1
vote
1answer
156 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
3
votes
2answers
132 views

Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...
6
votes
1answer
658 views

Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...
4
votes
2answers
276 views

Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$
3
votes
1answer
348 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $ $ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $ with both good ...
3
votes
2answers
201 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$ 2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ ...
2
votes
1answer
355 views

$\lambda$-invariant is constant for isogenous elliptic curves

How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$
13
votes
3answers
918 views

rational points on algebraic curves over $Q^{ab}$

Motivation: Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension ...
10
votes
2answers
1k views

Are Kato's zeta elements integral?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular ...
12
votes
1answer
705 views

P-adic L-functions of nonabelian twists of elliptic curves

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
17
votes
4answers
1k views

Does p-adic $L$- function determine the $L$ function

Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic ...