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Let $$A$$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $$A$$ does not have real multiplication by any real quadratic field with discriminant a multiple of $$p$$ . We prove that there are infinitely many places modulo which $$A$$ is isogenous to the product of two elliptic curves.
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Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K .
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- PAR ID:
- 10366115
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 10
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Given a smooth projective variety over a number field and an elementof its Brauer group, we consider the specialization of the Brauerclass at a place of good reduction for the variety and the class. Weare interested in the case of K3 surfaces.We show that a Brauer class on a very general polarized K3 surfaceover a number field becomes trivial after specialization at a set ofplaces of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derivedequivalent K3 surfaces which become derived equivalent after reductionat a positive proportion of places.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2022.14
