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 is defined over the rational numbers, then a prime is supersingular for E if the reduction of modulo is a supersingular elliptic curve over the residue field .

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

More generally, if is any global field—i.e., a finite extension either of or of —and is an abelian variety defined over , then a supersingular prime for A is a finite place of such that the reduction of modulo is a supersingular abelian variety.

• Supersingular prime (moonshine theory)

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• 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.