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Square-free orders for CM elliptic curves modulo <Emphasis Type="Italic">p</Emphasis>
Authors:Alina Carmen Cojocaru
Institution:(1) Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Str., 322 SEO, Chicago, IL 60607, USA;(2) The Institute of Mathematics of the Romanian Academy, Bucharest, Romania
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 $${p \nmid N}$$, px, 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].
Keywords:Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000)  Primary 11G05  Secondary 11N36  11R45
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