In Weil's short note entitled "*On the Riemann hypothesis in function-fields*"
he mentions the notion of the **complementary correspondence** associated to a given correspondence $T:C\rightarrow C$ where $C$ is an algebraic curve. He gives a reference to one of Severi's papers for the definition, which I did not bother to look at since I probably won't be able to understand the definition. 


**Q1**: So what is the definition of the complementary correspondence of $T$ in modern language? 


**Q2**: Is there a good modern reference where the theory of correspondences is developed from scratch and is presented in a geometrical and intuitive way (so that it conveys the intuition of the Italian's school without distorting it too much) ? 



On Severi's definition of the complementary correspondence
Given a correspondence $T \subset C \times C$ one defines an endomorphism of the Jacobian of $C$ by letting $T(P)$ be the divisor on $C$ cut out in $T$ by $\{P\}\times C$, extending by linearity and then taking linear equivalence classes. So the ring of endomorphisms is a quotient of the ring of correspondences. I think the complementary correspondence is a good representative of the equivalence class of the endomorphism $-T$ obtained by representing elements of the Jacobian as $\sum\_{i=1}^g P\_i - gP\_0$, for a fixed $P\_0$.


Modern books seem to focus on endomorphisms instead of correspondences (e.g. Mumford). There is a little bit of correspondences in Griffiths-Harris the way you want but it doesn't go very far.


