arXiv subjects

Filippo A. E. Nuccio

Publications and source records attributed to Filippo A. E. Nuccio.

8 recordsLinked to original sources

arXiv2023

Cohomology of normic systems and fake Z_p extensions

We set up a general framework to study Tate cohomology groups of Galois modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along $\mathbb{Z}_p$-extensions. As an application, we study the growth of class numbers along what we call "fake $\mathbb{Z}_p$-extensions of dihedral type". This study relies on a previous work, where we established a class number formula for dihedral extensions in terms of the cohomology groups of the units.

math.NT↗

arXiv2022

A formalization of Dedekind domains and class groups of global fields

Dedekind domains and their class groups are notions in commutative algebra that are essential in algebraic number theory. We formalized these structures and several fundamental properties, including number theoretic finiteness results for class groups, in the Lean prover as part of the mathlib mathematical library. This paper describes the formalization process, noting the idioms we found useful in our development and mathlib's decentralized collaboration processes involved in this project.

cs.LO↗

arXiv2021

Residual supersingular Iwasawa theory and signed Iwasawa invariants

For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a fine signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.

math.NT↗

arXiv2020

Coleman Map in Coleman Families

In this paper, we aimed at constructing a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result would be a non-ordinary generalization of a two-variable Coleman map for a given Hida deformation obtained by the second-named author.

math.NT↗

arXiv2019

Class number formula for dihedral extensions

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree $2q$, where $q$ is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and recovers similar formulas which have appeared in the literature as special cases. As a corollary of our main result we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

math.NT↗

arXiv2019

Représentations galoisiennes diédrales et formes à multiplication complexe

Pour une représentation galoisienne diédrale en caractéristique l on établit (sous certaines hypothèses) l'existence d'une newform à multiplication complexe, dont on contrôle le poids, le niveau et le caractère, telle que la représentation l-adique associée est congrue modulo l à celle de départ. Given a dihedral Galois representation in characteristic l, we establish (under some assumption) the existence of a CM newform, whose weight, level and Nebentypus we pin down, such that its l-adic representation is congruent modulo l to the one we started with.

math.NT↗

arXiv2008

Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.

math.NT↗

arXiv2008

On Jacobi Sums in $\mathbb Q(ζ_p)$

We study the p-adic behavior of Jacobi Sums for $\mathbb Q(ζ_p)$ and link this study to the p-Sylow subgroup of the ideal class group of $\mathbb Q(ζ_pà^+$

math.NT↗