Supersingular prime (for an elliptic curve) References Navigation menu[update]10.1007/BF0138898509033840329.120150448.100210585.14026e

Fermat (22n + 1)Mersenne (2p − 1)Double Mersenne (22p−1 − 1)Wagstaff (2p + 1)/3Proth (k·2n + 1)Factorial (n! ± 1)Primorial (pn# ± 1)Euclid (pn# + 1)Pythagorean (4n + 1)Pierpont (2m·3n + 1)Quartan (x4 + y4)Solinas (2m ± 2n ± 1)Cullen (n·2n + 1)Woodall (n·2n − 1)Cuban (x3 − y3)/(x − y)Carol (2n − 1)2 − 2Kynea (2n + 1)2 − 2Leyland (xy + yx)Thabit (3·2n − 1)Williams ((b−1)·bn − 1)Mills (⌊A3n⌋)WieferichpairWall–Sun–SunWolstenholmeWilsonLuckyFortunateRamanujanPillaiRegularStrongSternSupersingular (elliptic curve)Supersingular (moonshine theory)GoodSuperHiggsHighly cototientTwin (p, p + 2)Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …)Triplet (p, p + 2 or p + 4, p + 6)Quadruplet (p, p + 2, p + 6, p + 8)k−TupleCousin (p, p + 4)Sexy (p, p + 6)ChenSophie Germain (p, 2p + 1)Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)Safe (p, (p − 1)/2)Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)Balanced (consecutive p − n, p, p + n)Eisenstein primeGaussian primeProbable primeIndustrial-grade primeIllegal primeFormula for primesPrime gap


Classes of prime numbersAlgebraic number theoryUnsolved problems in mathematics


algebraic number theoryprime numberelliptic curverational numbersreductionsupersingular elliptic curveresidue fieldElkies (1987)Lang & Trotter (1976)global fieldfinite extensionabelian varietyfinite placeabelian variety




In algebraic number theory, a supersingular prime is a prime number with a certain relationship to a given elliptic curve. If the curve E defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field Fp.


Elkies (1987) showed that any elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero. Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of Xln⁡Xdisplaystyle frac sqrt Xln X, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2012[update], this conjecture is open.[needs update?]


More generally, if K is any global field—i.e., a finite extension either of Q or of Fp(t)—and A is an abelian variety defined over K, then a supersingular prime pdisplaystyle mathfrak p for A is a finite place of K such that the reduction of A modulo pdisplaystyle mathfrak p is a supersingular abelian variety.



References



  • Elkies, Noam D. (1987). "The existence of infinitely many supersingular primes for every elliptic curve over Q". Invent. Math. 89 (3): 561–567. doi:10.1007/BF01388985. MR 0903384..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em


  • Lang, Serge; Trotter, Hale F. (1976). Frobenius distributions in GL2-extensions. Lecture Notes in Mathematics. 504. New York: Springer-Verlag. ISBN 0-387-07550-X. Zbl 0329.12015.


  • Ogg, A. P. (1980). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey. The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979. Proc. Symp. Pure Math. 37. Providence, RI: American Mathematical Society. pp. 521–532. ISBN 0-8218-1440-0. Zbl 0448.10021.


  • Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. 106. New York: Springer-Verlag. ISBN 0-387-96203-4. Zbl 0585.14026.







Algebraic number theory, Classes of prime numbers, Unsolved problems in mathematicsUncategorized
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