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])

On The Cohomology of NC(−2)N_C(-2) in Positive Characteristic

Let $C \subset \mathbb{P}^3$ be a general Brill--Noether curve. A classical problem is to determine when $H^0(N_C(-2)) = 0$, which controls the quadric section of $C$. So far this problem has only been solved in characteristic zero, in which case $H^0(N_C(-2)) = 0$ with finitely many exceptions. In this note, we extend these results to positive characteristic, uncovering a wealth of new exceptions in characteristic 2

Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions

We compute the stable homology of the braid group with coefficients in any Schur functor applied to the integral reduced Burau representation. This may be considered as a hyperelliptic analogue of the Mumford conjecture (Madsen-Weiss theorem) with twisted coefficients. We relate the result to the function field case of conjectures of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of families of quadratic L-functions. In particular, we formulate a purely topological homological stability conjecture, which when combined with our calculations would imply a precise asymptotic formula for all moments in the rational function field case.Comment: 91 page

On Hyperelliptic Abelian Functions of Genus 3

The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions.Comment: 40 page

On the distribution of orders of Frobenius action on ℓ\ell-torsion of abelian surfaces

The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$. The idea of Schoof's algorithm is to compute the trace of Frobenius $t$ modulo primes $\ell$ and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order $r$ of the Frobenius action on $E[\ell]$ and of restricting the number $t \pmod{\ell}$ to enumerate by using the formula $t^2 \equiv q (\zeta + \zeta^{-1})^2 \pmod{\ell}$. Here $\zeta$ is a primitive $r$-th root of unity. In this paper, we generalize Atkin's formula to the general case of abelian variety of dimension $g$. Classically, finding of the order $r$ involves expensive computation of modular polynomials. We study the distribution of the Frobenius orders in case of abelian surfaces and $q \equiv 1 \pmod{\ell}$ in order to replace these expensive computations by probabilistic algorithms