Notes on an analogue of the Fontaine-Mazur conjecture

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.Comment: 12 pages; minor revisions according to referees' comment

The distribution of class groups of function fields

Using equidistribution results of Katz and a computation in finite symplectic groups, we give an explicit asymptotic formula for the proportion of curves C over a finite field for which the l-torsion of Jac(C) is isomorphic to a given abelian l-group. In doing so, we prove a conjecture of Friedman and WashingtonComment: To appear, JPA

A heuristic for the distribution of point counts for random curves over a finite field

How many rational points are there on a random algebraic curve of large genus $g$ over a given finite field $\mathbb{F}_q$? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean $q+1+1/(q-1)$. We prove a weaker version of this statement in which $g$ and $q$ tend to infinity, with $q$ much larger than $g$.Comment: 16 pages; v2: refereed version, Philosophical Transactions of the Royal Society A 201

The integral monodromy of hyperelliptic and trielliptic curves

We compute the \integ/\ell and \integ_\ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the \integ/\ell monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group \sp_{2g}(\integ/\ell). We prove that the \integ/\ell monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group \su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])

STRATIFICATIONS ON MODULI SPACES OF ABELIAN VARIETIES IN POSITIVE CHARACTERISTIC

Over a field of positive characteristic p, we consider moduli spaces of polarized abelian varieties equipped with an action by a ring unramified at p. Using deformation theory, we show that ordinary points are dense in each of the following situations: the polarization is separable; the polarization is mildly inseparable, and the ring of endomorphisms is a totally real number field; or the polarization is arbitrary, and the ring is a real quadratic field acting on abelian fourfolds. We introduce a new invariant which measures the extent to which a polarized Dieudonne module admits an isotropic splitting lifting the Hodge filtration, and use it to explain the singularities arising from mildly inseparable polarizations