Linear relations between modular form coefficients and non-ordinary primes

Abstract

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on SL2(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) = Sigma(infinity)(n=1), a(f) (n)(q)(n) is an element of S-k boolean AND O-L [[q]] be a normalized Hecke eigenform (note that q := e(2 pi iz)), and let k = delta(k) (mod 12), where delta(k) is an element of {4, 6, 8, 10, 14}. Reproducing earlier results of Hatada and Hida, if p is a prime for which k equivalent to J(k) (mod p - 1), and p subset of O-L is a prime ideal above p, a proof is given to show that a(f) (p) equivalent to 0 (mod p).