On mod $p$ singular modular forms II

TL;DR

This paper extends the concept of mod p^m singular Siegel modular forms to vector-valued forms, establishing a congruence relation between scalar weight and p-rank, with a simplified proof compared to previous work.

Contribution

It generalizes the notion of mod p^m singular Siegel modular forms to vector-valued cases and proves a congruence relation with a simpler proof.

Findings

Established a congruence mod (p-1)p^{m-1} between scalar weight and p-rank for vector-valued forms.

Extended the theory of mod p^m singular Siegel modular forms to a broader class.

Provided a simpler proof compared to previous scalar-valued case.

Abstract

We generalize the notion of mod $p^m$ singular Siegel modular forms of $p$-rank $r$ to the vector-valued case and we show that also in this case a congruence mod $(p-1)p^{m-1}$ between the scalar weight and the $p$-rank must hold. In some sense our proof is even simpler than the one we gave previously in the scaler valued case.