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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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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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Oriented Supersingular Elliptic Curves and Eichler Orders [PDF]
, 2023 Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic curve with level-$c$ structure, and the endomorphism ring $\text{End}(E,G)$ is isomorphic to an Eichler order with ...
Guanju Xiao, Zijian Zhou, Longjiang Qu
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Minimal CM Liftings of Supersingular Elliptic Curves [PDF]
Pure and Applied Mathematics Quarterly, 2008 In this paper, we prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 1 2 . We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 23 as suggested by Elkies.
Tonghai Yang
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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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Finding orientations of supersingular elliptic curves and quaternion orders [PDF]
Designs, Codes and CryptographyAbstractAn 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
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Gross lattices of supersingular elliptic curves
Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$.
Christelle Vincent +3 more
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Computing Isomorphisms between Products of Supersingular Elliptic Curves [PDF]
The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time, given the endomorphism rings of the curves involved.
Pierrick Gaudry +2 more
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A bound on the $μ$-invariants of supersingular elliptic curves [PDF]
Canadian Mathematical BulletinAbstractLet $E/\mathbb {Q}$ be an elliptic curve and let p be a prime of good supersingular reduction. Attached to E are pairs of Iwasawa invariants $\mu _p^\pm $ and $\lambda _p^\pm $ which encode arithmetic properties of E along the cyclotomic $\mathbb {Z}_p$ -extension of $\mathbb {Q}$ . A well-known conjecture of B.
Rylan Gajek-Leonard
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Efficient Supersingularity Testing of Elliptic Curves Using Legendre Curves
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2023 Yuji Hashimoto, Koji Nuida
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