Supersingular prime (algebraic number theory)

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In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E is 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.

Noam Elkies showed that every elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero (if E does not have complex multiplication). Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of  X ln ⁡ X {\displaystyle {\frac {\sqrt {X}}{\ln X}}} , using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open.

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  p {\displaystyle {\mathfrak {p}}} for A is a finite place of K such that the reduction of A modulo  p {\displaystyle {\mathfrak {p}}} is a supersingular abelian variety.

## See also[edit]

* Supersingular prime (moonshine theory)

## References[edit]

* Elkies, Noam D. (1987). "The existence of infinitely many supersingular primes for every elliptic curve over Q". Invent. Math. 89 (3): 561–567. Bibcode:1987InMat..89..561E. doi:10.1007/BF01388985. MR 0903384. S2CID 123646933.
* Lang, Serge; Trotter, Hale F. (1976). Frobenius distributions in GL2-extensions. Lecture Notes in Mathematics. Vol. 504\. New York: Springer-Verlag. ISBN 0-387-07550-X. Zbl 0329.12015.
* Ogg, A. P. (1980). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). 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. Vol. 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. Vol. 106\. New York: Springer-Verlag. ISBN 0-387-96203-4. Zbl 0585.14026.

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Prime number classes

By formula|

* Fermat (22n + 1)
* Mersenne (2p − 1)
* Double Mersenne (22p−1 − 1)
* Wagstaff (2p + 1)/3
* Proth (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)
* Leyland (xy + yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin

By property|

* Wieferich (pair)
* Wall–Sun–Sun
* Wolstenholme
* Wilson
* Lucky
* Fortunate
* Ramanujan
* Pillai
* Regular
* Strong
* Stern
* Supersingular (elliptic curve)
* Supersingular (moonshine theory)
* Good
* Super
* Higgs
* Highly cototient
* Unique

Base-dependent|

* Palindromic
* Emirp
* Repunit (10n − 1)/9
* Permutable
* Circular
* Truncatable
* Minimal
* Delicate
* Primeval
* Full reptend
* Unique
* Happy
* Self
* Smarandache–Wellin
* Strobogrammatic
* Dihedral
* Tetradic

Patterns|

* Twin (p, p + 2)
* Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
* Triplet (p, p + 2 or p + 4, p + 6)
* Quadruplet (p, p + 2, p + 6, p + 8)
* k-tuple
* Cousin (p, p + 4)
* Sexy (p, p + 6)
* Chen
* Sophie Germain/Safe (p, 2p + 1)
* Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
* Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
* Balanced (consecutive p − n, p, p + n)

By size|

* Mega (1,000,000+ digits)
* Largest known
* list

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

* Pseudoprime
* Catalan
* Elliptic
* Euler
* Euler–Jacobi
* Fermat
* Frobenius
* Lucas
* Perrin
* Somer–Lucas
* Strong
* Carmichael number
* Almost prime
* Semiprime
* Sphenic number
* Interprime
* Pernicious

Related topics|

* Probable prime
* Industrial-grade prime
* Illegal prime
* Formula for primes
* Prime gap

First 60 primes|

* 2
* 3
* 5
* 7
* 11
* 13
* 17
* 19
* 23
* 29
* 31
* 37
* 41
* 43
* 47
* 53
* 59
* 61
* 67
* 71
* 73
* 79
* 83
* 89
* 97
* 101
* 103
* 107
* 109
* 113
* 127
* 131
* 137
* 139
* 149
* 151
* 157
* 163
* 167
* 173
* 179
* 181
* 191
* 193
* 197
* 199
* 211
* 223
* 227
* 229
* 233
* 239
* 241
* 251
* 257
* 263
* 269
* 271
* 277
* 281

List of prime numbers
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