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.

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.

It is a fascinating result of Ulmer that the elliptic curve y^2=x^4+x^3+t^d attains arbitrarily large rank over $\bar{F_q}(t)$ as d varies over the positive integers. In this talk we will provide some new examples of this phenomenon and provide an overview of previous work in this area, particularly that of Ulmer and Berger.

We will discuss some of the local theory of rigid-analytic spaces including Tate's algebra, affinoid algebras, Washnitzer's algebra and dagger algebras. After we provide enough motivation we will discuss the results of research completed by myself and Professor Daqing Wan. The results of this research form a basis for generalizing Washnitzer's algebra.