Averages and moments associated to class numbers of imaginary quadratic fields

For any odd prime 𝓁, let $h𝓁(−d)$ denote the 𝓁-part of the class number of the imaginary quadratic field $Q(√−d)$. Nontrivial pointwise upper bounds are known only for $𝓁 = 3$; nontrivial upper bounds for averagesof $h𝓁(−d)$ have previously been known only for $𝓁 = 3, 5$. In this paper we prove nontrivial upper bounds for the average of $h𝓁(−d)$ for all primes $𝓁 ≥ 7$, as well as nontrivial upper bounds for certain higher moments for all primes $𝓁 ≥ 3$.