### Abstract

Let E be an elliptic curve without CM that is defined over a number field K. For all but finitely many non-Archimedean places v of K there is a reduction E(v) of E at v that is an elliptic curve over the residue field k(v) at v. The set of v's with ordinary E(v) has density 1 (Serre). For such v the endomorphism ring End(E(v)) of E(v) is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers N and M there are infinitely many non-Archimedean places v of K such that the discriminant Δ(v) of End(E(v)) is divisible by N and the ratio Δ(v)/N is relatively prime to NM. We also discuss similar questions for reductions of Abelian varieties. The subject of this paper was inspired by an exercise in Serre's "Abelian l-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.

Original language | English (US) |
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Pages (from-to) | 81-106 |

Number of pages | 26 |

Journal | St. Petersburg Mathematical Journal |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Applied Mathematics

### Cite this

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*St. Petersburg Mathematical Journal*, vol. 29, no. 1, pp. 81-106. https://doi.org/10.1090/spmj/1483

**Endomorphism rings of reductions of elliptic curves and Abelian varieties.** / Zarhin, Yu G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Endomorphism rings of reductions of elliptic curves and Abelian varieties

AU - Zarhin, Yu G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let E be an elliptic curve without CM that is defined over a number field K. For all but finitely many non-Archimedean places v of K there is a reduction E(v) of E at v that is an elliptic curve over the residue field k(v) at v. The set of v's with ordinary E(v) has density 1 (Serre). For such v the endomorphism ring End(E(v)) of E(v) is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers N and M there are infinitely many non-Archimedean places v of K such that the discriminant Δ(v) of End(E(v)) is divisible by N and the ratio Δ(v)/N is relatively prime to NM. We also discuss similar questions for reductions of Abelian varieties. The subject of this paper was inspired by an exercise in Serre's "Abelian l-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.

AB - Let E be an elliptic curve without CM that is defined over a number field K. For all but finitely many non-Archimedean places v of K there is a reduction E(v) of E at v that is an elliptic curve over the residue field k(v) at v. The set of v's with ordinary E(v) has density 1 (Serre). For such v the endomorphism ring End(E(v)) of E(v) is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers N and M there are infinitely many non-Archimedean places v of K such that the discriminant Δ(v) of End(E(v)) is divisible by N and the ratio Δ(v)/N is relatively prime to NM. We also discuss similar questions for reductions of Abelian varieties. The subject of this paper was inspired by an exercise in Serre's "Abelian l-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.

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UR - http://www.scopus.com/inward/citedby.url?scp=85040089636&partnerID=8YFLogxK

U2 - 10.1090/spmj/1483

DO - 10.1090/spmj/1483

M3 - Article

AN - SCOPUS:85040089636

VL - 29

SP - 81

EP - 106

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 1

ER -