Power residues of Fourier coefficients of modular forms
| dc.creator | Weston, Tom | |
| dc.date | 2003-09-30 | |
| dc.date | 2003-10-09 | |
| dc.date.accessioned | 2026-07-07T05:01:32Z | |
| dc.date.available | 2026-07-07T05:01:32Z | |
| dc.description | 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. | |
| dc.description | Expanded to include applications to abelian extensions of imaginary quadratic fields. Several incorrect statements also eliminated | |
| dc.identifier | https://arxiv.org/abs/math/0309475 | |
| dc.identifier | http://arxiv.org/abs/math/0309475 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68708 | |
| dc.subject | Number Theory | |
| dc.subject | 11F30; 11G15;11A15 | |
| dc.title | Power residues of Fourier coefficients of modular forms | |
| dc.type | text |