Endomorphism rings of reductions of elliptic curves and Abelian varieties

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)81-106
Number of pages26
JournalSt. Petersburg Mathematical Journal
Volume29
Issue number1
DOIs
StatePublished - Jan 1 2018

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Endomorphism Ring
Abelian Variety
Elliptic Curves
Relatively prime
Imaginary Quadratic Field
Divisible
Discriminant
Number field
Exercise
Integer

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

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

In: St. Petersburg Mathematical Journal, Vol. 29, No. 1, 01.01.2018, p. 81-106.

Research output: Contribution to journalArticle

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