Selberg's orthogonality conjecture and symmetric power L-functions

• Published in: Journal of number theory Vol. 238; pp. 967 - 977
• Main Author: Wong, Peng-Jie
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2022
• Subjects: Selberg's orthogonality conjecture; Symmetric power L-functions; Selberg's orthogonality conjecture; 11M41; Symmetric power L-functions; 11F66
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2021.11.001

Let π be a cuspidal representation of GL2(AQ) defined by a non-CM holomorphic newform of weight w≥2, and let K/Q be a totally real Galois extension with Galois group G. In this article, under Selberg's orthogonality conjecture, we show that for any irreducible character χ of G, the twisted symmetric power L-function L(s,Symmπ×χ) is a primitive function in the Selberg class, and it is automorphic subject to further the solvability of K/Q. The key new idea is to apply the work of Barnet-Lamb, Geraghty, Harris, and Taylor on the potential automorphy of Symmπ.