Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T15:26:10.959Z Has data issue: false hasContentIssue false
  • English
  • Français

Cusp forms on the exceptional group of type $E_{7}$

Part of: Lie groups Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  07 September 2015

Henry H. Kim  and
Takuya Yamauchi
Henry H. Kim
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada Korea Institute for Advanced Study, Seoul, Korea email henrykim@math.toronto.edu
Takuya Yamauchi
Affiliation:
Department of Mathematics, Faculty of Education, Kagoshima University, Korimoto 1-20-6, Kagoshima 890-0065, Japan Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada email yamauchi@edu.kagoshima-u.ac.jp, tyama@math.toronto.edu
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.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

Keywords

exceptional group of type E7Ikeda liftEisenstein seriesLanglands functoriality

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables)
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 20G41: Exceptional groups
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 223 - 254
Copyright
© The Authors 2015 

References

Baily, W. L. Jr., An exceptional arithmetic group and its Eisenstein series, Ann. of Math. (2) 91 (1970), 512–549.CrossRefGoogle Scholar
Baily, W. L. Jr. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domain, Ann. of Math. (2) 84 (1966), 442–528.CrossRefGoogle Scholar
Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, part 1 (American Mathematics Society, Providence, RI, 1977), 189–207.Google Scholar
Bourbaki, N., Lie groups and Lie algebras (Springer, Berlin, 2002), ch. 4–6.CrossRefGoogle Scholar
Carter, R. W., Simple groups of Lie type, Pure and Applied Mathematics, vol. 28 (John Wiley & Sons, 1972).Google Scholar
Casselman, W., Introduction to admissible representations of p-adic groups, http://www.math.ubc.ca/∼cass/.Google Scholar
Cogdell, J. and Michel, P., On the complex moments of symmetric power L-functions at s = 1, Int. Math. Res. Not. IMRN 31 (2004), 1561–1617.Google Scholar
Cogdell, J. and Piatetski-Shapiro, I., Base change for the Saito–Kurokawa representations of PGSp(4), J. Number Theory 30 (1988), 298–320.CrossRefGoogle Scholar
Coxeter, H. S. M., Integral Cayley numbers, Duke Math. J. 13 (1946), 561–578.CrossRefGoogle Scholar
Freudenthal, H., Zur ebenen Oktavengeometrie, Indag. Math. (N.S.) 15 (1953), 195–200.CrossRefGoogle Scholar
Ikeda, T., On the theory of Jacobi forms and Fourier–Jacobi coefficients of Eisenstein series, J. Math. Kyoto Univ. 34 (1994), 615–636.Google Scholar
Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2) 154 (2001), 641–681.CrossRefGoogle Scholar
Ikeda, T., On the lifting of Hermitian modular forms, Compositio Math. 144 (2008), 1107–1154.CrossRefGoogle Scholar
Ikeda, T., On the lifting of automorphic representation of $\text{PGL}_{2}(\mathbb{A})$to $\text{Sp}_{2n}(\mathbb{A})$of $\widetilde{\text{Sp}_{2n+1}(\mathbb{A})}$over a totally real algebraic field, available at:https://www.math.kyoto-u.ac.jp/∼ikeda/adelic1.pdf.Google Scholar
Ikeda, T. and Hiraga, K., On the Kohnen plus space for Hilbert modular forms of half-integral weight I, Compositio Math. 149 (2013), 1963–2010.Google Scholar
Karel, M., On certain Eisenstein series and their Fourier coefficients, PhD thesis, University of Chicago (1972).Google Scholar
Karel, M., Fourier coefficients of certain Eisenstein series, Ann. of Math. (2) 99 (1974), 176–202.CrossRefGoogle Scholar
Kim, H., Exceptional modular form of weight 4 on an exceptional domain contained in ℂ27, Rev. Mat. Iberoam. 9 (1993), 139–200.CrossRefGoogle Scholar
Kim, H., On local L-functions and normalized intertwining operators, Canad. J. Math. 57 (2005), 535–597.CrossRefGoogle Scholar
Knapp, A. W., Representation theory of semisimple groups (Princeton University Press, 1986).CrossRefGoogle Scholar
Krieg, A., Jacobi forms of several variables and the Maass space, J. Number Theory 56 (1996), 242–255.CrossRefGoogle Scholar
Kudla, S., Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), 361–401.CrossRefGoogle Scholar
Kudla, S., Some extensions of the Siegel–Weil formula, in Eisenstein series and applications, Progress in Mathematics, vol. 258 (Birkhäuser, Boston, MA, 2008), 205–237.CrossRefGoogle Scholar
Kurokwa, N., Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49 (1978), 149–165.CrossRefGoogle Scholar
Langlands, R., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, vol. 544 (Springer, 1976).CrossRefGoogle Scholar
Piatetski-Shapiro, I. I., On the Saito–Kurokawa lifting, Invent. Math. 71 (1983), 309–338.CrossRefGoogle Scholar
Steinberg, R., Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University (1968).Google Scholar
Takase, K., On two-fold covering group of Sp(n, ℝ) and automorphic factor of weight 1/2, Comment. Math. Univ. St. Pauli 45 (1996), 117–145.Google Scholar
Weissman, M., The Fourier–Jacobi map and small representations, Represent. Theory 7 (2003), 275–299.CrossRefGoogle Scholar
Yamana, S., On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups, J. Number Theory 130 (2010), 2480–2527.CrossRefGoogle Scholar