In this paper, we consider sums of class numbers of the type ∑m ≡ a (mod p) H (4n − m2), where p is an odd prime, n ∈ ℕ, and a ∈ ℤ. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for p = 5 and 7, we then prove a conjecture of Brown et al. in the case that n = ℓ is prime.

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.

While investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$ . It was later determined by Kohnen and Zagier that the generating function for the function $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.