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Modular forms of half-integral weight on exceptional groups
Published online by Cambridge University Press: 22 February 2024
Abstract
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by 
${\pm }1$. We analyze the minimal modular form 
$\Theta _{F_4}$ on the double cover of 
$F_4$, following Loke–Savin and Ginzburg. Using 
$\Theta _{F_4}$, we define a modular form of weight 
$\tfrac {1}{2}$ on (the double cover of) 
$G_2$. We prove that the Fourier coefficients of this modular form on 
$G_2$ see the 
$2$-torsion in the narrow class groups of totally real cubic fields.
Keywords
MSC classification
Secondary:
20G41: Exceptional groups
- Type
- Research Article
- Information
- Copyright
- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
Footnotes
SL has been supported by an AMS-Simons Travel Award and by NSF grant DMS-1902865. AP has been supported by the Simons Foundation via Collaboration Grant number 585147, by the NSF via grant numbers 2101888 and 2144021, and by an AMS Centennial Research Fellowship.
References
Adams, J., Barbasch, D., Paul, A., Trapa, P. and Vogan, D. A. Jr., Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), 701–751; MR 2291917.Google Scholar
Bernstein, I. N. and Zelevinsky, A. V., Induced representations of reductive 
$p$-adic groups. I, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; MR 579172.CrossRefGoogle Scholar
$p$-adic groups. I, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; MR 579172.CrossRefGoogle ScholarBrylinski, J.-L. and Deligne, P., Central extensions of reductive groups by ${\boldsymbol K}_2$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 5–85; MR 1896177.CrossRefGoogle Scholar
${\boldsymbol K}_2$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 5–85; MR 1896177.CrossRefGoogle ScholarBump, D., Friedberg, S. and Ginzburg, D., Small representations for odd orthogonal groups, Int. Math. Res. Not. IMRN 2003 (2003), 1363–1393; MR 1968295.Google Scholar
Casselman, W., Introduction to the theory of admissible representations of $p$-adic reductive groups, Preprint (1995), https://personal.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf.Google Scholar
$p$-adic reductive groups, Preprint (1995), https://personal.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf.Google ScholarChinta, G., Friedberg, S. and Hoffstein, J., Double Dirichlet series and theta functions, in Contributions in analytic and algebraic number theory, Springer Proceedings in Mathematics, vol. 9 (Springer, New York, 2012), 149–170; MR 3060459.CrossRefGoogle Scholar
Cohen, H., Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 271–285; MR 382192.CrossRefGoogle Scholar
$L$-functions of quadratic characters, Math. Ann. 217 (1975), 271–285; MR 382192.CrossRefGoogle ScholarDalal, R., Counting discrete, level- $1$, quaternionic automorphic representations on $G_2$, J. Inst. Math. Jussieu (2023), doi: 10.1017/S1474748023000476.CrossRefGoogle Scholar
$1$, quaternionic automorphic representations on 
$G_2$, J. Inst. Math. Jussieu (2023), doi: 10.1017/S1474748023000476.CrossRefGoogle ScholarDeligne, P., Extensions centrales de groupes algébriques simplement connexes et cohomologie galoisienne, Publ. Math. Inst. Hautes Études Sci. 84 (1996), 35–89; MR 1441006.Google Scholar
Deligne, P. and Gross, B. H., On the exceptional series, and its descendants, C. R. Math. Acad. Sci. Paris 335 (2002), 877–881; MR 1952563.CrossRefGoogle Scholar
Friedberg, S. and Ginzburg, D., Descent and theta functions for metaplectic groups, J. Eur. Math. Soc. (JEMS) 20 (2018), 1913–1957; MR 3854895.CrossRefGoogle Scholar
