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Ignazio Longhi

Publications and source records attributed to Ignazio Longhi.

18 recordsLinked to original sources
arXiv2026

Procounting measures and the Bateman--Horn conjecture

Let $D$ be the ring of $S$-integers in a global field and $\da$ its profinite completion. We propose a profinite version of the Bateman--Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations. Our approach is based on the new notion of procounting measure: a distribution on $\da$ which should be seen as a profinite analogue of the counting function for a subset of $\R$. This allows us to deal with subsets of $\da$ having Haar measure $0$ (corresponding to density zero in $\R$).

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arXiv2023

The algebra $\mathbb{Z}_\ell[[\mathbb{Z}_p^d]]$ and applications to Iwasawa theory

Let $\ell$ and $p$ be distinct primes, and let $\G$ be an abelian pro-$p$-group. We study the structure of the algebra $\L:=\Z_\ell[[\G]]$ and of $\L$-modules. The algebra $\L$ turns out to be a direct product of copies of ring of integers of cyclotomic extensions of $\Q_\ell$ and this induces a similar decomposition for a family of $\L$-modules. Inside this family we define Sinnott modules and provide characteristic ideals and formulas \`a la Iwasawa for orders and ranks of their quotients. When $\G\simeq \Z_p^d$\, is the Galois group of an extension of global fields, $\ell$-class groups and (duals of) $\ell$-Selmer groups provide examples of Sinnott modules and our formulas vastly extend results of L. Washington and W. Sinnott on $\ell$-class groups in $\Z_p$-extensions. Moreover, for global function fields of positive characteristic we use the specialization of a Stickelberger series to define an element in $\L$ which interpolates special values of Artin $L$-functions. With this element and the characteristic ideal of $\ell$-class groups we formulate an Iwasawa Main Conjecture for this setting and prove some special cases of it for relevant $\Z_p$-extensions.

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arXiv2023

Heights and transcendence of $p$--adic continued fractions

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.

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arXiv2022

Coset topologies on $\mathbb{Z}$ and arithmetic applications

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.

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arXiv2020

Densities on Dedekind domains, completions and Haar measure

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.

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arXiv2019

On the $\mu$-invariants of abelian varieties over function fields of positive characteristic

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.

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arXiv2015

Aspects of Iwasawa theory over function fields

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.

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arXiv2015

Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(θ)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[θ]\,$), 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 $β(j)$ (i.e., on the values of the Goss $ζ$-function at negative integers).

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arXiv2014

Characteristic ideals and Iwasawa theory

Let $Ł$ be a non-noetherian Krull domain which is the inverse limit of noetherian Krull domains $Ł_d$ and let $M$ be a finitely generated $Ł$-module which is the inverse limit of $Ł_d$-modules $M_d\,$. Under certain hypotheses on the rings $Ł_d$ and on the modules $M_d\,$, we define a pro-characteristic ideal for $M$ in $Ł$, 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 $\Gal(\calf/F)\simeq\Z_p^\infty$.

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arXiv2014

Characteristic ideals and Selmer groups

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.

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arXiv2014

Pontryagin duality for Iwasawa modules and abelian varieties

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 $A$.

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arXiv2013

On the Iwasawa Main conjecture of abelian varieties over function fields

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.

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arXiv2009

Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

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.

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arXiv2004

Teitelbaum's exceptional zero conjecture in the function field case

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.

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