Note sur la conjecture de Leopoldt
英文摘要:
We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units
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Jean-François Jaulent
We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We use logarithmic {\ell}-class groups to take a new view on Greenberg's conjecture about Iwasawa {\ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldt's conjecture, we prove that Greenberg's conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{\ell}-extensions of K. As an illustration of our approach, in the special case where the prime {\ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group …
We compute the capitulation kernel for the module of Bertrandias-Payan in an arbitrary $\ell$-extension of number fields which satisfies the Leopoldt conjecture. As a consequence we relate the existence of extensions with trivial such module to the classical problem of locally cyclotomic towers.
In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.
In the present article, we study the conjecture of Sharifi on the surjectivity of the map $\varpi_θ$. Here $θ$ is a primitive even Dirichlet character of conductor $Np$, which is exceptional in the sense of Ohta. After localizing at the prime ideal $\mathfrak{p}$ of the Iwasawa algebra related to the trivial zero of the Kubota\textendash Leopoldt $p$-adic $L$-function $L_p(s,θ^{-1}ω^2)$, we compute the image of $\varpi_{θ,\mathfrak{p}}$ in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of…
We show that the assumption of a weak form of the Hardy-Littlewood conjecture on the Goldbach problem suffices to disprove the possible existence of exceptional zeros of Dirichlet L-functions. This strengthens a result of the authors named in the title.
We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field
We associate to a filtered $φ$-module $D$ a sub-$Z_p[[x]]$-module of convergent series on the open unit disk in which the $p$-adic $L$-functions of the Galois representation associated to $D$ and the $Z_p$-cyclotomic extension live (if they exist). This generalizes the already known case where $D$ is associated to an elliptic curve or a modular form.
We survey the classical results on the prime number theorem
The article revisits a result of Antonio Lei, David Loeffler, and Sarah Livia Zerbes concerning the structure of the image by the Iwasawa regulator map of the Iwasawa module associated with a semi-stable $p$-adic representation on an unramified finite extension of $Q_p$ and give a direct proof based on the results of "Représentations $p$-adiques et normes universelles: I. le cas cristallin" in the crystalline case and of "Théorie d'Iwasawa des représentations $p$-adiques semi-stables" in the semi-stable case.
Let $α\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^α$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.
We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group.
The limit points of the bass notes of arithmetic hyperbolic surfaces
本文证明了算术双曲曲面的低音音符的极限点恰好构成区间[0,1/4]。
Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.
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