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.