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Potential level-lowering for GSp(4)

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  30 January 2009

Claus M. Sorensen
Claus M. Sorensen
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA, ([email protected]).
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Abstract

In this article, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place ω, and forms with a tamely ramified principal series at ω, Thus, after base change to a finite solvable totally real extension, one can often lower the level at ω. For the proof, we first establish an analogue of the Jacquet–Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity modulo the centre, and split at all the finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups.

Keywords

Hilbert–Siegel modular formslevel-lowering congruencestrace formula

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F80: Galois representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 3 , July 2009 , pp. 595 - 622
Copyright
Copyright © Cambridge University Press 2009

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References

1.Arthur, J., The Selberg trace formula for groups of F-rank one, Annals Math. (2) 100 (1974), 326–385.CrossRefGoogle Scholar
2.Arthur, J., On local character relations, Selecta Math. 2(4) (1996), 501–579.CrossRefGoogle Scholar
3.Arthur, J., Towards a stable trace formula, in Proc. Int. Congress of Mathematicians, Volume 2, Berlin, 1998, Documenta Mathematica, Extra Volume II, pp. 507–517 (1998).CrossRefGoogle Scholar
4.Blasius, D. and Rogawski, J., Zeta functions of Shimura varieties, in Motives, Seattle, WA, 1991, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 2, pp. 525–571 (American Mathematical Society, Providence, RI, 1994).Google Scholar
5.Clozel, L., Harris, M. and Taylor, R., Automorphy for some l-adic lifts of automorphic mod l representations, preprint (available at www.math.harvard.edu/~rtaylor/).Google Scholar
6.Gan, W. T. and Takeda, S., The local Langlands conjecture for GSp(4), preprint (available at www.math.ucsd.edu/~wgan/).Google Scholar
7.Gelbart, S. and Jacquet, H., Forms of GL2 from the analytic point of view, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, Volume 33, Part 1, pp. 213–251 (American Mathematical Society, Providence, RI, 1979).CrossRefGoogle Scholar
8.Genestier, A. and Tilouine, J., Systemes de Taylor–Wiles pour GSp(4), in Formes automorphes, II, Le cas du group GSp(4), Astérisque, Volume 302, pp. 177–290 (Société Mathématiques de France, Paris, 2005).Google Scholar
9.Ghitza, A., Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms, J. Number Theory 106(2) (2004), 345–384.CrossRefGoogle Scholar
10.Gross, B., Algebraic modular forms, Israel J. Math. 113(1999), 61–93.CrossRefGoogle Scholar
11.Hales, T., Shalika germs on GSp4, Orbites unipotentes et representations, II, Astérisque 171–172(1989), 195–256.Google Scholar
12.Hales, T., The fundamental lemma for Sp(4), Proc. Am. Math. Soc. 125(1) (1997), 301–308.CrossRefGoogle Scholar
13.Harris, M., Introduction to The stable trace formula, Shimura varieties, and arithmetic applications, Volume 1 (available at www.math.jussieu.fr/~harris/), forthcoming.Google Scholar
14.Helm, D., Mazur's principle for U(2,1) Shimura varieties, preprint (available at www.math.harvard.edu/~dhelm/).Google Scholar
15.Ihara, Y., On modular curves over finite fields, in Discrete Subgroups of Lie Groups and Applications to Moduli, International Colloquium, Bombay, 1973, pp. 161–202 (Oxford University Press, 1975).Google Scholar
16.Keys, D., On the decomposition of reducible principal series representations of p-adic Chevalley groups, Pac. J. Math. 101(2) (1982), 351–388.CrossRefGoogle Scholar
17.Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Annalen 275(3) (1986), 365–399.CrossRefGoogle Scholar
