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A Remark on a Modular Analogue of the Sato--Tate Conjecture |
Canad. Math. Bull. 50(2007), 234-242
Published:2007-06-01
Printed: Jun 2007
Printed: Jun 2007
- Wentang Kuo
Abstract
The original Sato--Tate Conjecture concerns the angle distribution
of the eigenvalues arising from non-CM elliptic curves. In this paper,
we formulate a modular analogue of the Sato--Tate Conjecture and prove
that the angles arising from non-CM holomorphic Hecke
eigenforms with non-trivial central characters are not distributed
with respect to the Sate--Tate measure
for non-CM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
| Keywords: | $L$-functions, Elliptic curves, Sato--Tate |
| MSC Classifications: |
11F03 - Modular and automorphic functions 11F25 - Hecke-Petersson operators, differential operators (one variable) |