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Solvable base change

Laurent Clozel and Conjeeveram S. Rajan ORCID logo

Abstract

We determine the image and the fibers for solvable base change.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1638352

Funding statement: Clozel’s work was partially supported by the National Science Foundation under Grant No. DMS-1638352. The second author acknowledges the support of the Department of Atomic Energy, Government of India under project no. 12-RD-TFR-RT14001.

Acknowledgements

We would like to thank J.-L. Waldspurger for useful correspondence, and Christian Kaiser for a correction. Our work relies crucially on the stabilization of the twisted trace formula [14], due to Moeglin and Waldspurger. We note that one prerequisite for this proof is the so-called “weighted twisted fundamental lemma”, which at this point is still unproven.

References

[1] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Stud. 120, Princeton University Press, Princeton 1989. 10.1515/9781400882403Search in Google Scholar

[2] N. Bergeron and L. Clozel, Sur la cohomologie des variétés hyperboliques de dimension 7 trialitaires, Israel J. Math. 222 (2017), no. 1, 333–400. 10.1007/s11856-017-1593-9Search in Google Scholar

[3] A. Borel, Automorphic L-functions, Automorphic forms, representations and L-functions. Part 2 (Corvallis 1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 27–61. 10.1090/pspum/033.2/546608Search in Google Scholar

[4] L. Clozel, Base change for GL ( n ) , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley 1986), American Mathematical Society, Providence (1987), 791–797. Search in Google Scholar

[5] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. 10.2307/2374103Search in Google Scholar

[6] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. II, Amer. J. Math. 103 (1981), no. 4, 777–815. 10.2307/2374050Search in Google Scholar

[7] R. E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999). Search in Google Scholar

[8] J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL ( 2 ) , Canadian J. Math. 31 (1979), no. 4, 726–785. 10.4153/CJM-1979-070-3Search in Google Scholar

[9] R. P. Langlands, Base change for GL ( 2 ) , Ann. of Math. Stud. 96, Princeton University Press, Princeton 1980. Search in Google Scholar

[10] R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr. 31, American Mathematical Society, Providence (1989), 101–170. 10.1090/surv/031/03Search in Google Scholar

[11] R. P. Langlands, Representations of abelian algebraic groups, Pacific J. Math. 181 (1997), no. 3, 231–250. 10.2140/pjm.1997.181.231Search in Google Scholar

[12] E. Lapid and J. Rogawski, On twists of cuspidal representations of GL ( 2 ) , Forum Math. 10 (1998), no. 2, 175–197. Search in Google Scholar

[13] C. Moeglin and J. L. Waldspurger, Le spectre résiduel de GL ( n ) , Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), no. 4, 605–674. 10.24033/asens.1595Search in Google Scholar

[14] C. Moeglin and J. L. Waldspurger, Stabilisation de la formule des traces tordue. Vol. 1–2, Progr. Math. 316–317, Birkhäuser, Basel 2016. 10.1007/978-3-319-30049-8_1Search in Google Scholar

[15] C. S. Rajan, On the image and fibres of solvable base change, Math. Res. Lett. 9 (2002), no. 4, 499–508. 10.4310/MRL.2002.v9.n4.a9Search in Google Scholar

[16] J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: L-functions and Galois properties (Durham 1975), Academic Press, London (1977), 193–268. 10.1007/978-3-642-39816-2_110Search in Google Scholar

[17] J.-P. Serre, Local fields, Grad. Texts in Math. 67, Springer, New York 1979. 10.1007/978-1-4757-5673-9Search in Google Scholar

[18] J.-L. Waldspurger, Stabilisation de la partie géométrique de la formule des traces tordue, Proceedings of the International Congress of Mathematicians. Vol. II (Seoul 2014), Kyung Moon Sa, Seoul (2014), 487–504. Search in Google Scholar

Received: 2019-05-15
Revised: 2020-06-11
Published Online: 2020-08-05
Published in Print: 2021-03-01

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