Bulletin of the London Mathematical Society



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LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES


YOUNGJU CHOIE a1 1 , WINFRIED KOHNEN a2 and KEN ONO a3 1
a1 Department of Mathematics, Pohang Institute of Science and Technology, Pohang 790-784 Korea yjc@postech.ac.kr
a2 Universität Heidelberg, Mathematisches Institut, INF 288, D-69120 Heidelberg Germany winfried@mathi.uni-heidelberg.de
a3 Department of Mathematics, University of Wisconsin, Madison, WI 53706 USA ono@math.wisc.edu

Article author query
choie y   [Google Scholar] 
kohnen w   [Google Scholar] 
ono k   [Google Scholar] 
 

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 $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.

(Received April 7 2004)
(Revised July 7 2004)

Maths Classification

11F33 (primary); 11F11 (secondary).



Footnotes

1 The first author is partially supported by KOSEF R01-2003-00011596-0. The third author is grateful for the support of a grant from the National Science Foundation, and the generous support of the Alfred P. Sloan, David and Lucile Packard, H. I. Romnes, and John S. Guggenheim Fellowships.