Given an ordinary differential equation whose coefficients are
meromorphic functions of a complex variable, the only obstruction to
convergence of local solutions in a disc is the presence of
singularities within the disc. It was observed decades ago that this
fails if one replaces "complex" by "p-adic", e.g., consider the
exponential function. In recent work of Baldassarri, Poineau, Pulita,
and the speaker, it has emerged that the convergence properties of such
solutions in the p-adic case can be described quite simply in terms of
Berkovich analytic geometry. We will give this description (without
assuming any prior familiarity with Berkovich's theory) and mention some
applications to studying wild ramification of covers of p-adic curves.
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.
The local Langlands correspondence is a relationship between representations of the Galois group of a p-adic field F and the rerepresentations of GL_n(F). Understanding the behavior of the local Langlands correspondence as one varies Galois representations in families is an important ingredient in Emerton's recent proof of many cases of the Fontaine-Mazur conjecture. I will explain this question, and its connection to questions involving the Bernstein center, an algebra that acts naturally on a category of representations of GL_n(F).
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.
I'll start with definitions and basic properties of Brauer-Grothendieck groups and Brauer-Manin sets of algebraic varieties. After that I'll discuss several finiteness results for these groups with special reference to the case of abelian varieties and K3 surfaces.
This is a report on joint work with Alexei Skorobogatov.
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.
A subset $X$ in $\{0,1\}^n$ is a called an $\epsilon$-biased set if for any nonempty subset $T\subseteq [n]$ the following condition holds. Randomly choosing an element $x\in X$, the parity of its T-coordinates sum has bias at most $\epsilon$. This concept is essentially equivalent or close to expanders, pseudorandom generators and linear codes of certain parameters. For instance, viewing $X$ as a generator matrix, an $\epsilon$-biased set is equivalent to an $[|X|, n]_2$-linear code of relative distance at least $1/2-\epsilon$. For fixed $n$ and $\epsilon$, it's a challenging problem to construct smallest $\epsilon$-biased sets. In this talk we will first introduce several known constructions of $\epsilon$-bias sets with the methods from number theory and geometrical coding theory. Then we will present a construction with a conjecture, which is closely related to the subset sum problem over prime fields.
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.