Imaginary quadratic fields with $\ell$-torsion-free class groups and specified split primes

Abstract

Given an odd prime $\ell$ and finite set of odd primes $S_{+}$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell$ and which splits at every prime in $S_{+}$. Notably, we do not require that $p\not\equiv -1(\mathrm{mod}\ell )$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.