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Power Residues of Fourier Coefficients of Modular Forms

 Printed: Oct 2005
  • Tom Weston
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Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ 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 CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the 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 $p$.
MSC Classifications: 11F30, 11G15, 11A15 show english descriptions Fourier coefficients of automorphic forms
Complex multiplication and moduli of abelian varieties [See also 14K22]
Power residues, reciprocity
11F30 - Fourier coefficients of automorphic forms
11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22]
11A15 - Power residues, reciprocity

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