Results 1 to 10 of about 42,992 (209)
CM liftings of Supersingular Elliptic Curves [PDF]
Journal de Théorie des Nombres de Bordeaux, 2009 Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $|D|
Ben Kane
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Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves [PDF]
Journal of Mathematical Cryptology, 2021 Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽p2, if an imaginary quadratic order O can be embedded in End(E) and a ...
Xiao Guanju, Luo Lixia, Deng Yingpu
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Finding orientations of supersingular elliptic curves and quaternion orders [PDF]
Designs, Codes and Cryptography, 2023 AbstractAn oriented supersingular elliptic curve is a curve which is enhanced with the information of an endomorphism. Computing the full endomorphism ring of a supersingular elliptic curve is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $$\
Sarah Arpin +5 more
semanticscholar +7 more sources
On random sampling of supersingular elliptic curves
Annali di Matematica Pura ed Applicata (1923 -)Abstract We consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) elliptic curve and a random walk over some supersingular isogeny graph.
M G Mula, Nadir Murru, Federico Pintore
semanticscholar +4 more sources
Simultaneous supersingular reductions of CM elliptic curves [PDF]
Journal für die reine und angewandte Mathematik (Crelles Journal), 2021 We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become equidistributed)
Aka, Menny +3 more
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Supersingular elliptic curves [PDF]
, 2021 In the previous chapter, we showed that Brandt matrices for an order in a definite quaternion algebra B contain a wealth of arithmetic. In the special case where \({{\,\mathrm{disc}\,}}B=p\) is prime, there is a further beautiful connection between Brandt matrices and the theory of supersingular elliptic curves, arising from the following important ...
John Voight
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On supersingular elliptic curves and hypergeometric functions [PDF]
Involve, a Journal of Mathematics, 2012 The Legendre family of elliptic curves has the remarkable property that both its periods and its supersingular locus have descriptions in terms of the hypergeometric function [math] . In this work we study elliptic curves and elliptic integrals with respect to the hypergeometric functions [math] and [math] , and prove that the supersingular [math ...
Keenan Monks
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The main conjecture for CM elliptic curves at supersingular primes [PDF]
Annals of Mathematics, 2004 At a prime of ordinary reduction, the Iwasawa ``main conjecture'' for elliptic curves relates a Selmer group to a $p$-adic $L$-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the $p$-adic $L$-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of
Robert Pollack, Karl Rubin
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Supersingular Elliptic Curves and Moonshine [PDF]
Symmetry, Integrability and Geometry: Methods and Applications, 2019 We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other ...
Victor Manuel Aricheta
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On the plus and the minus Selmer groups for elliptic curves at supersingular primes [PDF]
Tokyo Journal of Mathematics, 2016 Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct the plus and the minus Selmer groups of $E$ over the cyclotomic $\mathbb Z_p$-extension in a more general setting ...
Takahiro Kitajima, Rei Otsuki
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