Power Residues of Fourier Coefficients of Modular Forms

Weston, Tom

Canadian journal of mathematics, Tome 57 (2005) no. 5, pp. 1102-1120

Résumé

Let $\rho :\,{{G}_{Q}}\,\to \,\text{G}{{\text{L}}_{n}}\left( {{Q}_{\ell }} \right)$ be a motivic $\ell $ -adic Galois representation. For fixed $m\,>\,1$ we initiate an investigation of the density of the set of primes $p$ such that the trace of the image of an arithmetic Frobenius at $p$ under $\rho $ is an $m$ -th power residue modulo $p$ . Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals $1/m$ whenever the image of $\rho $ is open. We further conjecture that for such $\rho $ the set of these primes $p$ is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain $m$ in the complementary case of modular forms of $\text{CM}$ -type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the $\text{CM}$ case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at $p$ in abelian extensions of imaginary quadratic fields unramified away from $p$ .

Détail

DOI : 10.4153/CJM-2005-042-5

Mots-clés : Primary: 11F30, secondary: 11G15, 11A15

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