2024 ORDINARY PRIMES FOR SOME VARIETIES WITH EXTRA ENDOMORPHISMS
Francesc Fité
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Publ. Mat. 68(1): 27-40 (2024). DOI: 10.5565/PUBLMAT6812402

Abstract

Let A be an abelian variety defined over a number field and of dimension g. When g2, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for A. In higher dimension very little is known. We show that if g=3 and A has multiplication by an imaginary quadratic field E, then there exists a nonzero density set of ordinary primes for A. We reach the same conclusion if g=4 and the pair (A,E) has signature (2,2). We also obtain partial results when g=3 and A has multiplication by a totally real cubic field. We show that our methods also apply to certain abelian varieties of Albert type IV of higher dimension. These results are derived from an extended version of the -adic methods of Katz, Ogus, and Serre in the presence of extra endomorphisms.

Acknowledgements

Thanks to Bjorn Poonen for explaining the content of [8, Proposition 2.7.1] to the attendants of the MIT number theory group meeting. Thanks to Drew Sutherland for drawing my attention to the question considered in this note in the case of Picard curves. Thanks to Xavier Guitart for helpful discussions and for his comments and corrections to a previous version of this manuscript, as well as for alerting me to the existence of [17]. I was financially supported by the Simons Foundation grant 550033.

Dedication

Dedicat a Jordi Quer amb gratitud

Citation

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Francesc Fité. "ORDINARY PRIMES FOR SOME VARIETIES WITH EXTRA ENDOMORPHISMS." Publ. Mat. 68 (1) 27 - 40, 2024. https://doi.org/10.5565/PUBLMAT6812402

Information

Received: 2 June 2021; Accepted: 11 April 2023; Published: 2024
First available in Project Euclid: 25 December 2023

MathSciNet: MR4682722
Digital Object Identifier: 10.5565/PUBLMAT6812402

Subjects:
Primary: 11G10 , 11R45

Keywords: endomorphism algebras , ordinary abelian varieties , λ-adic representations

Rights: Copyright © 2024 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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