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Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  02 February 2016

Yingkun Li
Yingkun Li*
Affiliation:
Fachbereich Mathematik, TU Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany email li@mathematik.tu-darmstadt.de
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Abstract

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In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

Keywords

harmonic Maass formautomorphic Green’s functionGalois representation

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80: Galois representations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1159 - 1197
Copyright
© The Author 2016 

References

Booker, A., Strömbergsson, A. and Venkatesh, A., Effective computation of Maass cusp forms, Int. Math. Res. Not. IMRN 2006 (2006), doi:10.1155/IMRN/2006/71281.Google Scholar
Bringmann, K. and Ono, K., Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA 104 (2007), 3725–3731.Google Scholar
Bruinier, J. and Bundschuh, M., On Borcherds products associated with lattices of prime discriminant, Ramanujan J. 7 (2003), 49–61.Google Scholar
Bruinier, J. and Funke, J., On two geometric theta lifts, Duke Math. J. 125 (2004), 45–90.Google Scholar
Bruinier, J., Kudla, S. and Yang, T. H., Special values of Green functions at big CM points, Int. Math. Res. Not. IMRN 2012 (2012), 1917–1967.Google Scholar
Bruinier, J. and Ono, K., Heegner divisors, L-functions and harmonic weak Maass forms, Ann. of Math. (2) 172 (2010), 2135–2181.Google Scholar
Bruinier, J. and Yang, T. H., CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288.Google Scholar
Bruinier, Y. and Yang, T. H., Twisted Borcherds products on Hilbert modular surfaces and their CM values, Amer. J. Math. 129 (2007), 807–841.Google Scholar
Bruinier, J. and Yang, T. H., Faltings heights of CM cycles and derivatives of L-functions, Invent. Math. 177 (2009), 631–681.CrossRefGoogle Scholar
Cohen, H., A lifting of modular forms in one variable to Hilbert modular forms in two variables, in Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Mathematics, vol. 627 (Springer, Berlin, 1977), 175–196.CrossRefGoogle Scholar
Cohen, H., Diaz y Diaz, F. and Olivier, M., Computing ray class groups, conductors and discriminants, Math. Comput. 67 (1998), 773–795.Google Scholar
Cohen, H. and Oesterlé, J., Dimensions des espaces de formes modulaires, in Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Mathematics, vol. 627 (Springer, Berlin, 1977), 69–78; (in French).Google Scholar
Darmon, H. and Dasgupta, S., Elliptic units for real quadratic fields, Ann. of Math. (2) 163 (2006), 301–346.CrossRefGoogle Scholar
Darmon, H., Lauder, A. and Rotger, V., Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields, Adv. Math. 283 (2015), 130–142.Google Scholar
Deligne, P. and Serre, J. P., Formes modulaires de poids 1, Ann. Sci. Éc. Norm. Supér. (4) 7 (1975), 507–530; (in French).Google Scholar
Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics, vol. 228 (Springer, New York, 2005).Google Scholar
Duke, W., Imamoḡlu, Ö. and Tóth, Á., Cycle integrals of the j-function and mock modular forms, Ann. of Math. (2) 173 (2011), 947–981.CrossRefGoogle Scholar
Duke, W. and Li, Y., Harmonic Maass forms of weight one, Duke Math. J. 164 (2015), 39–113.CrossRefGoogle Scholar
Ehlen, S., CM values of regularized theta lifts, PhD thesis, TU Darmstadt (2013).Google Scholar
Goren, E. and Lauter, K., Genus 2 curves with complex multiplication, Int. Math. Res. Not. IMRN 2012 (2012), 1068–1142.CrossRefGoogle Scholar
Gross, B. and Zagier, D., On singular moduli, J. reine angew. Math. 355 (1985), 191–220.Google Scholar
Gross, B. and Zagier, D., Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.Google Scholar
Hecke, E., Analytische Funktionen und algebraische Zahlen, zweiter Teil, Abh. Math. Semin. Univ. Hambg. 3 (1924), 231–236; Mathematische Werke (Vandenhoeck and Ruprecht, Göttingen, 1970), 381–404.CrossRefGoogle Scholar
Hejhal, D. A., The Selberg trace formula for PSL(2,ℝ), vol. 2, Lecture Notes in Mathematics, vol. 1001 (Springer, Berlin, 1983).CrossRefGoogle Scholar
Hirzebruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms of nebentypus, Invent. Math. 36 (1976), 57–113.Google Scholar
Kohnen, W., Newforms of half-integral weight, J. reine angew. Math. 333 (1982), 32–72.Google Scholar
Kudla, S., Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), 545–646.CrossRefGoogle Scholar
Lang, S., Algebraic number theory, Graduate Texts in Mathematics, vol. 110 (Springer, New York, 1994).CrossRefGoogle Scholar
Li, Y., Mock-modular forms of weight one, PhD thesis, UCLA (2013).Google Scholar
Manickam, M., Ramakrishnan, B. and Vasudevan, T. C., On the theory of newforms of half-integral weight, J. Number Theory 34 (1990), 210–224.Google Scholar
Okazaki, R., Inclusion of CM-field and divisibility of relative class numbers, Acta Arith. XCII.4 (2000), 319–338.CrossRefGoogle Scholar
Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory, Current Developments in Mathematics, vol. 2008 (International Press, Somerville, MA, 2009), 347–454.Google Scholar
Pei, T. Y., Eisenstein series of weight 3/2, I, