Extremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami

For even k ≥ 4 let f_k = 1 + O(q^{dim M_k}) be the extremal modular form of weight k for SL_2(ℤ). Bannai, Koike, Shinohara and Tagami conjectured that if k = 12μ and f_k mod p is a nonconstant power series in q^p (their Case (2)), then f_k(τ) ≡ g(p^r τ) (mod p) for an extremal modular form g of smaller weight and some r ≥ 1. We observe that this is a corollary of the classical theory of modular forms modulo p: the key input is the equality w(h^p) = p·w(h) for the filtration of a p-th power, which follows from Swinnerton-Dyer's structure theorem and underlies the theorem of Serre and Katz on the kernel of θ. For every even k ≥ 4 and every prime p in Case (2) we get f_k(τ) ≡ f_{k'}(pτ) (mod p) with k' = w(f_k mod p)/p, where 4 ≤ k' ≤ k/p and k' ≡ k (mod p − 1). So the conjecture holds with r = 1, and by iteration also with the largest possible r. For k = 12μ every prime in Case (2) satisfies 11 ≤ p ≤ 3μ and p ≠ 13, and Case (2) is decided by a finite test. Exact computations for all weights 12μ ≤ 7200 and all even k ≤ 2400, covering 9433 pairs (k, p) in Case (2), agree with these statements. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-782-001, with duplicate record OWR-782-002 (Oberwolfach Reports 1/2005, E. Bannai, joint work with M. Koike, M. Shinohara and M. Tagami, "Spherical designs, extremal lattices and the Fourier coefficients modulo p of the extremal modular forms", Conjecture 4).