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 ${\displaystyle E}$ is defined over the rational numbers, then a prime ${\displaystyle p}$ is supersingular for E if the reduction of ${\displaystyle E}$ modulo ${\displaystyle p}$ is a supersingular elliptic curve over the residue field ${\displaystyle \mathbb {F} _{p}}$.

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 ${\displaystyle E}$ does not have complex multiplication). Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound ${\displaystyle X}$ is within a constant multiple of ${\displaystyle {\sqrt {X}}/\log X}$, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open.

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

• Supersingular prime (moonshine theory)

• 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.