My research involves the study of number-theoretic problems via methods of algebraic geometry. More precisely, I am interested in the arithmetic properties of different kinds of modular forms and of the associated Galois representations, and in the geometric properties of the corresponding moduli spaces.
This page is a more vague version of my research statement, of which you can read a long form (9 pages) or a short form (5 pages). Some information can also be found in my CV.
An elliptic modular form of weight k and level N over the field of complex numbers is a homomorphic function on the upper-half plane, satisfying a certain growth condition at infinity, as well as a certain transformation property with respect to the action of determinant one two-by-two integral matrices which are congruent to the identity matrix modulo N. Equivalently, it is a global section of a certain line bundle on the modular curve X(N) parametrizing generalized elliptic curves with level N structure. Although more abstract, this second definition has several advantages. First, it allows us to consider elliptic modular forms over fields (or even rings) other than the complex numbers, for instance over finite fields (giving rise to modular forms mod p). Second and maybe more importantly, it makes available the machinery of algebraic geometry.
Many generalizations of the concept of modular form also enjoy algebraic-geometric interpretations: for instance, Siegel and Hilbert modular forms can be defined as global sections of vector bundles on certain moduli spaces of abelian varieties with additional structure. Modular forms and their generalizations are naturally endowed with a collection of commuting linear operators, called Hecke operators. They give rise to systems of eigenvalues which have wonderful arithmetic properties and are therefore of great interest in number theory.
One can associate to a Hecke eigenform f (mod p) a Galois representation (mod p), i.e. a continuous group homomorphism from the absolute Galois group of the rationals Q to GL(2) of the algebraic closure of the field with p elements. This representation turns out to be irreducible and odd (takes value -1 on any complex conjugation, and its trace at most Frobenius elements is precisely the corresponding Hecke eigenvalue of f. Conversely, Serre's conjecture predicts that any Galois representation (mod p) with these properties arises in this fashion from a Hecke eigenform (mod p). Khare and Wintenberger recently proved the level 1 case of this conjecture, and they announced similar results for odd levels.
As part of the Langlands philosophy, this correspondence between modular forms and Galois representations is expected to hold in much greater generality. This is a much more difficult problem, and at the moment there are only few partial results in this direction: for example, work of Laumon, Taylor, and Weissauer yields symplectic four-dimensional Galois representations associated to Siegel modular forms of genus 2, but it seems unlikely that their approach can be generalized to higher genera.
In this context, my research has focused mainly on the study of Hecke eigensystems (mod $p$), mostly for Siegel modular forms.
My approach to the study of eigensystems (following suggestions of de Jong and Gross) was to consider the restriction of modular forms (mod p) to the set of superspecial abelian varieties (these are natural generalizations of the notion of supersingular elliptic curves).
A publication list with abstracts.
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