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Apologies if this is not quite at the level of MathOverflow, but I have already asked at MSE and received no answer after two+ weeks and a bounty.

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.

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    $\begingroup$ (The last paragraph is Hartshorne Ex IV 4.15.) $\endgroup$
    – Piotr Achinger
    Commented Dec 14, 2021 at 7:37
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    $\begingroup$ Since pullback of line bundles agrees with pullback on $H^1(\mathcal{O}^*)$, you can use naturality of the short exact sequence $$ 0\to \mathcal{O}_X\to \mathcal{O}^*_{X[\varepsilon]}\to \mathcal{O}^*_X\to 0 $$ to conclude (where $X[\varepsilon] = X\times \operatorname{Spec}k[\varepsilon]/(\varepsilon^2)$). $\endgroup$
    – Piotr Achinger
    Commented Dec 14, 2021 at 7:39
  • $\begingroup$ @PiotrAchinger aha, that's quite nice. Thanks for the insight. $\endgroup$
    – Hank Scorpio
    Commented Dec 28, 2021 at 19:21

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