On mod $p$ singular modular forms II

TL;DR

This work extends the theory of mod $p^m$ singular Siegel modular forms to the vector-valued setting, showing that a rank constraint forces a congruence between the vector-weight and the $p$-adic level. The authors develop a framework based on partial Fourier series and controlled twists to avoid theta decompositions, culminating in the key result that for an odd prime $p$, a mod $p^m$ singular vector-valued form of rank $r<n$ satisfies $2k - r ≡ 0$ mod $(p-1)p^{m-1}$. The approach generalizes prior scalar results (BoKi) and is presented as conceptually simpler, with potential extensions to broader congruence settings and nebentypus characters. Overall, the paper clarifies how weight-rank congruences behave in the vector-valued Siegel setting and points toward wider applicability of these techniques in arithmetic automorphic forms.

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.