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We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in the 1970s and studied extensively from a Galois module structure point of view in…

We generalize some results of Greither and Popescu to a geometric Galois cover $X\rightarrow Y$ which appears naturally for example in extensions generated by $\mathfrak{p}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in…

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 fix motivic data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and an ableian $t$-module $E$, defined over a certain Dedekind subring of $F$. For this data,…

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module…

Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overline{\rho}: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the…

In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our…

Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois…

In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with…

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)…

Let $\ell$ and $p$ be distinct primes, and let $\G$ be an abelian pro-$p$-group. We study the structure of the algebra $\L:=\Z_\ell[[\G]]$ and of $\L$-modules. The algebra $\L$ turns out to be a direct product of copies of ring of integers…

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as…

We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we…

The authors establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. They consider fields K that are complete discrete…

For an abelian, CM extension $H/F$ of a totally real number field $F$, we improve upon the reformulation of the Equivariant Tamagawa Number Conjecture for the Artin motive $h_{H/F}$ by Atsuta-Kataoka in \cite{Atsuta-Kataoka-ETNC} and extend…

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue…

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…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$…

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark…