2010
DOI: 10.1090/conm/521/10277
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Group order formulas for reductions of CM elliptic curves

Abstract: Abstract. We give an overview of joint work with Karl Rubin on computing the number of points on reductions of elliptic curves with complex multiplication, including some of the history of the problem.

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Cited by 7 publications
(9 citation statements)
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References 26 publications
(50 reference statements)
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“…The splitting of the Frobenius polynomial of E is well known in this case, with early results dating back to Gauss. The two statements below follow easily from [59,Th. 2.5,2.6].…”
Section: B1 the Ordinary Casementioning
confidence: 99%
“…The splitting of the Frobenius polynomial of E is well known in this case, with early results dating back to Gauss. The two statements below follow easily from [59,Th. 2.5,2.6].…”
Section: B1 the Ordinary Casementioning
confidence: 99%
“…It is worth to mention that K. Rubin and A. Silverberg [RS09] generalized Stark's work to the case of all CM elliptic curves over arbitrary number fields. See [Si10] for a nice survey of their joint work with reasonable backgrounds. There were some partial generalizations before that, for instance W. Miller [Mi98] treated the case with CM by an imaginary quadratic field whose discriminant is even and not divisible by 3.…”
Section: Image Of Galois Representationsmentioning
confidence: 99%
“…The theory of complex multiplication gives a = π u +π u for some π u ∈ O such that π uπu = q (see e.g. Theorem 14.16 in [3], [10, §II.10], or [8] for a thorough discussion). As K ⊂ k, we have K = kK, and we let ψ = ψ E/K be the Grössencharacter associated to E and K (see [10,§II.9] or [4]).…”
Section: Computing the Local Constantmentioning
confidence: 99%
“…If we choose f so that −D is not a square (mod f ), f is inert in K/Q, and so f O K 7 Determined up to the correspondence of class field theory. 8 See p.483 of [10], with f = 1 (in Silverman's notation), for a Weierstrauss equation. 9 Specifically with Sage's interface to John Cremona's 'mwrank' and Denis Simon's 'simon_two_descent.'…”
Section: Dihedral Extensions Of Qmentioning
confidence: 99%