Papers
Topics
Authors
Recent
Search
2000 character limit reached

Fourier Coefficients and Algebraic Cusp Forms on $\mathrm{U}(2,n)$

Published 27 Apr 2024 in math.NT and math.RT | (2404.17743v1)

Abstract: We establish a theory of scalar Fourier coefficients for a class of non-holomorphic, automorphic forms on the quaternionic real Lie group $\mathrm{U}(2,n)$. By studying the theta lifts of holomorphic modular forms from $\mathrm{U}(1,1)$, we apply this theory to obtain examples of non-holomorphic cusp forms on $\mathrm{U}(2,n)$ whose Fourier coefficients are algebraic numbers.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (27)
  1. Daniel Bump. Automorphic forms and representations, volume 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.
  2. Fourier coefficients of modular forms on G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Duke Math. J., 115(1):105–169, 2002.
  3. The Gross-Prasad conjecture and local theta correspondence. Invent. Math., 206(3):705–799, 2016.
  4. L𝐿Litalic_L-functions and Fourier-Jacobi coefficients for the unitary group U⁢(3)U3{\rm U}(3)roman_U ( 3 ). Invent. Math., 105(3):445–472, 1991.
  5. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
  6. A distinguished family of unitary representations for the exceptional groups of real rank =4absent4=4= 4. In Lie theory and geometry, volume 123 of Progr. Math., pages 289–304. Birkhäuser Boston, Boston, MA, 1994.
  7. On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math., 481:73–123, 1996.
  8. Michael Harris. Arithmetic vector bundles and automorphic forms on Shimura varieties. II. Compositio Math., 60(3):323–378, 1986.
  9. Roger Howe. Transcending classical invariant theory. J. Amer. Math. Soc., 2(3):535–552, 1989.
  10. Atsushi Ichino. A regularized Siegel-Weil formula for unitary groups. Math. Z., 247(2):241–277, 2004.
  11. C. G. J. Jacobi. Fundamenta nova theoriae functionum ellipticarum, Gesammelte Werke, Bände II, pp.49-239. Chelsea Publishing Co., New York, 1969. Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften. Zweite Ausgabe.
  12. Whittaker functions for the large discrete series representations of SU⁢(2,1)SU21{\rm SU}(2,1)roman_SU ( 2 , 1 ) and related zeta integrals. Publ. Res. Inst. Math. Sci., 31(6):959–999, 1995.
  13. Jian-Shu Li. Theta lifting for unitary representations with nonzero cohomology. Duke Math. J., 61(3):913–937, 1990.
  14. Modular forms of half-integral weight on exceptional groups. Compositio Mathematica (to appear), 2022.
  15. The Weil representation, Maslov index and theta series, volume 6 of Progress in Mathematics. Birkhäuser, Boston, MA, 1980.
  16. Aaron Pollack. The Fourier expansion of modular forms on quaternionic exceptional groups. Duke Math. J., 169(7):1209–1280, 2020.
  17. Aaron Pollack. A quaternionic Saito-Kurokawa lift and cusp forms on G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Algebra Number Theory, 15(5):1213–1244, 2021.
  18. Aaron Pollack. Exceptional theta functions and arithmeticity of modular forms on g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, 2023.
  19. Dipendra Prasad. Weil representation, Howe duality, and the theta correspondence. In Theta functions: from the classical to the modern, volume 1 of CRM Proc. Lecture Notes, pages 105–127. Amer. Math. Soc., Providence, RI, 1993.
  20. I. I. Piatetski-Shapiro. On the Saito-Kurokawa lifting. Invent. Math., 71(2):309–338, 1983.
  21. S. Rallis. On the Howe duality conjecture. Compositio Math., 51(3):333–399, 1984.
  22. N. R. Wallach. On the constant term of a square integrable automorphic form. In Operator algebras and group representations, Vol. II (Neptun, 1980), volume 18 of Monogr. Stud. Math., pages 227–237. Pitman, Boston, MA, 1984.
  23. Nolan R. Wallach. Generalized Whittaker vectors for holomorphic and quaternionic representations. Comment. Math. Helv., 78(2):266–307, 2003.
  24. André Weil. Sur certains groupes d’opérateurs unitaires. Acta Math., 111:143–211, 1964.
  25. Martin H. Weissman. D4subscript𝐷4D_{4}italic_D start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT modular forms. Amer. J. Math., 128(4):849–898, 2006.
  26. Chenyan Wu. Irreducibility of theta lifting for unitary groups. J. Number Theory, 133(10):3296–3318, 2013.
  27. Hiroshi Yamashita. Embeddings of discrete series into induced representations of semisimple Lie groups. II. Generalized Whittaker models for SU⁢(2,2)SU22{\rm SU}(2,2)roman_SU ( 2 , 2 ). J. Math. Kyoto Univ., 31(2):543–571, 1991.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Authors (3)

  1. Anton Hilado 
  2. Finn McGlade 
  3. Pan Yan 

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 2 tweets with 0 likes about this paper.

Sign Up for Free