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Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?

This could immediately follow if their coholomogies (for the sheaf of regular functions) were Weil cohomologies. But the proof of this is also hard to find.

Yet, there are references to this fact in connection with the use of Picard-Fuchs equation and counting rational points on such curves.

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For endomorphisms of elliptic curves, the Lefschetz fixed-point theorem goes back to Hasse and Deuring in the 1930s. See Silverman III 8.6 for the proof. For abelian varieties, it was proved by Weil in the 1940s, which was one of the developments that led to etale cohomology.

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  • $\begingroup$ For abelian varieties, the proof can be found in Mumford's book with that title. $\endgroup$ – ACL Jan 4 '18 at 1:23
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I am confused by the reference to sheaves of regular functions, and suspect the OP is confused. If $E$ is an elliptic curve over $\mathbb{F}_p$, then the cohomology groups of the sheaf of regular functions on $E$ is an $\mathbb{F}_p$ vector space, and Weil cohomology theories are normally required to take values in a characteristic $0$ vector space. If you take a cohomology theory with values in a vector space of characteristic $p$, then you can only count fixed points modulo $p$. It is true that, for $X$ a projective variety over $\mathbb{F}_p$, we have $$\# X(\mathbb{F}_p) \equiv \sum_j (-1)^j\ \mathrm{Tr}(\mathrm{Frob}^{\ast} : H^j(X, \mathcal{O}) \to H^j(X, \mathcal{O})) \bmod p.$$ A reference for this is W. Fulton, A fixed point formula for varieties over finite fields. But I suspect that isn't what the OP wants.

If you want to count points, not just count modulo $p$, then you need some definition of $H^j$ which takes values in modules over a characteristic zero ring. For an elliptic curve, you can use the Tate module: $T_{\ell}(E) := \lim_{\infty \leftarrow n} E[\ell^n]$ where $\ell$ is a prime not equal to $p$ and the maps in the inverse limit are multiplication by $\ell$.

You can make an ad hoc definition $H^1(E) = T_{\ell}(E)$ and $H^2(E) = \bigwedge^2 H^1(E)$. With this definition, as anon says, Lefschetz trace goes back to the 1930's. (See Silverman III.7.1 for $T_{\ell}(E) \cong \mathbb{Z}_{\ell}^2$, so $\bigwedge^2 T_{\ell}(E) \cong \mathbb{Z}_{\ell}$. Silverman III.8.3 implies that $\mathrm{Frob}$ acts on $\bigwedge^2 T_{\ell}(E)$ by $p$. The computations in Silverman V.1 and V.2 make it clear that $\# E(\mathbb{F}_p) = 1-\mathrm{Tr}\left( \mathrm{Frob}^{\ast} T_{\ell}(E) \to T_{\ell}(E)\right) + p$.)

If you want to show this matches some other definition of $H^j$, you have to say what definition you are using.

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    $\begingroup$ 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. $\endgroup$ – Joe Silverman Jan 2 '18 at 21:46
  • $\begingroup$ Whoops! You're right, of course. $\endgroup$ – DES-SupportsMonicaAndTransfolk Jan 2 '18 at 21:47
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Milne's Etale Cohomology leaps to mind...

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