Twisted moments of automorphic \(L\)-functions. (English) Zbl 1214.11064
Let \(k\geq 2\) be an even integer. Let \(N\) be a squarefree integer such that the set \(H_k^*(N)\) of primitive forms of weight \(k\) over \(\Gamma_0(N)\) is not empty. To any \(f\in H_k^*(N)\) it is associated an automorphic \(L\)-function \(L(s,f)\), arising from the standard representation of \(\text{SU}(2)\). Denote by \(\Sigma^h\) the harmonic average. The authors present formulas for \(\Sigma_{f\in H_k^*(N)}^h L(\frac12,f)^2L(1,\text{Sym}^m f)^z\), \(z\in{\mathbb C}\), \(N\) sufficiently large.
Reviewer: Florin Nicolae (Berlin)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F12 | Automorphic forms, one variable |
References:
| [1] | Cogdell, James; Michel, Philippe, On the complex moments of symmetric power \(L\)-functions at \(s = 1\), Int. Math. Res. Not. IMRN, 31, 1561-1617 (2004), MR MR2035301 (2005f:11094) · Zbl 1093.11032 |
| [2] | Gelbart, Stephen; Jacquet, Hervé, A relation between automorphic representations of \(GL(2)\) and \(GL(3)\), Ann. Sci. École Norm. Sup. (4), 11, 4, 471-542 (1978), MR MR533066 (81e:10025) · Zbl 0406.10022 |
| [3] | Iwaniec, Henryk; Kowalski, Emmanuel, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53 (2004), American Mathematical Society: American Mathematical Society Providence, RI, MR MR2061214 (2005h:11005) · Zbl 1059.11001 |
| [4] | Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter, Low lying zeros of families of \(L\)-functions, Publ. Math. Inst. Hautes Études Sci., 2000, 91, 55-131 (2001), MR MR1828743 (2002h:11081) · Zbl 1012.11041 |
| [5] | Kim, Henry H., Functoriality for the exterior square of \(GL_4\) and the symmetric fourth of \(GL_2\), J. Amer. Math. Soc., 16, 1, 139-183 (2003), (electronic), with Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Kim and Peter Sarnak. MR MR1937203 (2003k:11083) · Zbl 1018.11024 |
| [6] | Kim, Henry H.; Shahidi, Freydoon, Cuspidality of symmetric powers with applications, Duke Math. J., 112, 1, 177-197 (2002), MR MR1890650 (2003a:11057) · Zbl 1074.11027 |
| [7] | Royer, Emmanuel; Wu, Jie, Special values of symmetric power \(L\)-functions and Hecke eigenvalues, J. Théor. Nombres Bordeaux, 19, 3, 703-753 (2007), MR MR2388795 · Zbl 1196.11071 |
| [8] | Tenenbaum, Gérald; Wu, Jie, Moyennes de certaines fonctions multiplicatives sur les entiers friables, J. Reine Angew. Math., 564, 119-166 (2003), MR MR2021037 (2004m:11151) · Zbl 1195.11132 |
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