Supersingular prime (algebraic number theory) explained

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). conjectured that the number of supersingular primes less than a bound X is within a constant multiple of

(\sqrt{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 F_p(t)—and A is an abelian variety defined over K, then a supersingular prime

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

See also

• Supersingular prime (moonshine theory)

References

• Noam Elkies . Noam D. . Elkies . The existence of infinitely many supersingular primes for every elliptic curve over Q . . 89 . 1987 . 3 . 561–567 . 10.1007/BF01388985 . 1987InMat..89..561E . 0903384. 123646933 .
• Book: Lang . Serge . Serge Lang. Trotter . Hale F. . Hale F. Trotter. 1976 . Frobenius distributions in GL₂-extensions . New York . . 0-387-07550-X . 0329.12015 . Lecture Notes in Mathematics . 504 .
• Book: Ogg, A. P. . Andrew Ogg

. Andrew Ogg . Modular Functions . The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979 . Bruce . Cooperstein . Geoffrey . Mason . Providence, RI . . 521–532 . 1980 . 0-8218-1440-0 . 0448.10021 . Proc. Symp. Pure Math. . 37 .

• Book: Silverman, Joseph H. . Joseph H. Silverman

. Joseph H. Silverman . 1986 . The Arithmetic of Elliptic Curves . . New York . 0-387-96203-4 . 0585.14026 . . 106 .