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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 $ K=\mathbb{Q}_{\infty} $ be the cyclotomic $ \mathbb{Z}_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X_{E}(\mathbb{Q}_{\infty}) $ of $ Sel_{E}(\mathbb{Q}_{\infty})_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism

$$ X_{E}(\mathbb{Q}_{\infty}) \sim (\bigoplus_{i=1}^{s}\Lambda/(f_{i}(T)^{a_{i}}))\bigoplus(\bigoplus_{j=1}^{t}\Lambda/(p^{\mu_j})) $$

Are there any known methods for computing these decompositions for elliptic curves over $\mathbb{Q}$ $?$

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1) 91b3 is an example with $\mu=2$ at $p=3$. Recall that the $\mu$-invariant (with respect to a prime $p$) only changes when there is an isogeny of degree $p$. More precisely it changes just be the quotient of the real Néron periods. Hence we are looking here for curves with cyclic isogenies of degree 9 defined over $\mathbb{Q}$ whose kernel is in $E(\mathbb{R})$. I guess one could parametrise them.

2) You have to assume that the reduction of $E$ is ordinary if you want the Selmer group to be torsion. Using the (know proven) main conjecture, one can get the characteristic series of $X_E$ to any precision (using sage, magma, ... all based on Pollacks earlier work for his tables). In practice this allows one to determine the decomposition very often. First note that there is (conjecturally) a curve in the isogeny class for which the $\mu$-invariant is zero. Better work with this one as you then know that $\mu=0$ by computing sufficiently many coefficients. Now the only trouble could come from repeated factors. Of course very often the characteristic series is a square-free polynomial. Conjecturally again the entries $a_i$ should all be zero, but it may be hard to prove that. Usually however, some information about the curve in the first few layers of the $\mathbb{Z}_p$-extension will suffice.

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  • $\begingroup$ I am trying to check the $\mu$-invariant of 91b3 in SAGE. \\ Input - E = EllipticCurve('91b3') E.padic_lseries(3).series(4) \\ Output - O(T^5) \\ Can you please tell me why am I getting this error ? $\endgroup$
    – Suman
    Commented Jan 5, 2014 at 19:54
  • $\begingroup$ This is not an error. You need to increase the precision .series(7) to get the coefficients. Of course, it is better to copmute the series for 91b1 and then to multiply with the change of periods, which in this case is 9. $\endgroup$
    – Chris Wuthrich
    Commented Jan 5, 2014 at 21:15
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Regarding 1), Robert Pollack's tables - http://math.bu.edu/people/rpollack/Data/data.html - will be probably useful.

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  • $\begingroup$ They are useful, but unfortunately not for the precise questions above. They only contain the information for the optimal curve, which is not going to have $\mu=2$. Also it does not tell you about all $f_i$ only the cyclotomic factors and the total degree $\lambda$. $\endgroup$
    – Chris Wuthrich
    Commented Jan 5, 2014 at 17:27

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