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Sturm-Like Bound for Square-Free Fourier Coefficients Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Critical Values of L-Functions for GL 3 ×GL 1 over a Totally Real Field

Authors : A. Raghuram, Gunja Sachdeva

Published in: L-Functions and Automorphic Forms

Publisher: Springer International Publishing

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Abstract

We prove an algebraicity result for all the critical values of L-functions for GL3 ×GL1 over a totally real field F, which we derive from the theory of Rankin–Selberg L-functions attached to pairs of automorphic representations on GL3 ×GL2. This is a generalization and refinement of the results of Mahnkopf (J. Reine Angew. Math. 497:91–112, 1998) and Geroldinger (Ramanujan J. 38(3):641–682, 2015).

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previous chapter A Note on the Growth of Nearly Holomorphic Vector-Valued Siegel Modular Forms
next chapter Indecomposable Harish-Chandra Modules for Jacobi Groups
Literature
1.
Bernshtein, I.N., Zelevinskii, V.: Representations of the group GL(n, F) where F is a non archimedean local field. Russ. Math. Surveys 31(3), 1–68 (1976)
2.
Blasius, D.: Period relations and critical values of L-functions. Pac. J. Math. Special Issue 183(3), 53–83 (1997). Olga Taussky-Todd, in Memoriam, eds. M. Aschbacher, D. Blasius, D. Ramakrishnan
3.
4.
Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. In: Proceedings of Symposium on Pure Mathematics, vol. XXXIII, part I, pp. 189–202. American Mathematical Society, Providence, RI (1979)
5.
Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Mathematical Surveys and Monographs, vol. 67, 2nd edn. American Mathematical Society, Providence, RI (2000)
6.
7.
Casselman, W., Shalika, J.: The unramified principal series of p-adic groups, II, The Whittaker function. Compositio Math. 41(2), 207–231 (1980)MathSciNetMATH
8.
Clozel, L.: Motifs et formes automorphes: applications du principe de fonctorialité. In: Clozel, L., Milne, J.S. (eds.) Automorphic Forms, Shimura Varieties, and L-Functions, vol. I. Perspective in Mathematics, vol. 10. (Ann Arbor, MI, 1988), pp. 77–159. Academic, Boston, MA (1990)
9.
Deligne, P.: Valeurs de fonctions L et périodes d’intégrales, With an appendix by N. Koblitz and A. Ogus. In: Proceedings of Symposium on Pure Mathematics, vol. XXXIII, part II, pp. 313–346. American Mathematics Society, Providence, RI (1979)
10.
Gan, W.T., Raghuram, A.: Arithmeticity for periods of automorphic forms. In: Automorphic Representations and L Functions, pp. 187–229. Tata Institute of Fundamental Research Studies in Mathematics, vol. 22. Tata Institute of Fundamental Research, Mumbai (2013)
11.
Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. École Norm. Sup. (4) 11(4), 471–542 (1978)
12.
Geroldinger, A.: p-adic automorphic L-functions on GL(3). Ramanujan J. 38(3), 641–682 (2015)
13.
Goodman, R., Wallach, N.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, Dordrecht (2009)
14.
Harder, G.: Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89(1), 37–118 (1987)
15.
Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Conducteur des représentations du groupe linéaire. (French) [Conductor of linear group representations]. Math. Ann. 256(2), 199–214 (1981)
16.
17.
Knapp, A.W.: Local Langlands correspondence: the Archimedean case. In: Motives. Proceedings of Symposium on Pure Mathematics, vol. 55, part 2, pp. 393–410. American Mathematical Society, Providence, RI (1994)
18.
Mahnkopf, J.: Modular symbols and values of L-functions on GL3. J. Reine Angew. Math. 497, 91–112 (1998)MathSciNetMATH
19.
20.
21.
Raghuram, A.: Critical values of Rankin-Selberg L-functions for GL n  ×GL n−1 and the symmetric cube L-functions for GL2. Forum Math. 28(3), 457–489 (2016)
22.
Raghuram, A., Shahidi, F.: Functoriality and special values of L-functions. In: Gan, W.T., Kudla, S., Tschinkel, Y. (eds.) Eisenstein Series and Applications. Progress in Mathematics, vol. 258. Birkhaüser, Boston (2008)
23.
Raghuram, A., Shahidi, F.: On certain period relations for cusp forms on GL n . Int. Math. Res. Not. (2008). https://​doi.​org/​10.​1093/​imrn/​rnn077
24.
Raghuram, A., Tanabe, N.: Notes on the arithmetic of Hilbert modular forms. J. Ramanujan Math. Soc. 26(3), 261–319 (2011)MathSciNetMATH
25.
Schmidt, R.: Some remarks on Local newforms for GL(2). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)MathSciNetMATH
26.
27.
28.
29.
Waldspurger, J.-L.: Quelques propriéteś arithmétiqués de certaines formes automorphes sur GL(2). Comput. Math. 54, 121–171 (1985)MATH
Metadata
Title
Critical Values of L-Functions for GL 3 ×GL 1 over a Totally Real Field
Authors
A. Raghuram
Gunja Sachdeva
Copyright Year
2017
Book
L-Functions and Automorphic Forms

Print ISBN: 978-3-319-69711-6

Electronic ISBN: 978-3-319-69712-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-69712-3_12

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