What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several ``Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over F_q. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.

To a Hilbert modular form one may attach a p-adic analytic
L-function interpolating certain special values of the usual L-function.
Conjectures in the style of Mazur, Tate and Teitelbaum prescribe the order
of vanishing and first Taylor coefficient of such p-adic L-functions, the
first coefficient being controlled by an L-invariant which has conjectural
(arithmetic) value defined by Greenberg and Benois. I will explain how to
compute arithmetic L-invariants for (critical, exceptional) symmetric
powers of non-CM Iwahori level Hilbert modular forms via triangulations on
eigenvarieties. This is based on joint work with Robert Harron.

Abstract: The comparison isomorphism in p-adic Hodge theory asserts that
in some sense, the p-adic etale cohomology and the algebraic de Rham cohomology
of a smooth proper variety over a finite extension of Q_p determine each
other. We propose an alternate interpretation in which the central
object is a standard auxiliary object in p-adic Hodge theory called a
(phi, Gamma)-module, from which p-adic etale cohomology and algebraic de
Rham cohomology are functorially derived using mechanisms introduced by
Fontaine. The hope is to then enrich this object to carry additional
structures especially for varieties defined over number fields; we
illustrate this by showing how to incorporate the rational structure of
de Rham cohomology. (This depends on joint work with Chris Davis.)

How many cusp form are there on SL(2), SL(n), or a more general (reductive or semisimple) linear algebraic group? Until a few years ago it was not known that there are infinitely many cusp forms on a group such as SL(n) beyond very small values of n.

Weyl's law refers to an asymptotic formula for the number of cusp forms on a given connected reductive group, in particular establishing their infinitude. I will discuss some work-in-progress, joint with Werner Mueller of University of Bonn, establishing Weyl's law with remainder terms for classical groups. Without remainder terms, this result was established, for spherical cusp forms, by Lindenstrauss and Venkatesh in a rather general setting.

Let $C$ be a curve over a local field whose reduction is totally
degenerate. We discuss the related problems of 1) determining the
group structure of the torsion subgroup of the Jacobian of $C$, and 2)
determining if a given line bundle on $C$ is divisible by a given
integer $r$. Under certain hypotheses on the reduction of $C$, we
exhibit explicit algorithms for answering these two questions.