Local to global principles for homomorphisms of abelian schemes

Author:

Gajda Wojciech,Petersen Sebastian

Abstract

AbstractLet A and B be abelian varieties defined over the function field k(S) of a smooth algebraic variety S/k. We establish criteria, in terms of restriction maps to subvarieties of S, for existence of various important classes of k(S)-homomorphisms from A to B, e.g., for existence of k(S)-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case when the base field is finite.

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics

Reference24 articles.

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2. O. Bültel, Construction of abelian varieties with given monodromy, Geometric and Functional Analysis 15 (2005), 634–696.

3. O. Bültel, Further counterexamples to Zarhin’s conjecture about micro weights, https://arxiv.org/abs/2207.08706.

4. A. Cadoret and A. Tamagawa, Ghosts and families of abelian varieties with a common isogeny factor, https://webusers.imj-prg.fr/∼anna.cadoret/Ghosts.pdf.

5. L. Dixon, M. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groups, Cambridge Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge, 1999.

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