Supersingular prime (algebraic number theory)

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) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of Xln⁡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 for A is a finite place of K such that the reduction of A modulo 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. Bibcode: 1987InMat..89..561E.
* 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.
* 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.
* Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. 106. New York: Springer-Verlag. ISBN 0-387-96203-4.

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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)
* Carol (2n − 1)2 − 2
* Kynea (2n \+ 1)2 − 2
* Leyland (xy \+ yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin
* Partitions
* Bell
* Motzkin

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

Base-dependent|

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

Patterns|

* Twin (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−Tuple
* Cousin (p, p \+ 4)
* Sexy (p, p \+ 6)
* Chen
* Sophie 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)

By size|

* Titanic (1,000+ digits)
* Gigantic (10,000+ digits)
* Mega (1,000,000+ digits)
* Largest known

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

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