Cyclic components of quotients of abelian varieties mod p

Cristian Virdol

doi:10.1016/j.jnt.2018.08.005

Cyclic components of quotients of abelian varieties mod p

Cristian Virdol

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by F_p the finite field of cardinality p. If A has good reduction at p, let A¯_p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γ_p be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient [Formula presented] has at most 2r−1 cyclic components.

Original language	English
Pages (from-to)	135-144
Number of pages	10
Journal	Journal of Number Theory
Volume	197
DOIs	https://doi.org/10.1016/j.jnt.2018.08.005
Publication status	Published - 2019 Apr

Bibliographical note

Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01056643).

Publisher Copyright:
© 2018 Elsevier Inc.

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1016/j.jnt.2018.08.005

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title = "Cyclic components of quotients of abelian varieties mod p",

abstract = "Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by Fp the finite field of cardinality p. If A has good reduction at p, let A¯p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γp be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient [Formula presented] has at most 2r−1 cyclic components.",

author = "Cristian Virdol",

note = "Funding Information: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01056643). Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2019",

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doi = "10.1016/j.jnt.2018.08.005",

language = "English",

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journal = "Journal of Number Theory",

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AU - Virdol, Cristian

N1 - Funding Information: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01056643). Publisher Copyright: © 2018 Elsevier Inc.

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N2 - Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by Fp the finite field of cardinality p. If A has good reduction at p, let A¯p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γp be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient [Formula presented] has at most 2r−1 cyclic components.

AB - Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by Fp the finite field of cardinality p. If A has good reduction at p, let A¯p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γp be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient [Formula presented] has at most 2r−1 cyclic components.

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Cyclic components of quotients of abelian varieties mod p

Abstract

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