University of Washington Number Theory Seminar
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.
- 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.