The Gras conjecture in function fields by Euler systems

Abstract

We use Euler systems to prove the Gras conjecture for groups generated by Stark units in global function fields. The techniques applied here are classical and go back to Thaine, Kolyvagin and Rubin. We obtain our Euler systems from the torsion points of sign-normalized Drinfel'd modules.

Computing class groups of function fields using stark units

“…The following lemma relates Kolyvagin derivative classes through the ϕ p map. [20] is stronger than what is stated here. It allows ℓ to divide q d∞ − 1 as long as ℓ does not divide [H m : k] and χ is of a certain form.…”

“…Let ord N (q) be the order of q in (Z/N Z) × . The following lemma without the requirement that P is of degree at most max{ord N (q), 2 log q (ℓ 4a+2 [F : k])} is proven in [20,Thm 4.7]. However, our computation needs the effective version stated below with the degree of P bounded.…”

“…Hayes [7] gave an explicit description of the α R in terms of the m-torsion points Λ ρ (m). In particular, λ q−1 m is a Stark unit in H m [7][ § 4,6] and S F is generated by µ(F ) and [20][ § 3] where f ⊂ O k ranges over non zero ideals and g ⊂ O k ranges over non trivial ideals coprime to f.…”

“…⊕ Z/d r Z which can be solved efficiently using the extended Euclidean algorithm. Since For every σ ∈ Gal(F (a)/F ) and every prime p dividing a, the class of Ψ(p) (σ−1)Da in F (a) × /(F (a) × ) N is fixed by Gal(F (a)/F ) [20,Lem 4.1].…”

“…g generates S F up to roots of unity and the product of two Euler systems is an Euler system, for every α ∈ S F , there exists an Euler system Ψ such that Ψ(e) = α[20, Cor 3.16]. If Ψ(e) = α, we call Ψ an Euler system starting from α.…”