Imaginary Quadratic Fields With ℓ-Torsion-Free Class Groups and Specified Split Primes

Given an odd prime and finite set of odd primes S₊, we prove the existence of an imaginary quadratic field whose class number is indivisible by and which splits at every prime in S₊. Notably, we do not require that p ≡ -1 \,\;(mod\, ) 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.