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 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 (1976) conjectured that the number of supersingular primes less than a bound X 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 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 for A is a finite place of K such that the reduction of A modulo 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. 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.