Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-14T04:26:59.343Z Has data issue: false hasContentIssue false
  • English
  • Français

RAISING THE LEVEL OF AUTOMORPHIC REPRESENTATIONS OF $\mathrm {GL}_{2n}$ OF UNITARY TYPE

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  09 February 2021

Christos Anastassiades  and
Jack A. Thorne Open the ORCID record for Jack A. Thorne [Opens in a new window]
Christos Anastassiades
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
Jack A. Thorne
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.

Keywords

Automorphic formsJacquet–Langlands correspondencecongruences between automorphic forms

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1421 - 1444
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Arthur, J., The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications, 61 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Arthur, J. and Clozel, L., Simple algebras, base change, and the advanced theory of the trace formula, Ann. Math. Stud. 120 (1989), xiv+230. Google Scholar
Badulescu, A. I., Jacquet-Langlands et unitarisabilité [Jacquet–Langlands correspondence and unitarisability], J. Inst. Math. Jussieu 6(3) (2007), 349–379. CrossRefGoogle Scholar
Badulescu, A. I., Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172(2) (2008), 383–438 (with an appendix by Neven Grbac).CrossRefGoogle Scholar
Bellaïche, J. and Graftieaux, P., Augmentation du niveau pour $U(3)$ [Level-raising for $U(3)$ ], Amer. J. Math. 128(2) (2006), 271–309. CrossRefGoogle Scholar
Caraiani, A., Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J. 161(12) (2012), 2311–2413.CrossRefGoogle Scholar
Casselman, W., The unramified principal series of $p$ -adic groups. I. The spherical function, Compositio Math. 40(3) (1980), 387–406.Google Scholar
Casselman, W., A new nonunitarity argument for $p$ -adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math . 28(3) (1981), 907–928.Google Scholar
Clozel, L., On Ribet’s level-raising theorem for $U(3)$ , Amer. J. Math. 122(6) (2000), 1265–1287.CrossRefGoogle Scholar
Clozel, L., Purity reigns supreme, Int. Math. Res. Not. IMRN (2013), no. 2, 328–346.CrossRefGoogle Scholar
Clozel, L., Harris, M. and Labesse, J.-P., Endoscopic transfer, in On the Stabilization of the Trace Formula, pp. 475–496 (International Press, Somerville, MA, 2011).Google Scholar
Clozel, L., Harris, M. and Taylor, R., Automorphy for some $l$ -adic lifts of automorphic mod $l$ Galois representations, Publ. Math. 108 (2008), 1–181 (with Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras).CrossRefGoogle Scholar
Clozel, L. and Thorne, J. A., Level-raising and symmetric power functoriality, I, Compos. Math. 150(5) (2014), 729–748.CrossRefGoogle Scholar
Deligne, P., Kazhdan, D. and Vignéras, M.-F., Représentations des algèbres centrales simples $p$ -adiques [Representations of reductive groups over a local field] (Hermann, Paris, 1984).Google Scholar
Diamond, F. and Taylor, R., Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115(3) (1994), 435–462.CrossRefGoogle Scholar
Gee, T., Automorphic lifts of prescribed types, Math. Ann. 350(1) (2011), 107–144.CrossRefGoogle Scholar
Geraghty, D., Modularity lifting theorems for ordinary Galois representations, Math. Ann. 373(3–4) (2019), 1341–1427.CrossRefGoogle Scholar
Gross, B. H., Algebraic modular forms, Israel J. Math. 113 (1999), 61–93.CrossRefGoogle Scholar
Herzig, F., Koziol, K. and Vignéras, M.-F., On the existence of admissible supersingular representations of $p$ -adic reductive groups. Forum Math. Sigma 8 (2020), e2, pp 73. With an appendix by Sug Woo Shin.CrossRefGoogle Scholar
Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103(4) (1981), 777–815.CrossRefGoogle Scholar
Kaletha, T., The local Langlands conjectures for non-quasi-split groups, in Families of Automorphic Forms and the Trace Formula, pp. 217–257 (Springer, Cham, Switzerland, 2016).CrossRefGoogle Scholar
Kaletha, T., Minguez, A., Shin, S. W. and White, P.-J., Endoscopic classification of representations: inner forms of unitary groups. arXiv:1409.3731.Google Scholar
Kaletha, T. and Taïbi, O., Global rigid forms vs isocrystals. arXiv:1812.11373.Google Scholar
Karnataki, A., Level-raising for automorphic forms on $\mathrm{GL}(n)$ . Preprint.Google Scholar
Kneser, M., Lectures on Galois Cohomology of Classical Groups, Tata Institute of Fundamental Research Lectures on Mathematics, 47 (Tata Institute of Fundamental Research, Bombay, 1969). With an appendix by T. A. Springer, notes by P. Jothilingam.Google Scholar
Labesse, J.-P., Changement de base CM et séries discrètes [CM base change and discrete series], in On the Stabilization of the Trace Formula, pp. 429–470 (International Press, Somerville, MA, 2011).Google Scholar
Mínguez, A., Unramified representations of unitary groups, in On the Stabilization of the Trace Formula, pp. 389–410 (International Press, Somerville, MA, 2011).Google Scholar
Moeglin, C. and Waldspurger, J.-L., Le spectre résiduel de $\mathrm{GL}(n)$ [The residual spectrum of $\mathrm{GL}(n)$ ], Ann. Sci. École Norm. Sup. (4) 22(4) (1989), 605–674.Google Scholar
Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235(1108) (2015), vi+248.Google Scholar
Mazanti Sorensen, C., A generalization of level-raising congruences for algebraic modular forms, Ann. Inst. Fourier (Grenoble) 56(6) (2006), 1735–1766.CrossRefGoogle Scholar
Nakayama, T. and Matsushima, Y., Über die multiplikative Gruppe einer $p$ -adischen Divisionsalgebra [On the multiplicative group of $p$ -adic division algebras], Proc. Imp. Acad. Tokyo 19 (1943), 622–628. Google Scholar
Newton, J. and Thorne, J. A., Symmetric power functoriality for holomorphic modular forms. Preprint.Google Scholar
Ribet, K. A., Congruence relations between modular forms, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, pp. 503–514 (PWN, Warsaw, 1984).Google Scholar
Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$ -adic groups, Ann. Math. (2) 132(2) (1990), 273–330.CrossRefGoogle Scholar
Tadić, M., Induced representations of $\mathrm{GL}\left(n,A\right)$ for $p$ -adic division algebras $A$ , J. Reine Angew. Math. 405 (1990), 48–77.Google Scholar
Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98(2) (1989), 265–280.CrossRefGoogle Scholar
Taylor, R., Automorphy for some $l$ -adic lifts of automorphic mod $l$ Galois representations. II, Pub. Math. IHES 108 (2008), 183–239.CrossRefGoogle Scholar
Thorne, J. A., Raising the level for ${\mathrm{GL}}_n$ , Forum Math. Sigma 2 (2014), e16, 35.CrossRefGoogle Scholar