Carlitz a défini pour les corps de fonctions l’analogue du réel et Goss l’analogue des fonctions de Dirichlet. Nous prouvons dans un cas particulier qu’il existe des valeurs entières et des caractères pour lesquels peut être rationnel, algébrique ou bien transcendant.
We describe here two sets of generators of an ideal , of finite index inside the square of the augmentation ideal of , associated to the Dirichlet character of the finite group . That peculiar ideal first appeared in questions related to the computation of class number formulas for abelian non ramified extensions of -fields cf. [2] and [3], satisfying certain special conditions which are outlined in the introduction of [1]. A rough idea of these formulas is given in §§2 and 6.
In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the...
Let be an absolutely simple abelian variety over a number field; we study whether the reductions tend to be simple, too. We show that if is a definite quaternion algebra, then the reduction is geometrically isogenous to the self-product of an absolutely simple abelian variety for in a set of positive density, while if is of Mumford type, then is simple for almost all . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...
1. Motivation. Let J₀(N) denote the Jacobian of the modular curve X₀(N) parametrizing pairs of N-isogenous elliptic curves. The simple factors of J₀(N) have real multiplication, that is to say that the endomorphism ring of a simple factor A contains an order in a totally real number field of degree dim A. We shall sometimes abbreviate "real multiplication" to "RM" and say that A has maximal RM by the totally real field F if A has an action of the full ring of integers of F. We say that a...
For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group is a subgroup of the Galois cohomology group , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which is isomorphic...