Some congruence property of modular forms (Q1382627)

The author discusses the existence of a Siegel modular form \(F\) of degree \(n\) and weight \(p-1\) satisfying \(F\equiv 1\pmod p\) componentwise in the Fourier expansion. If \(p>n+2\) is a regular prime, i.e. \(p\) does not divide the class number of \(\mathbb{Q} (e^{2 \pi i/p})\), then the Siegel-Eisenstein series \(E^{(n)}_{p-1}\) possesses this property. The author shows that \(p=16\;843\) and \(p=2\;124\;679\) are the only primes in the range \(3\leq p<4 \cdot 10^6\) satisfying \(E^{(2)}_{p-1} \not\equiv 1\pmod p\). In these two cases he constructs a different Siegel modular form \(F\) of degree 2 and weight \(p-1\) satisfying \(F\equiv 1\pmod p\). These particular \(F\) arise from the Siegel-Eisenstein series as a Hermitian modular form for \(\mathbb{Q} (\sqrt{-3})\) by restricting the domain of definition to the Siegel half-space. The last section refers to applications to mod \(p\) modular forms.