Linear Relations Between Modular form Coefficients and Non‐Ordinary Primes

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on SL 2 (Z). As a consequence, spaces M k are identified, in which there are universal p ‐divisibility properties for certain p ‐power coefficients. As a corollary, let f ( z ) = ∑ n = 1 ∞a f( n ) q n ∈ S k ∩ O L [ [ q ] ]be a normalized Hecke eigenform (note that q:=e 2πiz) , and let k ≡ δ( k ) (mod 12), where δ( k ) ∈ {4, 6, 8, 10, 14}. Reproducing earlier results of Hatada and Hida, if p is a prime for which k ≡ δ( k ) (mod p −1), and p ⊂ O L is a prime ideal above p , a proof is given to show that a f ( p ) ≡ 0 (mod p). 2000 Mathematics Subject Classification 11F33 (primary), 11F11 (secondary).