Abstract
This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of \(\mathbb Q\)—in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If \(k=\mathbb F_{q}(T)\) and \(k_{\infty }\) is the analytic completion of k, we introduce the quantum modular invariant
as a multivalued, discontinuous modular invariant function. Then if \(K=k(f)\subset k_{\infty }\) is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field \(H_{\mathcal {O}_{K}}\) (associated to \(\mathcal {O}_{K}=\) integral closure of \(\mathbb F_{q}[T]\) in K) is generated over K by the product of the multivalues of \(j^\mathrm{qt}(f)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the case \(j(\upmu )=0,12^{3}\), one must use multiples of \(\wp _{\upmu }^{3}\) resp. \(\wp _{\upmu }^{2}\).
A quadratic extension K/k is real if the place at \(\infty \) splits completely, otherwise it is called complex.
The problem is that \({\mathcal {O}}_{L}\) has in general an infinite unit group (by Dirichlet’s Theorem on units), so that its images by the place embeddings are indiscrete. That is, apart from the complex quadratic case, \({\mathcal {O}}_{L}\) has rank \(>1\) and so its class fields cannot be treated by Hayes’ rank 1 techniques.
Since \(A_{\infty _{1}}\) is Dedekind, for any non-zero \(x\in {\mathfrak {a}}\), there exists \(y\in {\mathfrak {a}}\) with \({\mathfrak {a}}=(x,y)\). See for example Theorem 17 of [15], page 61.
References
Castaño Bernard, C., Gendron, T.M.: Modular invariant of quantum tori. Proc. Lond. Math. Soc. 109(4), 1014–1049 (2014)
Cohn, P.M.: Algebraic Numbers and Algebraic Functions. Chapman and Hall/CRC, London (1991)
Demangos, L., Gendron, T.M.: Quantum \(j\)-Invariant in positive characteristic I: definition and convergence. Arch. Math. 107 (1), 23–35 (2016). Correction, Arch. Math. 111 (4), 443–447 (2018)
Demangos, L., Gendron, T.M.: Quantum \(j\)-Invariant in positive characteristic II: formulas and values at the quadratics. Arch. Math. 107(2), 159–166 (2016)
Demangos, L., Gendron, T.M.: Quantum Drinfeld modules and ray class fields of real quadratic global function fields, arXiv:1709.05337 (2017)
Demangos, L., Gendron, T.M.: Modular invariant of rank 1 Drinfeld modules and class field generation. To appear in Journal of Number Theory
Gekeler, E.-U.: Zur Arithmetik von Drinfeld Moduln. Math. Ann. 262, 167–182 (1983)
Gekeler, E.-U.: On the coefficients of Drinfeld modular forms. Invent. Math. 93(3), 667–700 (1988)
Gendron, T.M., Leichtnam, E., Lochak, P.: Modules de Drinfeld quasicristallins. arXiv:1912.12323 (2019)
Goss, D.: Basic Structures of Function Field Arithmetic. Springer, Berlin (1998)
Hayes, D.R.: A brief introduction to Drinfeld modules. In: Goss, D., Hayes, D. R., Rosen, M. I. (eds.) The Arithmetic of Function Fields, vol. 2, pp. 313–402. Ohio State U. Mathematical Research Institute Publications, Walter de Gruyter, Berlin (1992)
Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic Curves over a Finite Field. Princeton Series in Applied Mathematics, Princeton University Press, Princeton (2008)
Liu, Q.: Geometry, Algebraic, Curves, Arithmetic: Oxford Graduate Texts in Mathematics 6. Oxford University Press, Oxford (2002)
Manin, Y.I.: Real multiplication and noncommutative geometry. In: Laudal, O.A., Piene, R. (eds.) The Legacy of Niels Henrik Abel, pp. 685–727. Springer, New York (2004)
Marcus, D.A.: Fields, Number: Universitext. Springer, New York (1995)
Neukirch, J.: Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften 322, Springer, Berlin(1999)
Pink, R.:Private communication, (21 October 2017)
Rosen, M.I.: The Hilbert class field in function fields. Expo. Math 5, 365–378 (1987)
Schappacher, N.: On the history of Hilbert’s 12th problem. A comedy of errors. Séminaires et Congrès 3, Société Mathématique de France, 243–273 (1998)
Serre, J.-P.: Complex multiplication. In: Cassels, J.W.S., Frölich, A. (eds.) Algebraic Number Theory, 2nd edn, pp. 293–296. London Mathematical Society, London (2010)
Shu, L.: Kummer’s criterion over global function fields. J. Number Theory 24, 319–359 (1994)
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)
Thakur, D.S.: Function Field Arithmetic. World Scientific, Singapore (2004)
Villa Salvador, G.D.: Topics in the Theory of Algebraic Function Fields. Birkhäuser, Boston (2006)
Acknowledgements
We thank the referee for taking the time to carefully read this paper and making suggestions for its improvement. We also thank Federico Pellarin and Dinesh Thakur, who each responded to questions which came up during the writing up of this work. We would like to express our gratitude to the Instituto de Matemáticas (Unidad Cuernavaca) of the Universidad Nacional Autónoma de México, as well as the University of Stellenbosch (and particularly Florian Breuer) for their generous support of L. Demangos during his stays at each institution.
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of David Goss.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Demangos, L., Gendron, T.M. Quantum modular invariant and Hilbert class fields of real quadratic global function fields. Sel. Math. New Ser. 27, 13 (2021). https://doi.org/10.1007/s00029-021-00619-4
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-021-00619-4