For an elliptic curve over , a prime of good reduction of is said to be supersingular with respect to if the reduced elliptic curve has no points of order over the algebraic closure of the prime field ; this is the case if and only if the ring of multiplicators of is a (noncommutative) maximal order in the quaternion algebra .
The author, “thinking quaternionically”, establishes the existence of infinitely many supersingular primes with respect to a given elliptic curve over , a fact not previously known for non-CM curves. He extends this result to elliptic curves over any algebraic number field of odd degree over . The method of proof essentially depends on work of M. Deuring [Abh. Math. Semin. Hansische Univ. 14, 197–272 (1941; Zbl 0025.02003)].