المؤلفون: Bandini, Andrea, Longhi, Ignazio

مصطلحات موضوعية: Mathematics - Number Theory, 11R23, 11R29

الوصف: 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

الوصول الحر: http://arxiv.org/abs/2312.04666Test

المؤلفون: Longhi, Ignazio, Murru, Nadir, Saettone, Francesco Maria

مصطلحات موضوعية: Mathematics - Number Theory, 11J70, 11J87

الوصف: 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: 13 pages, correction of minor mistakes in the previous version

الوصول الحر: http://arxiv.org/abs/2302.04017Test

المؤلفون: Longhi, Ignazio, Mu, Yunzhu, Saettone, Francesco Maria

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - General Topology

الوصف: 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

الوصول الحر: http://arxiv.org/abs/2202.13478Test

المؤلفون: Demangos, Luca, Longhi, Ignazio

مصطلحات موضوعية: Mathematics - Number Theory

الوصف: 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

الوصول الحر: http://arxiv.org/abs/2009.04229Test

المؤلفون: Lai, King-Fai, Longhi, Ignazio, Suzuki, Takashi, Tan, Ki-Seng, Trihan, Fabien

المصدر: Alg. Number Th. 15 (2021) 863-907

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 11R23, 11G10, 11S40, 14J27

الوصف: 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

الوصول الحر: http://arxiv.org/abs/1909.00511Test

المؤلفون: Demangos, L, Longhi, Ignazio

المساهمون: Demangos, L, Longhi, Ignazio

مصطلحات موضوعية: Densitie, Haar measure, global field, k-free values of polynomials

الوصف: In this work we investigate the general relation between the density of a subset of the ring of integers D of a general global field and the Haar measure of its closure in the profinite completion (D) over cap. We then study a specific family of sets, the preimages of k-free elements (for any given k is an element of N\{0, 1}) via one variable polynomial maps, showing that under some hypotheses their asymptotic density always exists and it is precisely the Haar measure of the closure in (D) over cap of their set.

العلاقة: info:eu-repo/semantics/altIdentifier/wos/WOS:000673241100001; volume:45; issue:9; firstpage:1373; lastpage:1397; numberofpages:25; journal:QUAESTIONES MATHEMATICAE; https://hdl.handle.net/2318/1931632Test; info:eu-repo/semantics/altIdentifier/scopus/2-s2.0-85110903874

الإتاحة: https://doi.org/10.2989/16073606.2021.1945701Test
https://hdl.handle.net/2318/1931632Test

المؤلفون: Demangos, Luca, Longhi, Ignazio

المصدر: Mathematische Zeitschrift ; volume 306, issue 2 ; ISSN 0025-5874 1432-1823

مصطلحات موضوعية: General Mathematics

الإتاحة: https://doi.org/10.1007/s00209-023-03415-2Test

المؤلفون: Anglès, Bruno, Bandini, Andrea, Bars, Francesc, Longhi, Ignazio

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 11M38, 11R23, 11G20, 14G10, 11R60, 11G45, 11G40, 14F30

الوصف: 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

الوصول الحر: http://arxiv.org/abs/1412.5957Test

المؤلفون: Bandini, Andrea, Bars, Francesc, Longhi, Ignazio

المساهمون: Gebhard Böckle, David Go, Urs Hartl, Matthew Papanikolas, Bandini, Andrea, Bars, Francesc, Longhi, Ignazio

العلاقة: ispartofseries:EMS Series of Congress Reports; ispartofbook:t-motives: Hodge structures, transcendence and other motivic aspects; firstpage:375; lastpage:416; numberofpages:42; alleditors:Gebhard Böckle; David Goss; Urs Hartl; Matthew Papanikolas; https://hdl.handle.net/2318/1931636Test

الإتاحة: https://hdl.handle.net/2318/1931636Test

المؤلفون: Lai, King Fai, Longhi, Ignazio, Tan, Ki-Seng, Trihan, Fabien

مصطلحات موضوعية: Mathematics - Number Theory

الوصف: 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

الوصول الحر: http://arxiv.org/abs/1406.6128Test