In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
By an elliptic curve over a scheme $S$, we mean a proper smooth morphism $p: E \to S$, whose geometric fibres are connected curves of genus one, together with a section $e : S \to E$.
Now this is a quite reasonable definition, which coincides with the usual notion of an elliptic curve when $S$ is the spectrum of a field. However, it does not seem (to me) to follow directly from the definition that such an elliptic curve over $S$ should be a group scheme over $S$ (an obviously desirable property which Katz seems to take as an obvious fact). When $S= \text{Spec }k$, I understand that this essentially follows from Riemann-Roch…
So, what principle allows one to come to this conclusion in the general case?
Thank you!
Best Answer
The argument that allows you to show that an elliptic curves defined as you say is a group scheme, and even a commutative one is the construction of a functorial and natural isomorphism $E(T) \rightarrow Pic_{E/S}^0(T)$ for every $S$-scheme $T$. This allows to see the functor $T \mapsto E(T)$ as a funtor in group, and since this functor is representable by $E$, this gives a structure of group scheme on $E$, which is the one you are looking for.
Essentially this map is defined as follows: one attached to a point in $E(T)$, that is a $T$-section of $E_T$, the invertible sheaf of divisor this section minus the trivial section $e_T$ (obtained by base change from the section $e$ which is part of the definition). To prove that this map is an isomorphism, one essentially reduces, using base change theorems for direct image in coherent cohomology, to the case of a field, where it becomes a consequence of Riemann-Roch. Note that by definition, the trivial section $e_T$ is send to to the trivial sheaf, so that $e$ is the neutral section of the group scheme structure on $E$, as desired.
Of course there are many details to deal with to make this argument a complete one, but this is done with great care in the beginning of the (unique) book by Katz and Mazur.