Square-free orders for CM elliptic curves modulo <Emphasis Type="Italic">p</Emphasis> |
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Authors: | Alina Carmen Cojocaru |
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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 |
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Abstract: | Let E be an elliptic curve defined over , of conductor N, and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes
, 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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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 11G05 Secondary 11N36 11R45 |
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