Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed

Let $F_q$ be a finite field of characteristic $p=2,3$. We give the number of irreducible polynomials x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x] with $a_{m-1}$ and $a_{m-3}$ prescribed for any given $m$ if $p=2$, and with $a_{m-1}$ and $a_1$ prescribed for $m=1,...,10$ if $p=2,3$.Comment: 17 pages, Part of the results was presented at the Polynomials over Finite Fields and Applications workshop at Banff International Research Station, Canad

Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.Comment: 30 pages (AMSTeX). Minor typos are correcte

Weight enumerators of Reed-Muller codes from cubic curves and their duals

Let $\mathbb{F}_q$ be a finite field of characteristic not equal to $2$ or $3$. We compute the weight enumerators of some projective and affine Reed-Muller codes of order $3$ over $\mathbb{F}_q$. These weight enumerators answer enumerative questions about plane cubic curves. We apply the MacWilliams theorem to give formulas for coefficients of the weight enumerator of the duals of these codes. We see how traces of Hecke operators acting on spaces of cusp forms for $\operatorname{SL}_2(\mathbb{Z})$ play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math., 201

Irreducible polynomials over F2r\mathbb{F}_{2^r} with three prescribed coefficients

For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz.Comment: Incorporated referee comments. Accepted for publication in Finite Fields App