We obtain a condition describing when the quasimodular forms given by the Bloch-Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator ∆. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials. * 1 As quasimodular forms were not yet defined, Schoeneberg only showed that θP is modular if P is harmonic. However, for every polynomial P it follows that θP is quasimodular by decomposing P as in Formula (1) below.

We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko–Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the solutions is the polynomial dependence of the index parameter. The results yield an explicit conjectural description for all double ramification cycle integrals in the Gromov–Witten theory of K3 surfaces.

We consider a system of polynomials f 1 , . . . , f R ∈ Z[x 1 , . . . , x n ] of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch [Bir62] we find quantitative asymptotics (in terms of the maximum of the absolute value of the coefficients of these polynomials) for the number of integer zeros of this system within a growing box. Using a quantitative version of the Nullstellensatz, we obtain a quantitative strong approximation result, i.e. an upper bound on the smallest integer zero provided the system of polynomials is non-singular.