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.



Correspondences on curves and their induced maps on differentials?
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.


