### 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 |
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Journal | Journal of Number Theory |

DOIs | |

Publication status | Accepted/In press - 2018 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*. https://doi.org/10.1016/j.jnt.2018.08.005

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**Cyclic components of quotients of abelian varieties mod p.** / Virdol, Cristian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Cyclic components of quotients of abelian varieties mod p

AU - Virdol, Cristian

PY - 2018/1/1

Y1 - 2018/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85053675309&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053675309&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2018.08.005

DO - 10.1016/j.jnt.2018.08.005

M3 - Article

AN - SCOPUS:85053675309

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -