Related papers: \ell-adic class field theory for regular local rin…

In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to…

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism \rho: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class…

We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent…

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian…

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local…

The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen--Lenstra--Martinet heuristic on ideal class groups. To that end, we show that…

We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing…

Given a real abelian field F with group G and an odd prime number {\ell}, we define the circular subgroup of the pro-{\ell}-group of logarithmic units and we show that for any Galois morphism $\rho$ from the pro-{\ell}-group of logarithmic…

It is shown that the Galois closure of the henselization of a one dimensional local field arising in geometric and arithmetic situation is separably closed.

This article investigates congruences of $\mathfrak{p}$-adic representations arising from effective $A$-motives defined over a global function field $K$. We give a criterion for two congruent $\mathfrak{p}$-adic representations coming from…

A field $K$ is quasi-classical $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ Henselian admissible discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of…

We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.

Neukirch developed abstract class field theory in his famous book "Class Field Theory". We show that it is possible to derive Jaulent's '-adic class field from Neukirch's framework. The proof requires in both cases (local case and global…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\ell$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a functional equation for the resulting…

We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen's proof of Mazur's Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with…

The $\ell$-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in $\ell$-adic numbers, which arises from Galois actions on $\ell$-adic \'etale paths on ${\mathbb P}^1 \backslash \{0,1,\infty\}$. In…

Let $K$ be a $p$-adic local field. In this work we study a special kind of $p$-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms.…

The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This version deals with a more general situation that is a priori much more difficult. In this paper, we…