Abstract
We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.
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A. Ash and D. Ginzburg, p-adic L-functions for GL(2n). Inventiones mathematicae 116 (1994), 27–73.
C. J. Bushnell and G. Henniart, The Local Langlands Conjecture for GL(2), Grundlehren der mathematischen Wissenschaften, Vol. 335, Springer Berlin–Heidelberg, 2006.
A. Borel, Stable real cohomology of arithmetic groups. II, in Manifolds and Lie Groups, Progress in Mathematics, Vol. 14, Birkhäuser Boston, MA, 1981, pp. 21–55.
D. Banerjee and A. Raghuram, p-adic L-Functions for GLn, in p-Adic Aspects of Modular Forms, World Scientific, Hackensack, NJ, 2016, pp. 2327–277.
K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer, New York–Berlin, 1982.
A. Borel and J-P. Serre, Corners and arithmetic groups, Commentarii Mathematici Helvetici 48 (1973), 436–491.
A. Borel and J-P. Serre, Cohomologie d’immeubles et de groupes S-arithmétiques, Topology 15 (1976), 211–232.
A. Borel and N. R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Mathematical surveys and monographs, Vol. 67, American Mathematical Society, providence, RI, 2000.
L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, in Automorphic Forms, Shimura Varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159.
A. Dabrowski, p-adic L-functions of Hilbert modular forms, Annales de l’institut Fourier 44 (1994), 1025–1041.
H. Darmon, Integration on Hp × H and arithmetic applications, Université de Grenoble. Annals of Mathematics 154 (2001), 589–639.
S. Friedberg and H. Jacquet, Linear periods, Journal für die Reine und Angewandte Mathematik 443 (1993), 91–139.
E. Groβe-Klönne, On the universal module of p-adic spherical Hecke algebras, American Journal of Mathematics 136 (2014), 599–652.
W. T. Gan and A. Raghuram, Arithmeticity for periods of automorphic forms, in Proceedings of the International Colloquium on Automorphic Representations and L-functions, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 22, Tata Institute of Fundamental Research, Mumbai, 2013, pp. 187–229.
H. Grobner and A. Raghuram, On the arithmetic of Shalika models and the critical values of L-functions for GL2n, American Journal of Mathematics 136 (2014), 599–652.
G. Harder, Eisenstein cohomology of arithmetic groups. The case GL2, Inventiones mathematicae 89 (1987), 37–118.
F. Januszewski, Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields, Journal für die Reine und Angewandte Mathematik 653 (2011), 1–45.
F. Januszewski, On p-adic L-functions for GL(n) × GL(n - 1) cover totally real fields, International Mathematics Research Notices 2015 (2015), 7884–7949.
H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika, Conducteur des représentations du groupe linéaire, Mathematische Annalen 256 (1981), 199–214.
H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Mathematica 102 (1996), 65–123.
H. Jacquet and J. Shalika, Exterior square L-functions, in Automorphic Forms, Shimura Varieties, and L-functions. Vol. II (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 11, Academic Press, Boston, MA, 1990, pp. 143–226.
S. Kato, On eigenspaces of the Hecke algebra with respect to a good maximal compact subgroup of a p-adic reductive group, Mathematische Annalen 257 (1981), 1–7.
H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, Journal of the American Mathematical Society 16 (2003), 139–183.
D. Kazhdan, B. Mazur and C.-G. Schmidt, Relative modular symbols and Rankin–Selberg convolutions, Journal für die Reine und Angewandte Mathematik 519 (2000), 97–141.
H. H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and the symmetric cube for GL2. Annals of Mathematics 155 (2002), 837–893.
J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, New York, 2003.
J. Mahnkopf, Eisenstein Cohomology and the Construction of p-adic analytic Lfunctions, Compositio Mathematica 124 (2000), 253–304.
Ju. I. Manin, Periods of cusp forms, and p-adic Hecke series, Matematicheskiĭ Sbornik 92 (1973), 378–401, 503.
Ju. I. Manin, Non-Archimedean integration and p-adic Jacquet–Langlands Lfunctions, Uspehi Matematičeskih Nauk 31 (1976), 5–54.
B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Inventiones Mathematicae 25 (1974), 1–61.
B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84 (1986), 1–48.
C. Nien, Uniqueness of Shalika models, Canadian Journal of Mathematics 61 (2009), 1325–1340.
R. Ollivier, Resolutions for principal series representations of p-adic GLn, Münster Journal of Mathematics 7 (2014), 225–240.
A. A. Panchishkin, Admissible non-Archimedean standard zeta functions associated with Siegel modular forms, in Motives (Seattle, WA, 1991). Part 2, Proceedings of Symposia in Pure Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1994, pp. 251–292.
M. Reeder, Modular symbols and the Steinberg representation, in Cohomology of Arithmetic Groups and Automorphic Forms, Lecture Notes in Mathematics, Vol. 1447, Springer, Berlin–Heidelberg, 1990, pp. 287–302.
A. Raghuram and F. Shahidi, Functoriality and special values of l-functions, in Eisenstein Series and Applications, Progress in Mathematics, Vol. 258, Birkhäuser Boston, MA, 2008, pp. 271–293.
C.-G. Schmidt, Relative modular symbols and p-adic Rankin–Selberg convolutions, Inventiones Mathematicae 112 (1993), 31–76.
J.-P. Serre, Cohomologie des groupes discrets, in Séminaire Bourbaki. 23ème année (1970/71) Exposés 382–399, Lecture Notes in Mathematics, Vol. 244, Springer, Berlin, 1971, pp. 337–350.
M. Spieβ, On special zeros of p-adic L-functions of Hilbert modular forms, Inventiones mathematicae 196 (2014), 69–138.
P. Schneider and U. Stuhler, Resolutions for smooth representations of the general linear group over a local field, Journal f´ur die reine und angewandte Mathematik 436 (1993), 19–32.
B. Sun, Cohomologically induced distinguished representations and cohomological test vectors, preprint, http://arxiv.org/abs/1111.2636, 2011.
M.-F. Vignéras, A criterion for integral structures and coefficient systems on the tree of - PGL(2, F), Pure and Applied Mathematics Quarterly 4 (2008), 1291–1316.
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Transactions of the American Mathematical Society 36 (1934), 63–89.
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Gehrmann, L. On Shalika models and p-adic L-functions. Isr. J. Math. 226, 237–294 (2018). https://doi.org/10.1007/s11856-018-1694-0
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DOI: https://doi.org/10.1007/s11856-018-1694-0