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

Abstract

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.