On the Fourier coefficients of modular forms. II
type: Item
instance of: scholarly article
title: On the Fourier coefficients of modular forms. II
zbMATH Open document ID: 0856.11022
DOI: 10.1007/BF01446299
author: Douglas Ulmer
published in: Mathematische Annalen
publication date: 1996-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/165409
review text: This paper is a continuation of [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, 129-160 (1995; Zbl 0827.11024)]. It is concerned with the study of the \(p\)-adic valuations of eigenvalues of the Hecke operator \(U_p\) acting on certain spaces of cusp forms of level divisible by \(p\) for the congruence subgroup \(\Gamma_1 (pN)\) of \(\text{SL}_2 (\mathbb{Z})\), \(p\nmid N\). For a positive integer \(M\), a non-negative integer \(k\), and a commutative \(\mathbb{Q}\)-algebra \(R\), let \(S_{k+2} (\Gamma_1 (M); R)= S_{k+2} (\Gamma_1 (M); \mathbb{Q}) \otimes_\mathbb{Q} R\), with \(S_{k+2} (\Gamma_1 (M); \mathbb{Q})\) the \(\mathbb{Q}\)-vector space of cusp forms (for \(\Gamma_1 (M)\)) of weight \(k+2\) all of whose Fourier coefficients at the standard cusp \(\infty\) are rational numbers. On the space \(S_{k+2} (\Gamma_1 (M); R)\) one has an action of Hecke operators \(T_\ell\) for all primes \(\ell \nmid M\), \(U_\ell\) for \(\ell \mid M\), and the diamond operators \(\langle d\rangle_M\) for \(d\in (\mathbb{Z}/ M\mathbb{Z})^\times\). In particular, for \(M= pN\), \(p\nmid N\), and \(R= \mathbb{Q}_p\), one has a decomposition \[ S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) \cong \bigoplus^{p- 2}_{j= 0} S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^j), \] where \(\chi: (\mathbb{Z}/ p\mathbb{Z} )^\times\to \mathbb{Z}_p\) is the Teichmüller character, and where \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi)\) consists of those \(f\in S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p)\) such that \(\langle d\rangle_p f= \chi(d) f\) for all \(d\in (\mathbb{Z}/ p\mathbb{Z})^\times\). In the sequel \(p\) will always be an odd prime, \(N\geq 5\), \(0\leq k< p\), and \(a\) will be an integer with \(0< a< p-1\). Then the action of \(U_p\) on \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^a)\) (which is a space of newforms at \(p\)) is semi-simple with eigenvalues algebraic integers of absolute value \(p^{(k+ 1)/2}\). \(U_p\) also acts on \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p )^{p-\text{old}}\), the space …
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 861918
zbMATH Keywords: Galois representation
zbMATH Keywords: Gauss-Manin connection
zbMATH Keywords: cohomology groups of motives
zbMATH Keywords: \(p\)-adic valuations of eigenvalues of the Hecke operator
zbMATH Keywords: spaces of cusp forms
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: diamond operators
zbMATH Keywords: Newton polygon
zbMATH Keywords: Hodge polygon
zbMATH Keywords: contact polygon
zbMATH Keywords: modular forms
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 861918