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 weak version of the Gross conjecture here. 1. Overview of this paper Let k be a global field and K/k be a finite abelian extension with Galois group G. Let S be a finite nonempty set of places of k which contains all archimedean places and places ramified in K. Let T be a finite nonempty set disjoint from S. Let US,T be the set of all S-units which is congruent to 1 (mod p) for all places p ∈ T . The Dirichlet unit theorem asserts that the unit group US,T is a finitely generated abelian group with rank n = |S| − 1. In the function field case, US,T is furthermore free. By a careful choice of T (for example, T contains places of different characteristics) in the number field, one can also assume that US,T free. Let Y be the free abelian group generated by S and let X be the kernel of the degree map Y → Z. Then X is also a free abelian group with rank n. On one hand, consider the following function