Basic Algorithms for Rational Function Fields

Algorithms for 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.

Algorithms for Rational Agents

Class numbers of some abelian extensions of rational function fields

Let P be a monic irreducible polynomial. In this paper we generalize the determinant formula for h(K Pn) of Bae and Kang and the formula for h−(KPn ) of Jung and Ahn to any subfields K of the cyclotomic function field KPn . By using these formulas, we calculate the class numbers h −(K), h(K+) of all subfields K of KP when q and deg(P ) are small.