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Moments of the central L-values of the Asai lifts

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  04 March 2024

Wenzhi Luo
Wenzhi Luo*
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States
*
e-mail: [email protected]
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Abstract

We study some analytic properties of the Asai lifts associated with cuspidal Hilbert modular forms, and prove sharp bounds for the second moment of their central L-values.

Keywords

Hilbert modular formAsai liftcentral L-valuesPetersson formula

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F30: Fourier coefficients of automorphic forms 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 796 - 804
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This research is partially supported by a Simons Foundation Collaboration Grant.

References

Asai, T., On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method . Math. Ann. 226(1977), no. 1, 81–94.CrossRefGoogle Scholar
Garrett, P. B., Holomorphic Hilbert modular forms, Wadsworth Inc., Monterey, California, 1990.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Tables of integrals, series and products, Academic Press, New York, 1965.Google Scholar
Hoffstein, J. and Lockhart, P., Coefficients of Maass forms and the Siegel zero, appendix by D. Goldfeld, J. Hoffstein, and D. Lieman, an effective zero free region . Ann. Math. 140(1994), 161–181.CrossRefGoogle Scholar
Iwaniec, H., Small eigenvalues of Laplacian for ${\varGamma}_0(N)$ . Acta Arith. 56(1990), no. 1, 65–82.CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
Krishnamurthy, M., The Asai transfer to $G{L}_4$ via the Langlands–Shahidi method . Int. Math. Res. Not. IMRN 2003(2003), no. 41, 2221–2254.CrossRefGoogle Scholar
Luo, W., Poincaré series and Hilbert modular forms. Rankin memorial issues . Ramanujan J. 7(2003), nos. 1–3, 129–140.CrossRefGoogle Scholar
Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and theorems for the special functions of mathematical physics, Springer, New York, 1966.CrossRefGoogle Scholar
Prasad, D. and Ramakrishnan, D., On the cuspidality criterion for the Asai transfer to $GL(4)$ , Appendix A to the paper ‘Determination of cusp forms on $GL(2)$ by coefficients restricted to quadratic subfields’ by M. Krishnamurthy . J. Number Theory 132(2012), no. 6, 1376–1383.Google Scholar
Ramakrishnan, D., Modularity of solvable Artin representations of $GO(4)$ -type . Int. Math. Res. Not. IMRN 2002(2002), no. 1, 1–54.CrossRefGoogle Scholar
Shimizu, H., On discontinuous groups acting on a product of upper half planes . Ann. of Math. 77(1963), 33–71.CrossRefGoogle Scholar
Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98(1989), 265–280.CrossRefGoogle Scholar