# University of Washington Number Theory Seminar

## Fall 2009

The NT Seminar meets Mondays, 2:30-3:20 in Padelford C401, unless otherwise noted.

- November 23: Robert Miller (UW)
TITLE:

**Explicit 3-descent**- ABSTRACT: Given an elliptic curve E defined over a number field K, we can compute the p-torsion part of the Shafarevich-Tate group explicitly using calculations in algebras over K. In this talk, I will explain the general theory for odd p, and work out all of the details for p=3.

- November 16: Alyson Deines (UW)
Title:

**A gentle introduction to Hilbert modular forms, part 2**- Abstract: A Hilbert modular form is a generalization of a modular form. Instead of being a holomorphic function on the upper half plane, they are holomorphic functions on n copies of the upper half plane. In this talk I will explain a few special properties of holomorphic functions in n variables, how cusps are generalized, Koecher's principle, and congruence subgroups.

- November 2: Alyson Deines (UW)
Title:

**A gentle introduction to Hilbert modular forms**- Abstract: TBA

- October 26: William Stein (UW)
Title:

**Computing Kolyvagin's Euler System of Heegner Points**- Abstract: We describe an algorithm for explicitly computing Kolyvagin's Euler system of Heegner points. It involves using rational quaternion algebras to efficiently and explicitly compute the reduction of Kolyvagin classes modulo a prime. This yields the first ever provable confirmation of Kolyvagin's conjecture for elliptic curves of rank 2.

- October 12: Ralph Greenberg (UW)
Title:

**Galois properties of elliptic curves with an isogeny.**Abstract: We consider an elliptic curve defined over the field of rational numbers. Assume that has an isogeny of prime degree defined over . There is a homomorphism which describes the action of the absolute Galois group of on the group of torsion points on of -power order. The object of this talk is to discuss various theorems which describe the image of the above homomorphism rather precisely.

- October 5: Salman Baig (UW)
Title:

**-functions of Twisted Elliptic Curves over Function Fields**Abstract: The -function of an elliptic curve over a function field with non-constant -invariant is a polynomial who degree is determined by the places of bad reduction of the elliptic curve. We will discuss how to compute this polynomial explicitly, as well as the -functions of quadratic twists of the elliptic curve. Time permitting, we will consider some data collected in this setting for a few specific families of twisted elliptic curves.

- October 2: Robert Bradshaw and Tom Boothby (UW)
Title:

**Computing Conjectural Integers to High Precision**Abstract: According to the Birch and Swinnerton-Dyer conjecture, the order of the Tate-Shafarevich group can be expressed as the ratio of several invariants of the curve. However, this ratio involves two numbers which are expected to be irrational; namely and . In the case , this ratio has never been proven to be rational. Recently, we have provably computed the ratio (1.000...) to 10kbits of precision for such a curve. We present a provable algorithm with runtime, where is the desired precision, and progress towards a provably correct algorithm.

## Winter 2008

February 26, David Freeman, Berkeley Slides

Title: Constructing abelian varieties for pairing-based cryptography Abstract: In recent years, the Weil and Tate pairings on abelian varieties over finite fields have been used to construct a vast number of new and useful cryptosystems. The abelian varieties used in these systems must have small embedding degree with respect to a large prime-order subgroup. Such ``pairing-friendly'' abelian varieties are rare and thus require specific constructions. In this talk we describe two of our recent contributions to the catalogue of pairing-friendly abelian varieties: (1) ordinary elliptic curves of prime order with embedding degree 10, and (2) ordinary abelian varieties of arbitrary dimension over $\mathbb{F}_p$ having arbitrary embedding degree with respect to a prime subgroup of size significantly smaller than $p$. Both results require finding curves whose Jacobians complex multiplication by a specified CM field; making this step feasible while maintaining the pairing-friendly property is the difficult part of such constructions. The second result is joint work with P. Stevenhagen and M. Streng (Leiden University).

February 19, 2008, John Voight, University of Vermont, Slides

TITLE: Number Field Enumeration ABSTRACT: How quickly can one enumerate number fields of fixed degree with bounded absolute discriminant? We discuss some mathematically and computationally interesting aspects of this question. For totally real number fields, a particular case of interest, we exhibit an algorithm which improves upon known methods by the use of elementary calculus (Rolle's theorem and Lagrange multipliers). TITLE: Shimura curves of genus at most two ABSTRACT: Shimura curves are generalizations of modular curves, where the matrix ring is replaced by a quaternion algebra over a totally real field. Recently, Long, Maclachlan, and Reid proved that the number of Shimura curves of bounded genus is finite. In this talk, we describe a method to explicitly enumerate all Shimura curves of genus at most 2. We examine some of the mathematically and computationally interesting aspects of this problem in turn.

## Fall 2007

- November 27, 2007, Soroosh Yazdani
Title: Szpiro's Conjecture and Level Lowering Speaker: Soroosh Yazdani (McMaster University) Location: Padelford C401 at 4:10pm on Tuesday, November 27, 2007 Abstract: Let $E/\QQ$ be an elliptic curve over the rationals. Two invariants attached to such elliptic curves are the minimal discriminant of $E$, $\Delta_E$, and the conductor of $E$, $N_E$. One knows that $N_E | \Delta_E$. Szpiro's conjecture states that for any $\epsilon>0$ there exists constant $C_\epsilon > 0$ such that for any elliptic curve $E/\QQ$ we have \[ |\Delta_E| < C_\epsilon (N_E)^{6+\epsilon}. \] In this talk, I will look at a similar conjecture that is implied by Szpiro. Specifically, if N_E=Mp with p large, then one expects $v_p(\Delta_E) \leq 6$. I will show how general level lowering results on modular forms can prove this conjecture for small values of $M$.

October 16, 2007, William Stein, UW,

**Convergence and the Sato-Tate conjecture**Notes, etc.November 20, 2007, Bill Casselman, UBC, Vancouver,

**Dirichlet's evaluation of the sign of Gauss sums**(or**Dirichlet does distributions**)In the Disquisiiones Arithmeticae, Gauss mentioned that quadratic reciprocity followed from an evaluation of the sign of certain quadratic Gauss sums. At that time he had found a great deal of empirical evidence for such an evaluation, but was able only a few years later to prove his conjecture. His proof was purely algebraic, but somewhat indirect, and dealt separately with different cases. The second evaluation of these signs was by Dirichlet, many years later. His very beautiful proof is much more direct than Gauss', albeit much less elementary. It relies on the theory of Fourier series, which as rigourous mathematics was due entirely to him, and Fresnel integrals, which had been introduced only recently in the theory of diffraction. I shall present a modern version of Dirichlet's proof, which seems to be new.