Selberg's orthogonality conjecture and symmetric power L-functions
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 symm...
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| Published in | Journal of number theory Vol. 238; pp. 967 - 977 |
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| Format | Journal Article |
| Language | English |
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01.09.2022
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Wong, Peng-Jie |
| Author_xml | – sequence: 1 givenname: Peng-Jie surname: Wong fullname: Wong, Peng-Jie email: pengjie.wong@uleth.ca organization: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada |
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| Cites_doi | 10.2307/1969183 10.1353/ajm.2006.0042 10.2977/PRIMS/31 10.1215/S0012-7094-93-07225-0 10.4007/annals.2010.171.779 10.1007/s10240-008-0016-1 10.1215/00127094-3714971 10.1515/form.2005.17.3.493 10.1016/j.jnt.2018.09.010 10.1090/S0894-0347-2010-00689-3 10.4007/annals.2014.179.2.3 10.1090/S0273-0979-1994-00479-3 10.1007/s10240-008-0015-2 10.4007/annals.2011.173.3.4 |
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| References | Barnet-Lamb, Gee, Geraghty, Taylor (br0030) 2014; 179 Taylor (br0190) 2008; 108 Kaczorowski, Perelli (br0120) 2011; 173 Murty (br0140) 1994; 31 Clozel, Harris, Taylor (br0070) 2008; 108 Ramakrishnan (br0160) 2009 Barnet-Lamb, Geraghty, Harris, Taylor (br0040) 2011; 47 Arthur, Clozel (br0010) 1990 Conrey, Ghosh (br0090) 1993; 72 Barnet-Lamb, Gee, Geraghty (br0020) 2011; 24 Brumley (br0060) 2006; 128 Selberg (br0170) 1992 Kaczorowski (br0110) 2006 Brauer (br0050) 1947; 48 Liu, Ye (br0130) 2005; 17 Serre (br0180) 1989 Newton, Thorne (br0150) 2021 Clozel, Thorne (br0080) 2017; 166 Wong (br0200) 2019; 196 Harris, Shepherd-Barron, Taylor (br0100) 2010; 171 Harris (10.1016/j.jnt.2021.11.001_br0100) 2010; 171 Brauer (10.1016/j.jnt.2021.11.001_br0050) 1947; 48 Clozel (10.1016/j.jnt.2021.11.001_br0070) 2008; 108 Kaczorowski (10.1016/j.jnt.2021.11.001_br0110) 2006 Selberg (10.1016/j.jnt.2021.11.001_br0170) 1992 Liu (10.1016/j.jnt.2021.11.001_br0130) 2005; 17 Barnet-Lamb (10.1016/j.jnt.2021.11.001_br0020) 2011; 24 Clozel (10.1016/j.jnt.2021.11.001_br0080) 2017; 166 Conrey (10.1016/j.jnt.2021.11.001_br0090) 1993; 72 Serre (10.1016/j.jnt.2021.11.001_br0180) 1989 Taylor (10.1016/j.jnt.2021.11.001_br0190) 2008; 108 Arthur (10.1016/j.jnt.2021.11.001_br0010) 1990 Kaczorowski (10.1016/j.jnt.2021.11.001_br0120) 2011; 173 Brumley (10.1016/j.jnt.2021.11.001_br0060) 2006; 128 Barnet-Lamb (10.1016/j.jnt.2021.11.001_br0040) 2011; 47 Newton (10.1016/j.jnt.2021.11.001_br0150) 2021 Wong (10.1016/j.jnt.2021.11.001_br0200) 2019; 196 Murty (10.1016/j.jnt.2021.11.001_br0140) 1994; 31 Ramakrishnan (10.1016/j.jnt.2021.11.001_br0160) 2009 Barnet-Lamb (10.1016/j.jnt.2021.11.001_br0030) 2014; 179 |
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| Snippet | 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... |
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| Title | Selberg's orthogonality conjecture and symmetric power L-functions |
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