On the Fourier coefficients of modular forms. II
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 of forms which are old at \(p\). The eigenvalues of \(U_p\) are again algebraic integers of absolute value \(p^{(k+ 1)/2}\). Write \[ S= S(k, b)= \begin{cases} S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^b) &\text{ if }b\not\equiv 0\pmod {p-1}\\ S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p )^{p-\text{old}} &\text{ if } b\equiv 0 \pmod {p-1}. \end{cases} \] For an interval \(I\subset \mathbb{R}\) let \(S_I= \bigoplus_{\lambda\in I} S_\lambda\), where \(S_\lambda \subset S\) is such that all of the eigenvalues of \(U_p\) on \(S_\lambda\) have slope \(\lambda\). In [loc. cit.] it was shown that, for \(N\geq 5\) and \(0< a< p-1\), the Newton polygon \(P(k, a)= \text{det} (1- U_p T|S(k,a))\) is bounded below by an explicit Hodge polygon. For any Newton polygon its associated contact polygon is defined as the highest Hodge polygon lying on or below it and having the same endpoints. In the context of [loc. cit.], one defines \(t^i= t^i (k, a)\), \(i= 1, \dots, k\), as the number of units that the slope \(i\) edge of the Hodge polygon should be raised so that it meets the slope \(i\) edge of the contact polygon of \(P(k, a)\) (or equivalently, so that it meets the Newton polygon). Let \(i\) be an integer \(1\leq i\leq k\). As a first result one has that \(t^i (k, a)= 0\) if one has \(i\leq a\) and \(k+ 1- i\leq p- 1- a\). For \(i= a+1\), \(t^i (k, a) =0\) if and only if \(S_2 (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^{k+ 1-i} )_{(0, 1)}= 0\). If \(k+ 1-i= p-a\) then \(t^i (k, a)= 0\) if and only if \(S_2 (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^{-i} )_{(0, 1)} =0\). For \(i> a+ 1\) or \(k+ 1-i> p-a\) one has \(t^i (k, a)> 0\). The main results of the paper give the dimensions of \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}) (\chi^a)_I\) for suitable values of \(k\), \(a\) and the interval \(I\subset \mathbb{R}\). The following cases are dealt with: (i) \(i\leq a\), \(k+ 1- i\leq p- 1-a\), and \(I= [i]\); (ii) \(i+1\leq a\), \(k+ 1- i\leq p- 1- a\), and \(I= (i, i+1)\); (iii) \(i=0\), \(a\leq p-1- k\), and \(I= (i, i+1)\); (iv) \(i=k\), \(a\geq k\), and \(I= (i, i+1)\); (v) \(k>1\), \(a= (p-1)- (k-1)\), \(i=1\), and \(I= (0, 1)\); (vi) \(k> 1\), \(a= k-1\), \(i=k\), and \(I= (0, 1)\). Actually, (i), (ii), (iii) and (iv) also hold for \(N\leq 2\wedge k\equiv a\pmod 2\). These results follow from a comparison of modular forms and cohomology, much in the spirit of [loc. cit.], in a motivic setting. Various complicated expressions for suitable (crystalline) cohomology groups of the motive at hand are proved. These proofs take up the greater part of the text.
\(p\)-adic valuations of eigenvalues of the Hecke operator
scientific article; zbMATH DE number 861918
scientific article