Abstract
Let 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.
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Acknowledgments
The authors were supported by a research grant UMO-2018/31/B/ST1/01474 of the National Centre of Sciences of Poland. S. P. thanks the Mathematics Department at Adam Mickiewicz University in Pozna´n for hospitality during research visits. We thank J¸edrzej Garnek, Marc Hindry and Bartosz Naskr¸ecki for useful discussions on the topic of this paper. Finally we want to thank the anonymous referee for a careful reading of the manuscript and for valualble comments.
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Dedicated to Moshe Jarden with admiration on the occasion of his 80th birthday
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Gajda, W., Petersen, S. Local to global principles for homomorphisms of abelian schemes. Isr. J. Math. 257, 281–312 (2023). https://doi.org/10.1007/s11856-023-2542-4
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DOI: https://doi.org/10.1007/s11856-023-2542-4