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For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of theclass number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$.Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivialupper bounds for averages of $h_\ell(-d)$ have previously been known only for$\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of$h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper boundsfor certain higher moments for all primes $\ell \geq 3$.

Averages and moments associated to class numbers of imaginary quadratic fields