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Jan 2, 2018 at 21:47 comment added David E Speyer Whoops! You're right, of course.
Jan 2, 2018 at 21:46 comment added Joe Silverman Formally, one should define $H_1(E,\mathbb Z_\ell)=T_\ell(E)$ and then define $H^1(E,\mathbb Z_\ell)$ to be the dual, ditto with $H^2$. And to get a vector space, instead of just a $\mathbb Z_\ell$-module, one needs to tensor $\otimes_{\mathbb Z_\ell}\mathbb Q_\ell$. But for computing traces, obviously, it doesn't matter which you use.
Jan 2, 2018 at 21:33 history answered David E Speyer CC BY-SA 3.0