The Gross conjecture over rational function fields

摘要

We study the Gross conjecture for the cyclotomic function field extension k(Λ_f)/k where k = F_g(t) is the rational function field and f is a monic polynomial in F_q[t]. We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.