The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\). (Q1094491)

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