For a subset D in an abelian group A, the subset
sum problem for D is to determine if D has a subset S which
sums to a given element of A. This is a well known NP-complete
problem, arising from diverse applications in coding theory,
cryptography and complexity theory. In this series of two
expository talks, we discuss and outline an emerging theory
of this subset sum problem by allowing D to have some
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.
We construct explicit genus 2 hyperelliptic curves whose Jacobian varieties have complex multiplication and are defined over an explicit algebraic number field. For these Jacobians, we give a formula for the number of points on their reductions modulo primes of good reduction. The construction and results can be viewed as a dimension 2 generalization of results of H. Stark. These formulas have application to cryptography and the CM method in dimension 2.
Endoscopy structures of automorphic forms was one of the
basic structures discovered through the Arthur-Selberg trace formula method to establish the Langlands functoriality for classical groups.
In this talk, we will discuss my recent work on characterization of the endoscopy structure in terms of the order of pole at s=1 of certain L-functions, and in terms of a family of periods of automorphic forms, which was discovered jointly with David Ginzburg. At the end, I may discuss how to contruct the
endoscopy transfer by integral operators, which is
a joint work with Ginzburg and Soudry.
Variables Separated Equations and Finite Simple Groups: Davenport's
problem is to figure out the nature of two polynomials over a number
field having the same ranges on almost all residue class fields of the
number field. Solving this problem initiated the monodromy method.
That included two new tools: the B(ranch)C(ycle)L(emma) and the
Hurwitz monodromy group. By walking through Davenport's problem with
hindsight, variables separated equations let us simplify lessons on
using these tools. We attend to these general questions:
1. What allows us to produce branch cycles, and what was their effect
on the Genus 0 Problem (of Guralnick/Thompson)?
2. What is in the kernel of the Chow motive map, and how much is it
captured by using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?
Each phrase addresses formulating problems based on equations. We seem
to need explicit algebraic equations. Yet why, and how much do we lose/
gain in using more easily manipulated surrogates for them? To make
this clear we consider the difference in the result for Davenport's
Problem and that for its formulation over finite fields, using a
technique of R. Abhyankar.