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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   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 (1976) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of ${\displaystyle {\frac {\sqrt {X}}{\ln X}}}$, 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 ${\displaystyle {\mathfrak {p}}}$ for A is a finite place of K such that the reduction of A modulo ${\displaystyle {\mathfrak {p}}}$ is a supersingular abelian variety .

See also [ edit ]

• Supersingular prime (moonshine theory)

References [ edit ]

• 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 GL ₂ -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 .