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Let $E$ be a supersingular elliptic curve over a finite field of characteristic $p$, and $\mathbb{F}_q\supset \mathbb{F}_{p^2}$ be a finite field large enough such that all (absolute) endomorphisms of $E$ is defined over $\mathbb{F}_q$. We write $G$ for the absolute Galois group of $\mathbb{F}_q$. It is well known that the Frobenius automorphism $\varphi$ is a (topological) generator of $G$. Let us fix a prime $\ell\neq p$. By the Tate Conjecture, $$\mathrm{End}(E)\otimes\mathbb{Q}_\ell=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))^G=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))^\varphi. $$

Since $E$ is supersingular, $\mathrm{End}(E)$ is an order in a quaternion algebra. In particular, $\mathrm{rank}_\mathbb{Z}(\mathrm{End}(E))=4$. So $$ \mathrm{End}(E)\otimes\mathbb{Q}_\ell=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))$$ It follows that $\varphi$ is in the center of $\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))$. In other words, $\varphi$ is a scalar. This clearly leads to a contradiction (say, with the Riemann Hypothesis). Where did the argument go wrong?

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This question is a fabulous instance of the phenomenon that surely many people have observed, many times over: the mistake in the argument is at the unique point where the author writes the word "clearly". –  Kevin Buzzard Jan 14 '11 at 11:58
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I think it's a well-known entry in the mathematical thesaurus that "clearly" is a synonym for "I should probably check that." –  JSE Jan 14 '11 at 14:24
    
Let $E/\mathbb F_p$ be an elliptic curve, with $p\ge5.$ Then $E$ is super-singular if and only if the Frob. trace $a_p$ on the Tate module is 0 (see Silverman I, an exercise in the finite field chapter). If $\alpha,\bar{\alpha}$ are the Frob. eigenvalues, then $\alpha+\bar{\alpha}=a_p=0\Rightarrow\alpha$ is purely imaginary. Hence the Frob. eigenvalues of $E\otimes\mathbb F_{p^2}$ are equal and are real (as Emerton pointed out in his answer). I learned this from B. Poonen. –  shenghao May 20 '11 at 12:24

1 Answer 1

There is nothing wrong. A typical case will be $\mathbb F_q = \mathbb F_{p^2}$ and $\varphi = -p$. Since $q^{1/2} = (p^2)^{1/2} = p$, this is consistent with RH.

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Thank you. For some reason I always thought for the elliptic curves, the eigenvalues of the Frobenius is in $\mathbb{C}-\mathbb{R}$. I guess this is an instance where it is not. –  Jiangwei Xue Jan 14 '11 at 11:37
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Jiangwei, I recommend to you in the strongest terms that you spend a few days doing explicit computations with the chord-and-tangent algorithm for adding points on an elliptic curve. A specific supersingular curve, and a specific $p$. And then a specific ordinary one over the same base. –  Lubin Jan 14 '11 at 19:16

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