Supersingular prime (algebraic number theory)

Template:Short description Template:No footnotes 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). Template:Harvtxt conjectured that the number of supersingular primes less than a bound X is within a constant multiple of $\frac{\sqrt{X}}{\ln X}$, 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 $𝔭$ for A is a finite place of K such that the reduction of A modulo $𝔭$ is a supersingular abelian variety.

See also

• Supersingular prime (moonshine theory)

References

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