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A Remark on a Modular Analogue of the Sato--Tate Conjecture

Open Access article
 Printed: Jun 2007
  • Wentang Kuo
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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 $L$-functions, Elliptic curves, Sato--Tate
MSC Classifications: 11F03, 11F25 show english descriptions Modular and automorphic functions
Hecke-Petersson operators, differential operators (one variable)
11F03 - Modular and automorphic functions
11F25 - Hecke-Petersson operators, differential operators (one variable)

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