Bruno Anglès

Arithmetic of positive characteristic L-series values in Tate algebras

The second author has recently introduced a new class of L-series in the arithmetic theory of function fields over finite fields. We show that the value at one of these L-series encode arithmetic informations of certain Drinfeld modules defined over Tate algebras. This enables us to generalize Andersons log-algebraicity Theorem and Taelmans Herbrand-Ribet Theorem.

Arithmetic of characteristic

Recently, the second author has associated a finite Fq[T]-module H to the Carlitz module over a finite extension of Fq(T). This module is an analogue of the ideal class group of a number field. In this paper, we study the Galois module structure of this module H for ‘cyclotomic’ extensions of Fq(T). We obtain function field analogues of some classical results on cyclotomic number fields, such as the p-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module H to Andersons module of circular units, and give a negative answer to Andersons Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second authors class number formula for the Carlitz module.

p

In this paper we give a bound for the Iwasawa lambda invariant of an abelian number field attached to the cyclotomic Z_p-extension of that field. We also give some properties of Iwaswa power series attached to p-adic L-functions.

special

In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤp-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained.

L

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

-values

We recover a result of Iwasawa on the p-adic logarithm of principal units of ℚp(ζpn+1) by studying the value at s = 1 of p-adic L-functions.