Power residues of Fourier coefficients of modular forms

Details Power residues of Fourier coefficients of modular forms Power residues of Fourier coefficients of modular forms

2003-09-30

arxiv.org/abs/math/0309475v2

Mathematics

Number Theory

Expanded to include applications to abelian extensions of imaginary quadratic fields. Several incorrect statements also elimin

Scientific paper

Let r : G_Q -> GL_n Q_l be a motivic l-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 r 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 r is open. We further conjecture that for such r 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.

Affiliated with

Also associated with

No associations

Rating

Power residues of Fourier coefficients of modular forms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Power residues of Fourier coefficients of modular forms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Power residues of Fourier coefficients of modular forms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-280271