BENOIT TRIHAN FABIEN

Let A be an abelian variety over a global function field K of characteristic p. We study the µ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate–Shafarevich group (which is now the µ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse–Weil L-function of A/K assuming the conjectural Birch–Swinnerton-Dyer formula. Our next result is to prove this µ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the “µ = 0” locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic p > 0 under a semisimplicity hypothesis.

We establish the Iwasawa main onje ture for semistable abelian varieties over a function field of charateristi p under certain restrictive assumptions. Namely we consider p-torsion free p-adic Lie extensions of the base field which contain the constant Zp-extension and are everywhere unramifield. Under the usual μ = 0 hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of p-adic cohomology [TV] together with the trace formulas of [EL1].

We prove a functional equation for two projective systems of finite abelian p-groups, {an} and {abn}, endowed with an action of ℤdp such that an can be identified with the Pontryagin dual of bn for all n. Let K be a global field. Let L be a ℤdp-extension of K (d ≥ 1), unramified outside a finite set of places. Let A be an abelian variety over K. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of A.

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over Z(p)(d)-extensions of function fields ramifying at a finite set of places.

We prove the Iwasawa Main Conjecture over the arithmetic -extension for semistable abelian varieties over function fields of characteristic .

Let A/K be an elliptic curve over a global field of characteristic p &gt 0. We provide an example where the Pontrjagin dual of the Selmer group of A over a Γ := ℤp-extension L/K is not a torsion ℤp[[Γ]]-module and show that the Iwasawa Main Conjecture for A/L holds nevertheless.

Let A/K be an abelian variety over a function field of characteristic p &gt 0 and let l be a prime number (l = p allowed). We prove the following: the parity of the corank rl of the l-discrete Selmer group of A/K coincides with the parity of the order at s = 1 of the Hasse-Weil L-function of A/K. We also prove the analogous parity result for pure l-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin-Tate conjecture. © The Author(s) 2014.

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p > 3. We also present partial results on the l-parity conjecture for primes l not equal p.

<title>Abstract</title>We study a (<italic>p</italic>-adic) geometric analogue for abelian varieties over a function field of characteristic <italic>p</italic> of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

We show that the Dieudonne crystal associated to a Barsotti-Tate group with potentially semistable reduction over a smooth curve is overconvergent. As a corollary, we obtain the rationality of the L-function associated to this group.

Let X be a proper smooth curve over a perfect field of characteristic p > 0 and U an open dense subscheme of X. We prove that convergent F-isocrystals on U are overconvergent under the condition that they are overconvergent at each point in X/U. Using this criterion, we show that the higher direct images Rifcrys* script O sign V by a proper smooth morphism of schemes f : V → U are overconvergent.

We generalize the study of the syntomic cohomology of [17] to the case of open varieties and with coefficients.

We give an expression of the zeroes and poles of the L-function of an overconvergent F-isocrystal defined on an open variety when the geometric situation lifts to the characteristic 0 and under the hypothesis that the connexion of the isocrystal is regular at infinity and satisfies a condition on the residues. © 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

We give m? expression of the zeroes and poles of the L-function of an overconvergent F-isocrystal defined on an open variety when the geometric situation lifts to the characteristic 0 and under the hypothesis that the connexion of the isocrystal is regular at infinity and satisfies a condition on the residues. (C) 2001 Academie des sciences/Editions scientifiques ct medicales Elsevier SAS.

Let X a proper and smooth scheme over a finite field, D a divisor with normal crossings and U the complementary of D in X. We give a description of the zeroes and poles of the Zeta function of these three schemes, when they lift to the characteristic 0. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.

The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonné crystalline theory ([B-B-M]) to "crystal of level m" in the sense of Berthelot. We show that the unit-root L-function of the Dieudonné crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.

The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonne crystalline theory ([B-B-M]) to "crystal of level m" in the sense of Berthelot. We show that the unit-root L-function of the Dieudonne crystal associated to G can be expressed in terms of the syntomic cohomology of the Err group of G by the constant sheaf.