Related papers: A Note on the Main Conjecture over Q

The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that…

This paper sets up a framework to organize anticyclotomic Iwasawa theory in the context of the Gan-Gross-Prasad conjecture for unitary groups. We propose multiple main conjectures depending on archimedean weight interlacing conditions,…

We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd…

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…

Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(\Gamma_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois…

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $p$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $K$ an imaginary quadratic field with all primes dividing $Np$ split.…

Let $p$ be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional $p$-adic Lie extension whose Galois group has an abelian Sylow $p$-subgroup.…

The main conjecture of Iwasawa theory is a conjecture on the relation between a Selmer group and a conjectural $p$-adic $L$-function. This conjectural $p$-adic $L$-function is expected to satisfy a conjectural functional equation in a…

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 this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient…

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…

Let $K$ be a imaginary quadratic field where the prime $p$ splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for $p$-nearly-ordinary families associated to $\mathrm{GL}_2\times…

Let $E/\mathbb{Q}$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$,…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex L-functions, typified by the conjecture of…

Assuming that Iwasawa's $\mu_{K/k}$-invariant vanishes, we prove the 'main conjecture' of equivariant Iwasawa theory, at odd prime numbers $l$, for arbitrary extensions $K/k$ of totally real number fields, up to its uniqueness assertion.

We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We…

The goal of this article is to obtain a proof of the Main conjectures of Iwasawa theory for rational elliptic curves over anticyclotomic extensions of imaginary quadratic fields, under mild arithmetic assumptions, both in the case where the…

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places.