Supersingular prime (algebraic number theory)

In the context of elliptic curves and algebraic number theory, a supersingular prime is a prime number p with the property that the reduction modulo p of a given elliptic curve has an elliptic curve with a special property: it is supersingular. More precisely, let E be an elliptic curve defined over the rational numbers (or a number field). We say that p is a supersingular prime for E if the reduction of E modulo p, denoted E_p, is a supersingular elliptic curve over the finite field F_p or some finite extension of F_p.

An elliptic curve E_p over a finite field F_q (where q is a power of p) is said to be supersingular if the group of its F_q-rational points E_p (F_q) contains no points of order p. Equivalently, the endomorphism ring of E_p is an order in a quaternion algebra over Q (rather than an order in an imaginary quadratic field). Other equivalent definitions involve the vanishing of the Hasse invariant or the p-torsion points.

The set of supersingular primes for a given elliptic curve E is infinite, a result proven by Elkies in 1987. This answered a conjecture of Serre. The density of the supersingular primes is zero.

The notion of supersingularity is important in the study of modular forms and the arithmetic of elliptic curves. Supersingular primes play a role in determining the image of Galois representations associated with elliptic curves and in the distribution of Frobenius eigenvalues.

The concept can be generalized. For example, one can talk about supersingular abelian varieties. These are abelian varieties whose p-rank is zero.

The arithmetic properties of supersingular primes are an active area of research in number theory.