Gan, W. T., An automorphic theta module for quaternionic exceptional groups, Canad. J. Math. 52 (2000), 737–756; MR 1767400.CrossRefGoogle Scholar
Gan, W. T. and Gao, F., The Langlands–Weissman program for Brylinski–Deligne extensions, in L-groups and the Langlands program for covering groups, Astérisque, vol. 398 (Société Mathématique de France, 2018), 187–275; MR 3802419.Google Scholar
Gan, W. T., Gross, B. and Savin, G., Fourier coefficients of modular forms on $G_2$, Duke Math. J. 115 (2002), 105–169; MR 1932327.CrossRefGoogle Scholar
$G_2$, Duke Math. J. 115 (2002), 105–169; MR 1932327.CrossRefGoogle ScholarGan, W. T. and Savin, G., On minimal representations definitions and properties, Represent. Theory 9 (2005), 46–93; MR 2123125.Google Scholar
Gao, F., Distinguished theta representations for certain covering groups, Pacific J. Math. 290 (2017), 333–379.Google Scholar
Gelbart, S. S., Weil's representation and the spectrum of the metaplectic group, Lecture Notes in Mathematics, vol. 530 (Springer, Berlin–New York, 1976); MR 0424695.CrossRefGoogle Scholar
Gelbart, S. and Piatetski-Shapiro, I. I., Distinguished representations and modular forms of half-integral weight, Invent. Math. 59 (1980), 145–188; MR 577359.CrossRefGoogle Scholar
Ginzburg, D., Towards a classification of global integral constructions and functorial liftings using the small representations method, Adv. Math. 254 (2014), 157–186; MR 3161096.CrossRefGoogle Scholar
Ginzburg, D., On certain global constructions of automorphic forms related to a small representation of $F_4$, J. Number Theory 200 (2019), 1–95; MR 3944431.CrossRefGoogle Scholar
$F_4$, J. Number Theory 200 (2019), 1–95; MR 3944431.CrossRefGoogle ScholarGross, B. H., Some remarks on signs in functional equations, Ramanujan J. 7 (2003), 91–93; MR 2035794.CrossRefGoogle Scholar
Gross, B. H. and Wallach, N. R., A distinguished family of unitary representations for the exceptional groups of real rank =4, in Lie theory and geometry, Progress in Mathematics, vol. 123 (Birkhäuser, Boston, MA, 1994), 289–304; MR 1327538.CrossRefGoogle Scholar
Gross, B. H. and Wallach, N. R., On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996), 73–123; MR 1421947.Google Scholar
Huang, J.-S., Pandžić, P. and Savin, G., New dual pair correspondences, Duke Math. J. 82 (1996), 447–471; MR 1387237.CrossRefGoogle Scholar
Karasiewicz, E., A Hecke algebra on the double cover of a Chevalley group over $\Bbb {Q}_2$, Algebra Number Theory 15 (2021), 1729–1753; MR 4333663.CrossRefGoogle Scholar
$\Bbb {Q}_2$, Algebra Number Theory 15 (2021), 1729–1753; MR 4333663.CrossRefGoogle ScholarKazhdan, D. A. and Patterson, S. J., Metaplectic forms, Publ. Math. Inst. Hautes Études Sci. 59 (1984), 35–142; MR 743816.CrossRefGoogle Scholar
Leslie, S., A generalized theta lifting, CAP representations, and Arthur parameters, Trans. Amer. Math. Soc. 372 (2019), 5069–5121; MR 4009400.CrossRefGoogle Scholar
The LMFDB Collaboration, The L-functions and modular forms database, 2020, http://www.lmfdb.org [accessed 31 July 2020].Google Scholar
Loke, H. Y. and Savin, G., Modular forms on non-linear double covers of algebraic groups, Trans. Amer. Math. Soc. 362 (2010), 4901–4920; MR 2645055.CrossRefGoogle Scholar
Matsumoto, H., Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Supér. (4) 2 (1969), 1–62; MR 240214.CrossRefGoogle Scholar
Merkurćev, A. S. and Suslin, A. A., $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136; MR 675529.Google Scholar
$K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136; MR 675529.Google ScholarMilne, J. S., Algebraic groups: the theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, vol. 170 (Cambridge University Press, Cambridge, 2017); MR 3729270.CrossRefGoogle Scholar
