scholarly journals On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

2021 ◽  
Author(s):  
Francesc Castella ◽  
Giada Grossi ◽  
Jaehoon Lee ◽  
Christopher Skinner
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Theory

EULER CHARACTERISTICS AS INVARIANTS OF IWASAWA MODULES

2002 ◽  
Vol 85 (3) ◽  
pp. 634-658 ◽  
Author(s):  
SUSAN HOWSON
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Lie Group ◽  
Iwasawa Theory ◽  
Subject Classification ◽  
Finitely Generated ◽  
Euler Characteristics ◽  
Primary 16E10 ◽  
Iwasawa Algebra

If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

scholarly journals On the μ-invariants in Iwasawa theory of elliptic curves

2004 ◽  
Vol 30 (1) ◽  
pp. 1-28
Author(s):  
Yoshitaka HACHIMORI
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Theory

scholarly journals Iwasawa theory for modular forms at supersingular primes

2011 ◽  
Vol 147 (3) ◽  
pp. 803-838 ◽  
Author(s):  
Antonio Lei
Keyword(s):  
Elliptic Curves ◽  
Modular Forms ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Main Conjecture ◽  
Supersingular Primes

AbstractWe generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

ON μ-INVARIANTS OF SELMER GROUPS OF SOME CM ELLIPTIC CURVES

2013 ◽  
Vol 09 (05) ◽  
pp. 1199-1214
Author(s):  
CHANDRAKANT S. ARIBAM
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Torsion Points

We show that the Selmer groups defined over the cyclotomic ℤ3-extension of the field of 3-torsion points of the CM elliptic curves 256a1, 256a2, 256d1, 256d2, 121b1, 121b2 have μ-invariant equal to zero, verifying a conjecture in non-commutative Iwasawa theory.

scholarly journals CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH SUPERSINGULAR REDUCTION

2019 ◽  
Vol 7 ◽  
Author(s):  
ANTONIO LEI ◽  
BHARATHWAJ PALVANNAN
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Numerical Evidence ◽  
Codimension Two ◽  
Main Conjecture ◽  
Iwasawa Algebra

A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s $2$ -variable $p$ -adic $L$ -functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a $2$ -variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$ , defined over  $\mathbb{Q}$ , with good supersingular reduction at $p$ . On the analytic side, we consider eight pairs of $2$ -variable $p$ -adic $L$ -functions in this setup (four of the $2$ -variable $p$ -adic $L$ -functions have been constructed by Loeffler and a fifth $2$ -variable $p$ -adic $L$ -function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\mathbb{Z}_{p}^{2}$ -extension of $K$ . We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.

Iwasawa Theory of Modular Elliptic Curves of Analytic Rank at most 1

1994 ◽  
Vol 50 (2) ◽  
pp. 243-264 ◽  
Author(s):  
John Coates ◽  
Gary McConnell
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Theory
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