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Fields of rationality of cusp forms
- Source :
- Israel Journal of Mathematics. 222:973-1028
- Publication Year :
- 2017
- Publisher :
- Springer Science and Business Media LLC, 2017.
-
Abstract
- In this paper, we prove that for any totally real field $F$, weight $k$, and nebentypus character $\chi$, the proportion of Hilbert cusp forms over $F$ of weight $k$ and character $\chi$ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible $GL_2$ representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for $GL_2$; and third, a Plancherel equidsitribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin-Templier and mirrors work of Finis-Lapid-Mueller by introducing an explicit bound for certain families of orbital integrals.<br />Comment: 41 pages
- Subjects :
- Cusp (singularity)
Pure mathematics
Mathematics - Number Theory
Mathematics::Number Theory
General Mathematics
010102 general mathematics
Zero (complex analysis)
Field (mathematics)
Equidistribution theorem
01 natural sciences
Measure (mathematics)
Upper and lower bounds
Character (mathematics)
Bounded function
0103 physical sciences
FOS: Mathematics
Number Theory (math.NT)
010307 mathematical physics
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 15658511 and 00212172
- Volume :
- 222
- Database :
- OpenAIRE
- Journal :
- Israel Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....c3c55e05d6d54f0e093a23966a7764fa