheless, Gross-Wallach and Gan-Gross-Savin have defined a notion of modular forms on these groups, and (using key input of Wallach) proved abstractly that these modular forms admit a nice Fourier expansion. I will explain what these modular forms are, and what their Fourier expansions look like explicitly.

### Winter Quarter, 2018

### Fall Quarter, 2017

- October 21,
**Kirsten Eisentraeger** (Penn State University)

Oregon Number Theory Days
- November 7,
**Christopher Pinner** (Kansas State University)

The Lind-Lehmer Constant for Finite Abelian Groups
**Abstract**: In 2005 Doug Lind generalized the concept of Mahler measure to an arbitrary compact abelian group. For a given group one can ask for the minimal non-trivial measure; the counterpart of the classical Lehmer Problem for the usual Mahler measure. For a finite abelian group this corresponds to the smallest non-trivial integral group determinant. After a quick survey of existing results I will present some new congruences satisfied by the Lind Mahler measure for p-groups. These enable us to determine the minimal measure when the p-group has one particularly large component and to compute the minimal measures for many new families of small p-groups.

This is joint work with Mike Mossinghoff of Davison College. If there is time I will also mention some 3-group results from a summer undergraduate research project with Stian Clem which may hint at what is going on in general.

- November 9,
**Bianca Viray** (University of Washington)

**Special Time/Location:** 10am in Deady 210

Points of unusually low degree on the modular curves X_1(n)
**Abstract**: The Mordell-Weil theorem says that for any elliptic curve E over a number field K, the set E(K) is a finitely generated abelian group. In particular, the torsion subgroup of E(K) is finite. Amazingly, the size of the torsion subgroup can be bounded by a constant that depends only on [K:Q], i.e., that is independent of K and E; this is Merel’s celebrated Uniform Boundedness Theorem. This theorem can be rephrased as saying that for any positive integer d, the infinite tower of modular curves {X_1(n)}_{n} has only finitely many closed points of degree at most d. Work of Frey and Abramovich from around the same time combine to give an independent proof of a weaker result, that for any positive integer d, there are only finitely many positive integers n such that X_1(n) has infinitely many degree at most d points. Moreover, the work of Frey essentially characterizes these integers n in terms of the gonality of X_1(n) and the Q-rational points on Jac(X_1(n)). In this talk, we study complementary part of Merel’s theorem, that is, the modular curves X_1(n) which have finitely many (but nonempty) points of degree at most d.

- November 14,
**Yifeng Liu** (Northwestern University)

Special cycles, level raising, and Selmer groups
**Abstract**: In this talk, we will introduce recent progress on the Bloch-Kato conjecture about L-functions and Selmer groups. We will explain how the arithmetic geometry of Shimura varieties and special cycles is used in the problem through congruence of automorphic forms, known as level raising.

**Previous years**: 2017 2016