On Jacobi Sums in Q(ζp)
نویسندگان
چکیده
We study the p-adic behavior of Jacobi sums for Q(ζp) and link this study to the p-Sylow subgroup of the class group of Q(ζp) + and to some properties of the jacobian of the Fermat curve Xp + Y p = 1 over Fl 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(ζp) 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(ζp) ∗ generated by Jacobi sums. Let A be the p-Sylow subgroup of the class group of Q(ζp). If E denotes the group of units of Q(ζp), then if Vandiver’s Conjecture is true for p, by Kummer Theory, we must have A − pA →֒ Gal(Q(ζp)( √ E)/Q(ζp)). 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(ζp) ∗ which was already considered
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