Fix a prime $p$. If two elliptic curves over $\mathbb{Q}$ have the same p-adic Galois representation, then what relatinships do we know between them? Any references are welcome.
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$\begingroup$ Galois group of whom acting on whom? $\endgroup$– user138661Commented May 18, 2019 at 6:14
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1$\begingroup$ @schematic_boi The action of $Gal(\mathbb{\bar{Q}}/(\mathbb{Q})$ on the p-adic etale cohomology $\endgroup$– BonbonCommented May 18, 2019 at 6:31
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1$\begingroup$ OK, I am not an expert, not even a half of an expert, but maybe see here: mathoverflow.net/q/41931/138661 $\endgroup$– user138661Commented May 18, 2019 at 6:41
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2$\begingroup$ I think Tate module (for $p$) is dual to etale cohomology with $\mathbb{Z}_p$-coefficients, so if you consider a single prime $p$ and Tate modules are isomorphic (as Galois modules), then your curves are isogenous. $\endgroup$– user138661Commented May 18, 2019 at 6:43
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6$\begingroup$ I think you are asking about the Tate conjecture (Faltings isogeny theorem). $\endgroup$– Felipe VolochCommented May 18, 2019 at 9:08
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