In this talk, we show how to explicitly determine the zeta functions of
hyperelliptic curves of the form $y^2 = x^p-ax-b$ defined over a finite
field $GF(p^s}$ where $p$ is a prime. Joint work with Hui Xue and Lin
You.

Let G be a reductive group satisfying the Harish-Chandra condition defined over a totally real field F, E/F a finite cyclic extension of fields. With further assumptions on G, by constructing an explicit morphism between eigenvarities, we prove that every p-adic family of p-adic automorphic representations of G over F can be lifted to a family of p-adic automorphic representations of G over E such that , at every classical point, the lifting is just the classical weak base change lifting. The key ingredients in the theory are: (1) a twisted p-adic trace formula for G/E ; (2) a p-adic fundamental lemma and an equation between p-adic trace formula and twisted p-adic trace formula; (3) a second construction of a twisted eigenvariety of G/E.

In this talk, I will report on a joint project with Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort stratification is defined by the vanishing locus of the partial Hasse-invariants; it is an analog of the stratification of modular curve mod p by the ordinary locus and the supersingular locus. We give explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for F.

In this talk, I will talk about one approach to study the Diophantine equation f(x)=g(y), which combines the tools from Galois theory, algebraic geometry and group theory.
In particular, I will explain how the methods are used in the joint work with Mike Zieve on the equation ax^m+bx^n+c=dy^p+ey^q.

The ideas and methods above are also used in a recent theorem
of Carney-Hortsch-Zieve, which says that for any polynomial f(x) in Q[x], the map f: Q -> Q, a -> f(a) is at most 6-to-1 off a finite subset of Q. I will state a much more general conjecture on uniform boundedness of rational preimages of rational functions on number fields, of which a quite special case implies the theorems of Mazur and Merel on rational torsion points of elliptic curves.

The endomorphism rings of ordinary jacobians of genus two curves defined over finite
fields are orders in quartic CM fields. The conductor gap between two endomorphism rings is
defined as the largest prime number that divides the conductor of one endomorphism ring but not
the other. We call a genus two curve isolated, if its endomorphism ring has large conductor gap
(>=80 bits) with any other possible endomorphism rings. There is no known algorithm to explicitly
construct isogenies from an isolated curve to curves in other endomorphism classes. I will
explain results on criteria for a curve to be isolated, as well as the heuristic asymptotic
distribution of isolated genus two curves.