Timeline for Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field
Current License: CC BY-SA 4.0
5 events
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| Dec 6, 2019 at 8:22 | comment | added | Riccardo Pengo | Let me just add that if question (1) had a positive answer, then the torsion in $A(K^{\text{per}})$ would be finite. This has been proved recently by Ambrosi and D'Addezio (arxiv.org/abs/1811.08423, Theorem 1.2.2). | |
| Nov 5, 2019 at 18:48 | comment | added | Asvin | In my previous comment, you don't need to consider the Neron model, you can simply say that any map from $Spec K^{per}$ to the abelian variety extends to a map from the curve to the Abelian variety and proceed from there. I don't know what happens in higher transcendence degree. | |
| Nov 5, 2019 at 18:11 | comment | added | Asvin | (ctd...) and therefore the set of sections would be a subgroup of group of homomorphisms which is essentially described by the endomorphism of the Jacobian which is finite generated. So shouldn't it be true in general that over a curve, any non isotrivial Abelian variety has finitely generated Mordell Weil group? | |
| Nov 5, 2019 at 18:10 | comment | added | Asvin | If I understand correctly, can't you always reduce the problem to a minimal field over which the Abelian variety is not isotrivial? And if this field happens to be a finite field, then the group contains $A(\overline{\mathbb F_p})$ which is not finitely generated. On the other hand, if the Abelian variety is not isotrivial over the function field of a curve, we can extend it a neron model over the curve and then the Mordell Weil group would just be the set of sections of the Neron Model. Any map from a curve to an Abelian variety would factor through the Jacobian of the curve (ctd...) | |
| Nov 5, 2019 at 15:15 | history | asked | Joël | CC BY-SA 4.0 |