TITLE: Correspondences on curves and their induced maps on differentials?
QUESTION [3 upvotes]: How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.
More specifically, I was reading a paper that said roughly the following:
Let C be a curve and $\Gamma$ (with maps $\pi_i: \Gamma \rightarrow C$ for $i=1,2$) be a correspondence on $C$. The map $\pi_2$ is a double cover and $\tau: \Gamma \rightarrow \Gamma$ is a map that switches the elements of the fibers of $\pi_2$. Then the induced map on $H^0(C, \Omega^1)$ is given by:
$\omega \mapsto \pi_1^* \omega + \tau^* \pi_1^* \omega$, where the differentials of $C$ are identified with the ones of $\Gamma$ that are $\tau^*$-invariant.
I don't understand why the map on $H^0(C, \Omega^1)$ is what it is claimed to be (even assuming the mentioned identification). I guess this follows from general theory of correspondences but I don't know where I would find such a statement.

REPLY [5 votes]: This was pretty much answered in the comments by Jason Starr and Emerton, but to elaborate a bit, the simplest type of correspondence is (the graph of) a map $f:C_1\to C_2$.
In this case the action on $H^0(\Omega^1)$, or anything else, is by $f^*$. In general,
given $C_1\leftarrow \Gamma\to C_2$, with maps labeled $\pi_i$, the action on differentials
is by
$$H^0(C_2,\Omega^1)\stackrel{\pi_2^*}{\to} H^0(\Gamma,\Omega^1)\stackrel{\pi_{1*}}{\to} H^0(C_1,\Omega^1)$$
The second map is dual to $\pi_1^*$ under Serre duality. If $\Gamma\to C_1$ is a Galois
with group $G$ having order prime to the characteristic (which includes the case you care about), $\pi_{1*}$ is the projection
$$H^0(\Gamma,\Omega^1)\to H^0(\Gamma,\Omega^1)^G\cong H^0(C_1,\Omega^1)$$
given by averaging over the group.
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