Patterson, S. J., Whittaker models of generalized theta series1, in Seminar on number theory, Paris 1982–83 (Paris, 1982/1983), Progress in Mathematics, vol. 51 (Birkhäuser, Boston, MA, 1984), 199–232; MR 791596.Google Scholar
Pollack, A., Lifting laws and arithmetic invariant theory, Camb. J. Math. 6 (2018), 347–449; MR 3870360.CrossRefGoogle Scholar
Pollack, A., The Fourier expansion of modular forms on quaternionic exceptional groups, Duke Math. J. 169 (2020), 1209–1280; MR 4094735.CrossRefGoogle Scholar
Pollack, A., A quaternionic Saito–Kurokawa lift and cusp forms on $G_2$, Algebra Number Theory 15 (2021), 1213–1244; MR 4283102.CrossRefGoogle Scholar
$G_2$, Algebra Number Theory 15 (2021), 1213–1244; MR 4283102.CrossRefGoogle ScholarPollack, A., The minimal modular form on quaternionic $E_8$, J. Inst. Math. Jussieu 21 (2022), 603–636; MR 4386823.CrossRefGoogle Scholar
$E_8$, J. Inst. Math. Jussieu 21 (2022), 603–636; MR 4386823.CrossRefGoogle ScholarPollack, A., Modular forms on exceptional groups, Arizona Winter School notes (2022), https://swc-math.github.io/aws/2022/index.html.Google Scholar
Pollack, A., Modular forms on indefinite orthogonal groups of rank three, J. Number Theory 238 (2022), 611–675, with appendix ‘Next to minimal representation’ by Gordan Savin; MR 4430112.CrossRefGoogle Scholar
Sage Developers, Sagemath, the Sage Mathematics Software System (2022), https://www.sagemath.org.Google Scholar
Stein, M. R., Surjective stability in dimension $0$ for $K_{2}$ and related functors, Trans. Amer. Math. Soc. 178 (1973), 165–191; MR 327925.Google Scholar
$0$ for 
$K_{2}$ and related functors, Trans. Amer. Math. Soc. 178 (1973), 165–191; MR 327925.Google ScholarSteinberg, R., Lectures on Chevalley groups, University Lecture Series, vol. 66 (American Mathematical Society, Providence, RI, 2016); Notes prepared by John Faulkner and Robert Wilson, Revised and corrected edition of the 1968 original [MR 0466335], with a foreword by Robert R. Snapp; MR 3616493.CrossRefGoogle Scholar
Swaminathan, A., Average $2$-torsion in class groups of rings associated to binary $n$-ic forms, Preprint (2021).Google Scholar
$2$-torsion in class groups of rings associated to binary 
$n$-ic forms, Preprint (2021).Google ScholarTate, J., Relations between $K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274; MR 429837.Google Scholar
$K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274; MR 429837.Google ScholarThańg, N. Q., Number of connected components of groups of real points of adjoint groups, Comm. Algebra 28 (2000), 1097–1110; MR 1742643.Google Scholar
Wallach, N. R., Generalized Whittaker vectors for holomorphic and quaternionic representations, Comment. Math. Helv. 78 (2003), 266–307; MR 1988198.CrossRefGoogle Scholar
Weissman, M. H., $D_4$ modular forms, Amer. J. Math. 128 (2006), 849–898; MR 2251588.CrossRefGoogle Scholar
$D_4$ modular forms, Amer. J. Math. 128 (2006), 849–898; MR 2251588.CrossRefGoogle ScholarWeissman, M. H., Covers of tori over local and global fields, Amer. J. Math. 138 (2016), 1533–1573; MR 3595494.CrossRefGoogle Scholar
Weissman, M. H., L-groups and parameters for covering groups, in L-groups and the Langlands program for covering groups, Astérisque, vol. 398 (Société Mathématique de France, 2018), 33–186; MR 3802418.Google Scholar
Zagier, D., Nombres de classes et formes modulaires de poids $3/2$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A883–A886; MR 429750.Google Scholar
$3/2$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A883–A886; MR 429750.Google Scholar