On the μ-invariants in Iwasawa theory of elliptic curves

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.

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Iwasawa theory for modular forms at supersingular primes

Compositio Mathematica ◽

10.1112/s0010437x10005130 ◽

2011 ◽

Vol 147 (3) ◽

pp. 803-838 ◽

Cited By ~ 10

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.

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

International Journal of Number Theory ◽

10.1142/s1793042113500206 ◽

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.

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Iwasawa theory of elliptic curves at supersingular primes over ℤ p -extensions of number fields

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle.2006.069 ◽

2006 ◽

Vol 2006 (598) ◽

Cited By ~ 1

Author(s):

Adrian Iovita ◽

Robert Pollack

Keyword(s):

Elliptic Curves ◽

Number Fields ◽

Iwasawa Theory ◽

Supersingular Primes

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

Forum of Mathematics Sigma ◽

10.1017/fms.2019.17 ◽

2019 ◽

Vol 7 ◽

Cited By ~ 1

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.

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