Punctured Distributions in the Rational Function Fields

The Gross Conjecture over Rational Function Fields

We study the Gross Conjecture on the cyclotomic function field extension k(Λf )/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t]. We show the conjecture in the Fermat curve case(i.e., when f = t(t− 1)) by direct calculation. We also prove the case when f is irreducible which is analogous to Weil’s reciprocity law. In the general case, we manage to show the w...

Multiple Dirichlet Series over Rational Function Fields

We explicitly compute some double Dirichlet series constructed from n order Gauss sums over rational function fields. These turn out to be rational functions in q−s1 and q−s2 , where q is the size of the constant field. Key use is made of the group of 6 functional equations satisfied by these series.

Irregularities in the Distributions of Primes in Function Fields

We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals.

The Growth of Valuations on Rational Function Fields

Fq-rational places in algebraic function fields using Weierstrass semigroups

We present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of theWeierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields. © 2008 Elsevier B.V. All rights reserved.