Abstract view
Power Residues of Fourier Coefficients of Modular Forms


Published:20051001
Printed: Oct 2005
Abstract
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 Cebatorevstyle
Galoistheoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CMtype 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 $p$.