I want to read Manin proof of Mordell Conjecture over function fields.I understand most of the article but I have problems with "kernel theorem"and it's proof:
consider $A$ is an abelian variety over $K$ where $K$ is an extension of $\mathbb{C}$.if $K(A)$ is regular over $K$ Manin use derivations of $K$ and construct homomorphisms from $A_K$ to $K$,define intersection of kernels of all this homomorphisms $A^0_K$.kernel theorem says $P \in A^0_K$ iff there is an integer $d$ such that $dP \in tr_{K/C}(A)$.
I think my main problem is this:how can I "extend" A to a family of abelian varieties over $V$ where $V$ is a variety over $\mathbb{C}$ such that $ \mathbb{C}(V)=K$.
also I appreciate if you give a reference to a more modern account to proof of this theorem.
thanks to R. van Dobben de Bruyn my first problem is solved.here an english version of the theorems mentioned from EGA: http://www-math.mit.edu/~poonen/papers/Qpoints.pdf chapter 3.
I have one more problem :Manin claims point of $A_K$ is in 1_1 correspondence with morphisms from $V$ to $\mathscr{A}$.why?