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 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 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 [math]\displaystyle{ \frac{\sqrt{X}}{\ln X} }[/math], 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 F_p(t)—and A is an abelian variety defined over K, then a supersingular prime [math]\displaystyle{ \mathfrak{p} }[/math] for A is a finite place of K such that the reduction of A modulo [math]\displaystyle{ \mathfrak{p} }[/math] 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.