Supersingular prime (for an elliptic curve)
From HandWiki
In algebraic number theory, a supersingular prime is a prime number with a certain relationship to a given elliptic curve. If the curve E 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.
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 [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.
- 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.
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