# Iwasawa Theory for $p$-adic Representations

@inproceedings{Greenberg1989IwasawaTF, title={Iwasawa Theory for \$p\$-adic Representations}, author={Ralph Greenberg}, year={1989} }

Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet L-functions) and the ideal class groups of certain towers of cyclotomic fields. Probably the first hint of such a link is Kummer's well-known criterion for irregularity of primes. In Iwasawa's theory one defines for each prime p certain modules over the Iwasawa algebra A (which we will describe in Section…

## 231 Citations

### p-ADIC L-FUNCTIONS FOR GALOIS DEFORMATIONS AND RELATED PROBLEMS ON PERIODS

- Mathematics
- 2006

At the first half of this article, we present a conjecture (cf. Conjecture 1.10) to associate “the p-adic L-function” to a family of Galois representation. In recent years, we have plenty of examples…

### Iwasawa Theory for Artin Representations, I

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This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main…

### On Iwasawa invariants of modular forms with reducible and non-$p$-distinguished residual Galois representations

- Mathematics
- 2022

. In the present paper, we study the p -adic L -functions and the (strict) Selmer groups over Q ∞ , the cyclotomic Z p -extension of Q , of the p -adic weight one cusp forms f , obtained via the p…

### Characteristic elements in noncommutative Iwasawa theory

- Mathematics
- 2003

Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not…

### Haruzo Hida ’ s p-adic automorphic forms on Shimura varieties by

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- 2006

Three topics figure prominently in the modern higher arithmetic: zeta-functions, Galois representations, and automorphic forms or, equivalently, representations. The zeta-functions are attached to…

### The algebraic p-adic L-function and isogeny between families of Galois representations

- Mathematics
- 2008

### Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms

- Mathematics
- 2018

In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a…

### Towards a Twist Conjecture in Non-Commutative Iwasawa Theory

- Mathematics
- 2014

In this thesis we study three conjectures of K. Kato. The first one concerns p-adic Lie extensions F/Q with Galois group G containing the cyclotomic Z_p-extension Q_cyc and the existence of an…

### Special values of anticyclotomic $L$-functions

- Mathematics
- 2003

The object of this paper is to extend the results and methods of [Vat01], where it was shown how cases of a conjecture of Mazur on the behavior of L-functions in an anticyclotomic Zp-extension could…

### Ju l 2 02 1 Exterior Powers in Iwasawa Theory

- Mathematics
- 2021

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is…