Results 21 to 30 of about 38,603 (168)
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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The most efficient indifferentiable hashing to elliptic curves of j-invariant 1728
Journal of Mathematical Cryptology, 2022 This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic ...
Koshelev Dmitrii
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An efficient post-quantum KEM from CSIDH
Journal of Mathematical Cryptology, 2022 The SIDH and CSIDH are now the two most well-known post-quantum key exchange protocols from the supersingular isogeny-based cryptography, which have attracted much attention in recent years and served as the building blocks of other supersingular isogeny-
Qi Mingping
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Algebraic theories of power operations
Journal of Topology, Volume 16, Issue 4, Page 1543-1640, December 2023., 2023 Abstract We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well‐behaved theories of power operations for E∞$\mathbb {E}_\infty$ ring spectra.
William Balderrama
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Families of ϕ‐congruence subgroups of the modular group
Mathematika, Volume 69, Issue 4, Page 1104-1144, October 2023., 2023 Abstract We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ‐congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers.
Angelica Babei +2 more
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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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The SEA algorithm for endomorphisms of supersingular elliptic curves [PDF]
For a prime $p{\,>\,}3$ and a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$ with ${j(E)\notin\{0,1728\}}$, consider an endomorphism $α$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We prove that the trace of $α$ may be computed in $O(n^4(\log n)^2 + dLn^3)$ bit operations, where $n{\,=\,}\log(p ...
Travis Morrison +3 more
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On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring [PDF]
Mathematics of Computation, 2015 Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order.
K. Fung, B. Kane
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Class numbers, cyclic simple groups, and arithmetic
Journal of the London Mathematical Society, Volume 108, Issue 1, Page 238-272, July 2023., 2023 Abstract Here, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in a systematic way. To do this, we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms.
Miranda C. N. Cheng +2 more
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A two‐dimensional arithmetic André–Oort problem
Bulletin of the London Mathematical Society, Volume 55, Issue 3, Page 1459-1488, June 2023., 2023 Abstract We state and investigate an integral analogue of the André–Oort conjecture (in integral models of Shimura varieties). We establish an instance of this conjecture: the case of a modular curve, as a scheme over Z$\mathbf {Z}$. Our approach relies on equidistribution estimates related to subconvexity in analytic number theory and our result is ...
Rodolphe Richard
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