Related papers: Abstract $\ell$-adic $1$-motives and Tate's class

In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for…

In Part I we review some specific properties of the $\Lambda$-modules in Iwasawa theory, which add structure to the general properties of Noetherian $\Lambda$-torsion modules. Part II deals with Kummer theory and gives a detailed…

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible \'etale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with…

Given an odd prime number $p$ and a $p$-stabilized Artin representation $\rho$ over $\mathbb{Q}$, we introduce a family of $p$-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a $p$-adic Stark conjecture which…

We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these…

We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field…

We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods.…

The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial)…

For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal…

We explain how to reconstruct the category of Artin-Tate $\mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an…

In this paper, we prove the $\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\ell$-adic variant of a result of Matsumi [13].

We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative K-groups. Our Tamagawa numbers lie in an…

In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest…

This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number…

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the…

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis…

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…

Using integral $p$-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness…

We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main…

We show that the cyclotomic conjecture on the characteristic polynomial of T-ramified S-split Iwasawa modules introduced in a previous paper and satisfied by abelian fields governs the Z${\ell}$-rank of the submodule of fixed points for all…