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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?

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    $\begingroup$ This is what is sometimes referred to as 'spreading out', although sometimes you will read 'by a standard limit argument'. The most general situation is worked out in EGA IV$_3$, Thm. 8.8.2(ii). The idea is that if $A$ has some equations over $K$, then these equations live over some finitely generated subring $R \subseteq K$; we may take $R = \mathbb C[V]$ for some variety $V$ with fraction field $K$. Over some big open locus $U \subseteq V$, you still get a family of abelian varieties, so replacing $V$ by $U$ gives the result. $\endgroup$ Jul 13, 2018 at 15:36
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    $\begingroup$ The last statement of the argument outlined above uses Thm. 8.10.5(xii) and Thm. 8.8.2(i) of [loc. cit.] to show that over a nonempty open $U \subseteq V$, the scheme $\mathscr A|_U \to U$ is proper and admits a group structure extending the group structure on $A \to \operatorname{Spec} K$. $\endgroup$ Jul 13, 2018 at 15:50
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    $\begingroup$ Just to note that Manin's proof of the "theorem of the kernel" contained a gap, and that you should consult Coleman's article Manin's proof of the Mordell conjecture over function fields in Ens. Math., tome 36 (1990), pp. 393--427. $\endgroup$ Jul 13, 2018 at 17:52
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    $\begingroup$ @ali A $K$-rational point $P$ of $A$ extends, by the valuative criterion of properness applied to $\mathcal{A} \to V$, to a morphism $U_P \to \mathcal{A}$ over $V$, where $U_P$ is a dense open of $V$ (which depends on $P$) whose complement is of codimension at least two. Now, since $\mathcal{A}$ is an abelian scheme, the fibres of the structure morphism do not contain any rational curves. Therefore, there is no obstruction to extending the morphism $U_P\to \mathcal{A}$ to a morphism $V\to \mathcal{A}$. $\endgroup$ Jul 13, 2018 at 18:21
  • $\begingroup$ @ali I should add that $V$ should be regular for the above argument to be complete. $\endgroup$ Jul 13, 2018 at 18:32

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