On Jacobi Sums in $\mathbb Q(ζ_p)$

Abstract

We study the p-adic behavior of Jacobi sums for Q(ζ p ) and link this study to the p-Sylow subgroup of the class group of Q(ζ p ) + and to some properties of the jacobian of the Fermat curve X p + Y p = 1 over F ℓ where ℓ is a prime number distinct from p.Let p be a prime number, p ≥ 5. Iwasawa has shown that the p-adic properties of Jacobi sums for Q(ζ p ) are linked to Vandiver's Conjecture (see [5]). In this paper, we follow Iwasawa's ideas and study the p-adic properties of the subgroup J of Q(ζ p ) * generated by Jacobi sums.Let A be the p-Sylow subgroup of the class group of Q(ζ p ). If E denotes the group of units of Q(ζ p ), then if Vandiver's Conjecture is true for p, by Kummer Theory, we must have. Note that J is analoguous for the odd part to the group of cyclotomic units for the even part. We introduce a submodule W of Q(ζ p ) * which was already considered