Supersingular primes for points on $X_0(p)/w_p$

Abstract

For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for which the infinitude of supersingular primes is known. We give concrete examples illustrating how these techniques can be explicitly used to construct supersingular primes for such elliptic curves. Finally, we discuss generalizations to points defined over larger number fields and indicate the types of obstructions that arise for higher level modular curves.