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Let $\ell$ and $p$ be distinct primes, and let $\Gamma$ be an abelian pro-$p$-group. We study the structure of the algebra $\Lambda:=\mathbb{Z}_\ell[[\Gamma]]$ and of $\Lambda$-modules. In the case $\Gamma\simeq \mathbb{Z}_p^d$, we consider a $\mathbb{Z}_p^d$-extension $K/k$ of a global field $k$ and use the structure theorems to provide explicit formulas for the orders and $\ell$-ranks of certain Iwasawa modules (namely $\ell$-class groups and $\ell$-Selmer groups) associated with the finite subextensions of $K$. We apply this new approach to provide different proofs and generalizations of results of Washington and Sinnott on $\ell$-class groups., Comment: (Very) Preliminary version. Comments are welcome

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions., Comment: final version (with erratum), to appear in "Annali di Matematica Pura e Applicata"

We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on $\mathbb{Z}$, obtained from closed sets of the profinite completion $\hat{\mathbb{Z}}$. We also discuss various applications to number theory., Comment: Final version; to appear in Expositiones Mathematicae

Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is equal to the Haar measure of its closure in $\hat{D}$. In order to have a precise statement, we give a general definition of density which encompasses the most commonly used ones. Using it we provide a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. In another direction, we extend the Davenport-Erd\H{o}s theorem to every $D$ as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any $D$ the set of elements divisible by at most $k$ distinct primes has density 0 for any natural number $k$. Finally, we show that the closure of the set of prime elements of $D$ is the union of the group of units of $\hat{D}$ with a negligible part., Comment: 41 pages, no figures. Final version, accepted in Mathematische Zeitschrift

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $\mu$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-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 $\mu$-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 $\mu$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$\mu=0$" 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., Comment: Accepted for publication in Algebra & Number Theory. No changes in the text from v3. 47 pages

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(\theta)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[\theta]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss $\zeta$-function at negative integers)., Comment: Section 3 entirely rewritten

We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$., Comment: arXiv admin note: substantial text overlap with arXiv:1205.5945

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), 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 $A$., Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1205.5945

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1205.5945

Let $A$ be an abelian variety defined over a global field $F$ of positive characteristic $p$ and let $\calf/F$ be a $\Z_p^{\N}$-extension, unramified outside a finite set of places of $F$. Assuming that all ramified places are totally ramified, we define a pro-characteristic ideal associated to the Pontrjagin dual of the $p$-primary Selmer group of $A$, in order to formulate an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$ (which we also prove for a constant abelian variety). To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a $\Z_p^d$-extension $\calf_d/F$ and for any $\Z_p^{d-1}$-extension contained in $\calf_d\,$, and then use a limit process., Comment: 10 pages, version updated to be compatible with the modifications of arXiv:1310.0680 [math.NT]

Let $\L$ be a non-noetherian Krull domain which is the inverse limit of noetherian Krull domains $\L_d$ and let $M$ be a finitely generated $\L$-module which is the inverse limit of $\L_d$-modules $M_d\,$. Under certain hypotheses on the rings $\L_d$ and on the modules $M_d\,$, we define a pro-characteristic ideal for $M$ in $\L$, which should play the role of the usual characteristic ideals for finitely generated modules over noetherian Krull domains. We apply this to the study of Iwasawa modules (in particular of class groups) in a non-noetherian Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$, where $F$ is a function field of characteristic $p$ and $\Gal(\calf/F)\simeq\Z_p^\infty$., Comment: 15 pages, substantial chenges in exposition, new section 2.3

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation., Comment: 80 pages; many relevant changes all over the paper from v1. Among the most significant ones: new introduction; proof of the functional equation for Gamma systems in more cases and some applications to CM abelian varieties

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for $\mathbb{Z}_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism., Comment: Final version. To appear in Conference proceedings of the conference t-motives: Hodge structures, transcendence and other motivic aspects at Banff International Research Station (BIRS) by EMS

This paper is a brief survey on explicit reciprocity laws of Artin-Hasse-Iwasawa-Wiles type for the Kummer pairing on local fields., Comment: To appear in the Proceedings of the Segundas Jornada de Teor\'ia de N\'umeros, in the Library of Rev.Mat. Iberoamericana

Let $F$ be a function field of characteristic $p>0$, $\F/F$ a Galois extension with $Gal(\F/F)\simeq \Z_l^d$ (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of $\F$ via appropriate versions of Mazur's Control Theorem. As a consequence we prove that $Sel_E(\F)_r$ is a cofinitely generated (in some cases cotorsion) $\Z_r[[Gal(\F/F)]]$-module., Comment: Final version to appear in Annales de l'Institut Fourier

We introduce the formalism of Coleman's power series for rank 1 Drinfeld modules and apply it to formulate and prove the analogue of Wiles' explicit reciprocity law in this setting., Comment: To appear in JNT

The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained., Comment: 31 pages