The notion of genus is quite intuitive on compact Riemann surfaces, as it is a topological notion that can be visualized in this case.
On the other hand, when defining an algebraic curve in characteristic $p > 0$ (elliptic curve over $F_p$ for instance), we lose that intuition.
I've read that Model Theory can "transfer" some statements over $\mathbb{C}$ to statements over the algebraic closure of $F_p$ (with possible exceptions on finitely many $p$ and restrictions on the set of "transferable" statements) [Edit: See https://webusers.imj-prg.fr/~adrien.deloro/teaching-archive/Moskva-ACF.pdf Theorem 3.9 "Cross-Characteristic Transfer and Ax’s Theorem" for reference]
Now here are my questions :
- Could we obtain the notion of "genus" on algebraic curves over $F_p$ from the fact that this genus exists and can be expressed algebraically on compact Riemann surfaces ?
- Can we use Model Theory to deduce results about curves on $F_p$ based on what we know on compact Riemann surfaces ? How far could we go (Riemann-Roch, structure of an Elliptic Curve group, etc.) ?