Formal groups of elliptic curves with potential good supersingular reduction
نویسندگان
چکیده
منابع مشابه
Formal Groups of Elliptic Curves with Potential Good Supersingular Reduction
Let L be a number field and let E/L be an elliptic curve with potential supersingular reduction at a prime ideal ℘ of L above a rational prime p. In this article we describe a formula for the slopes of the Newton polygon associated to the multiplication-by-p map in the formal group of E, that only depends on the congruence class of p mod 12, the ℘-adic valuation of the discriminant of a model f...
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Let d ≥ 1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F )tors be the torsion subgroup of E(F ). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F , such that the size of E(F )tors is bounded by B(d). Moreover, Merel gave a bound (exponential in ...
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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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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: Pacific Journal of Mathematics
سال: 2013
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.2013.261.145