3
votes
1answer
40 views

Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
8
votes
0answers
107 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
7
votes
1answer
175 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
7
votes
1answer
298 views

Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...