The Weil Conjectures are one of the most beautiful theorems in mathematics. In the number field context zeta and L-functions are transcendental. It is well known, for example, that zeta(2)=pi^2/6. The values of these functions, even at integers, are not well understood. The Weil conjectures state the perhaps shocking result that the function field analogues of these functions are almost as simple as possible: they are rational functions. Further, they include the analogue of the Riemann Hypothesis for function fields. In this talk we will explore what the Weil conjectures say, as well as how they are proven.

We obtain explicit formulas for the number of non-isomorphic
elliptic curves with a given group structure (considered as an abstract abelian group).
Moreover, we give explicit formulas for the number of distinct group structures of all
elliptic curves over a finite field. We use these formulas to derive
some asymptotic estimates and tight upper and lower bounds for
various counting functions related to classification of elliptic
curves accordingly to their group structure. Finally, we present
results of some numerical tests which exhibit several interesting
phenomena in the distribution of group structures.

In this talk, we present the family of generalized Hessian curves.

The family of generalized Hessian curves covers more isomorphism classes of elliptic curves than Hessian curves.

We provide efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen curve parameters.

We also also present extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require $9\M+3\s$, where $\M$ is the cost of a field multiplication and $\s$ is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only $5\M+4\s$ when the curve is chosen with small parameters.

Fall 2010 MAA SoCal-Nev Section MeetingPlease visit http://sections.maa.org/socalnv/Meeting2010Fall.html for more information. Registration deadline is Tuesday, October 12 at 5:00pm.