18.Labesse, J.-P., Cohomologie, stabilisation et changement de base (with two appendices: Appendix A by L. Clozel and J.-P. Labesse; Appendix B by L. Breen), Astérisque, Volume 257 (Société Mathématiques de France, Paris, 1999).Google Scholar
19.Langlands, R., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Volume 544 (Springer, 1976).CrossRefGoogle Scholar
20.Langlands, R., Les debuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII, Volume 13 (Universite de Paris VII, UER de Mathématiques, Paris, 1983).Google Scholar
21.Langlands, R., On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical Surveys and Monographs, Volume 31, pp. 101–170 (American Mathematical Society, Providence, RI, 1989).CrossRefGoogle Scholar
22.Langlands, R. and Shelstad, D., Descent for transfer factors, in The Grothendieck Festschrift, Volume II, pp. 485–563, Progress in Mathematics, Volume 87 (Birkhäuser, Boston, MA, 1990).Google Scholar
23.Piatetski-Shapiro, I., On the Saito–Kurokawa lifting, Invent. Math. 71(2) (1983), 309–338.CrossRefGoogle Scholar
24.Platonov, V. and Rapinchuk, A., Algebraic groups and number theory (translated from the 1991 Russian original by Rowen, R.), Pure and Applied Mathematics, Volume 139 (Academic Press, 1994).CrossRefGoogle Scholar
25.Ramakrishnan, D., Pure motives and automorphic forms, in Motives, Seattle, WA, 1991, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 2, pp. 411–446 (American Mathematical Society, Providence, RI, 1994).Google Scholar
26.Ribet, K., Congruence relations between modular forms, in Proc. Int. Congress of Mathematicians, Volume 1, Warsaw, 1983, pp. 503–514 (PWN, Warsaw, 1984).Google Scholar
27.Ribet, K., Report on mod l representations of Gal(/ℚ), in Motives, Seattle, WA, 1991, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 2, pp. 639–676 (American Mathematical Society, Providence, RI, 1994).Google Scholar
28.Roberts, B., The non-Archimedean theta correspondence for GSp(2) and GO(4), Trans. Am. Math. Soc. 351(2) (1999), 781–811.CrossRefGoogle Scholar
29.Roche, A., Types and Hecke algebras for principal series representations of split reductive p-adic groups, Annales Scient. Éc. Norm. Sup. 31(3) (1998), 361–413.CrossRefGoogle Scholar
30.Shelstad, D., Characters and inner forms of a quasi-split group over ℝ, Compositio Math. 39(1) (1979), 11–45.Google Scholar
31.Shelstad, D., L-indistinguishability for real groups, Math. Annalen 259(3) (1982), 385–430.CrossRefGoogle Scholar
32.Skinner, C. and Wiles, A., Base change and a problem of Serre, Duke Math. J. 107(1) (2001), 15–25.CrossRefGoogle Scholar
33.Sorensen, C., Level-raising for Saito–Kurokawa forms, in preparation.Google Scholar
34.Soudry, D., The CAP representations of GSp(4,), J. Reine Angew. Math. 383 (1988), 87–108.Google Scholar
35.Tadic, M., Representations of p-adic symplectic groups, Compositio Math. 90(2) (1994), 123–181.Google Scholar
36.Taylor, R., Automorphy for some l-adic lifts of automorphic mod l representations, II, preprint (available at www.math.harvard.edu/~rtaylor/).Google Scholar
37.Waldspurger, J.-L., Le lemme fondamental implique le transfert, Compositio Math. 105(2) (1997), 153–236.CrossRefGoogle Scholar
38.Weissauer, R., Four dimensional Galois representations, in Formes automorphes, II, Le cas du group GSp(4), Astérisque, Volume 302, pp. 67–150 (Société Mathématiques de France, Paris, 2005).Google Scholar
39.Weissauer, R., Character identities and Galois representations related to the group GSp(4), preprint (available at www.mathi.uni-heidelberg.de/~weissaue/papers.html).Google Scholar
40.Weissauer, R., Endoscopy for GSp(4), preprint (available at www.mathi.uni-heidelberg.de/~weissaue/papers.html).Google Scholar
41.Wiles, A., Modular elliptic curves and Fermat's last theorem, Annals Math. 141(3) (1995), 443–551.CrossRefGoogle Scholar
42.Winarsky, N., Reducibility of principal series representations of p-adic Chevalley groups, Am. J. Math. 100(5) (1978), 941–956.CrossRefGoogle Scholar