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Square-free orders for CM elliptic curves modulo p

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Abstract

Let E be an elliptic curve defined over \({\mathbb Q}\), of conductor N, and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes \({p \nmid N}\), p ≤ x, for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of E modulo p is cyclic, first studied by Serre (1977). They are also related to the stronger problem about primitive points on E modulo p, formulated by Lang and Trotter (Bull Am Math Soc 83:289–292, 1977), and the one about the primality of the order of E modulo p, formulated by Koblitz [Pacific J. Math. 131(1):157–165, 1988].

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Str., 322 SEO, Chicago, IL, 60607, USA

    Alina Carmen Cojocaru

  2. The Institute of Mathematics of the Romanian Academy, Bucharest, Romania

    Alina Carmen Cojocaru

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  1. Alina Carmen Cojocaru
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Correspondence to Alina Carmen Cojocaru.

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Cojocaru, A.C. Square-free orders for CM elliptic curves modulo p . Math. Ann. 342, 587–615 (2008). https://doi.org/10.1007/s00208-008-0249-9

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