The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a
fundamental question to understand the singularities which...
On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates
$(x,y)$ in a given number field form a finitely generated abelian
group. One natural question is how the rank of this group changes
when changing the number field. For the simplest...
The theory of complex multiplication describes finite abelian
extensions of imaginary quadratic number fields using singular
moduli, which are special values of modular functions at CM points.
I will describe joint work with Henri Darmon in the...
The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy''
proof of stable functorial transfer from tori to ${\rm SL}(2)$, by
means of the Kuznetsov formula. In this talk, I will show that
there is a local statement that underlies this work...
In 1987, Barry Mazur and John Tate formulated refined
conjectures of the "Birch and Swinnerton-Dyer type", and one of
these conjectures was essentially proved in the prime conductor
case by Ehud de Shalit in 1995. One of the main objects in
de...
This talk is concerned with the radius of convergence of p-adic
families of modular forms --- q-series over a p-adic disc whose
specialization to certain integer points is the q-expansion of a
classical Hecke eigenform of level p. Numerical...
The study of eigenvarieties began with Coleman and Mazur, who
constructed the first eigencurve, a rigid analytic space
parametrizing $p$-adic modular Hecke eigenforms. Since then various
authors have constructed eigenvarieties for automorphic forms...
We study the function field analogue of a classical problem in
analytic number theory on the sums of the generalized divisor
function in short intervals, in the limit as the degrees of the
polynomials go to infinity. As a corollary, we calculate...
Affine Deligne-Lusztig varieties (ADLV) naturally arise in the
study of Shimura varieties and Rapoport-Zink spaces; their
irreducible components give rise to interesting algebraic cycles on
the special fiber of Shimura varieties. We prove a...
Hilbert's 12th problem is to provide explicit analytic formulae for
elements generating the maximal abelian extension of a given number
field. In this talk I will describe an approach to Hilbert’s 12th
that involves proving exact p-adic formulae...
