Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $$\mathrm {SU}_{2,2}({\mathcal {O}}_K)$$ SU 2 , 2 ( O K ) where K is the imaginary-quadratic number field of discriminant $$-d$$ - d , $$d \in \{4, 7,8,11,15,19,20,24\}$$ d ∈ { 4 , 7 , 8 , 11 , 15 , 19 , 20 , 24 } we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

PROOF OF SOME CONJECTURAL CONGRUENCES OF DA SILVA AND SELLERS

Let $p_{\{3, 3\}}(n)$ denote the number of $3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc.104(3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences using the theory of modular forms.

Modular iterated integrals associated with cusp forms

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.