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Let $X$ and $Y$ be elliptic curves (over an algebraically closed field, but no assumptions on the characteristic) with Jacobians $J_X$ and $J_Y$ respectively. Suppose $f:X\to Y$ is an isogeny, with dual isogeny $\widehat{f}:J_Y\to J_X$. How can I show that the map on tangent spaces $d\widehat{f}:T_0J_Y\to T_0J_X$ is $f^*:H^1(Y,\mathcal{O}_Y)\to H^1(X,\mathcal{O}_X)$? I already know that the tangent space to the identity of $J_X$ is $H^1(X,\mathcal{O}_X)$ via considering maps $\operatorname{Spec} k[\varepsilon]/\varepsilon^2\to J_X$ and using the universal property of the Jacobian variety, but I'm a little stumped on how to rigorously show the statement about the map.

Background: I'm trying to connect the two characterizations of the Hasse invariant of an elliptic curve in terms of the action of the Frobenius on $H^1$ and the separability of the dual of the Frobenius. Knowing this statement would finish the problem by the link between separability and the map on tangent spaces (plus maybe an argument that the map on tangent spaces has to be the same everywhere? I'm actually realizing I might need a little help with that too, as I'm writing this).

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  • $\begingroup$ @hm2020 No nontrivial ones, unfortunately - it's clear that if $f$ is the identity or constant, the result holds by computing both sides separately, but this doesn't really help me out for a solution to the general problem or the application I'm looking for. $\endgroup$ Dec 1, 2021 at 21:24
  • $\begingroup$ Example: If $k$ is a field and $S:=Spec(k)$ and $X/k$ is separated and of finite type, there is an exact sequence $0 \rightarrow H^1(S, f_*\mathbb{G}_m) \rightarrow Pic(X) \rightarrow Pic_{X/k}(k) \rightarrow H^2(S, f_*\mathbb{G}_m) \rightarrow H^2(X, \mathbb{G}_m)$. Here $Pic_{X/k}(k)$ is the $k$-rational points of the picard scheme $Pic_{X/k}$ ("Neron models", page 203). Moreover $H^1(X_{et}, \mathcal{O}_{X_{et}}^*) \cong Pic(X)$. $\endgroup$
    – hm2020
    Dec 2, 2021 at 14:31
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    $\begingroup$ @hm2020 I flipped through the book but did not see anything obviously applicable here. Most of the tools they discuss seem to be much too advanced to be reasonably applied to this problem - by analogy, I'm trying to build a birdhouse, and they're telling me how to construct a skyscraper. Your example is also unclear to me - why should this have anything to do with the problem at hand? Your "moreover" doesn't even appear in the exact sequence! $\endgroup$ Dec 2, 2021 at 20:18
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    $\begingroup$ @hm2020 Even if there were, what would the point be? It's okay if you don't have an answer to this problem (I don't either!) but your continued speculation and pinging me with things that are neither relevant nor explained is becoming tiresome. Please either provide relevant and explained comments or leave this post alone. $\endgroup$ Dec 5, 2021 at 10:19

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