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 $\SL_2(\ZZ)$. 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 $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.