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Let $E$ be an elliptic curve defined over a number field $F$ with supersingular reduction at an odd prime $p$. Let $E[p]$ denotes the set of $p$-torsion points of $E$ over an algebraic closure of $F$.

Statement - $E[p]$ is always irreducible for an elliptic curve $E$ with supersingular reduction at $p$.

Question - Is the above statement true for arbitrary $F$ or only when $F=\mathbb{Q}$, the set of rational numbers $?$

References are welcome.

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    $\begingroup$ It fails if you extend scalars to the splitting field of that Galois module! In general, the representation is irreducible even on the inertia group at any $p$-adic place of $F$ unramified over $\mathbf{Q}$ (since the $p$-torsion in the Neron model over the valuation ring is connected and the absolutely ramification index is $1\le p-1$). This is an application of Raynaud's 1974 work on finite flat groups over complete (or henselian) discrete valuation rings of mixed characteristic $(0,p)$, but was known to Serre via the formal group in his 1972 paper (Inventiones 15). $\endgroup$ – nfdc23 Aug 24 '16 at 19:17
  • $\begingroup$ I may have a fundamental misunderstanding, @nfdc23, but isn’t the correct bound on the ramification in this supersingular case $p^2-1$? $\endgroup$ – Lubin Aug 27 '16 at 14:57
  • $\begingroup$ @Lubin: You are misreading which ramification I am referring to. I am referring to the ramification index $e(F_{\mathfrak{p}}|\mathbf{Q}_p)$ being equal to 1 for a prime $\mathfrak{p}$ of $F$ over $p$ (since Raynaud's result is then applicable to the maximal unramified extension of $F_{\mathfrak{p}}$), and not to the ramification of the splitting field of $E[p]$ over $F_{\mathfrak{p}}$. $\endgroup$ – nfdc23 Aug 30 '16 at 13:25
  • $\begingroup$ @nfdc23, thanks for this. I suspected that I might be going off in the wrong direction. $\endgroup$ – Lubin Aug 30 '16 at 13:31

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