Statistics for products of traces of high powers of the frobenius class of hyperelliptic curves

We study the averages of products of traces of high powers of the Frobenius class of hyperelliptic curves of genus g over a fixed finite field. We show that for increasing genus g, the limiting expectation of these products equals to the expectation when the curve varies over the unitary symplectic group USp(2g). We also consider the scaling limit of linear statistics for eigenphases of the Frobenius class of hyperelliptic curves, and show that their first few moments are Gaussian

Square-full polynomials in short intervals and in arithmetic progressions

We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_{q}[T]$ of polynomials over a finite field $F_{q}$ of $q$ elements, in the limit $q\rightarrow\infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them

Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree <i>0</i>-functions in F_<i>q</i>[<i>t</i>]

We compute the variances of sums in arithmetic progressions of generalised -divisor functions related to certain -functions in q[], in the limit as q → ∞. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q → ∞, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual -divisor function, when the -function in question has degree one. They illustrate the role played by the degree of the -functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over q[], and we illustrate them by examining in some detail the generalised -divisor functions associated with the Legendre curve

Moments of quadratic twists of elliptic curve L-functions over function fields

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.Comment: 32 page

Sums of divisor functions in <b>F</b>_q[<i>t</i>] and matrix integrals

We study the mean square of sums of theandnbsp;kth divisor functionandnbsp;dk(n)andnbsp;over short intervals and arithmetic progressions for the rational function field over a finite field ofandnbsp;qandnbsp;elements. In the limit asandnbsp;qandrarr;andinfin;andnbsp;we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums ofandnbsp;dk(n)andnbsp;in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.</p