3F2-hypergeometric functions and supersingular elliptic curves
نویسندگان
چکیده
منابع مشابه
3f2 Hypergeometric Series and Periods of Elliptic Curves
We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.
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We give an algorithm that constructs, on input of a prime power q and an integer t, a supersingular elliptic curve over Fq with trace of Frobenius t in case such a curve exists. If GRH holds true, the expected run time of our algorithm is e O((log q)). We illustrate the algorithm by showing how to construct supersingular curves of prime order. Such curves can readily be used for pairing based c...
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Introduction. The wonderful book by Andrews, Askey, and Roy [2] is mainly devoted to special functions of hypergeometric type – to the plain and q-hypergeometric series and integrals. Shortly before its publication, examples of a third type of such functions, related to elliptic curves, began to appear. A systematic theory of elliptic hypergeometric functions was constructed in 2000-2004 over a...
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In this paper, we give a ‘direct’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal’ CM liftings of such supersingular elliptic curves is a best possible result. We also prove that the result of Elkies on ‘minimal’ CM liftings of all supersingular elliptic...
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Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine an explicit constant Dp so that |D| > Dp implies that the map is necessarily surjective and then we compu...
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ژورنال
عنوان ژورنال: Involve, a Journal of Mathematics
سال: 2015
ISSN: 1944-4184,1944-4176
DOI: 10.2140/involve.2015.8.481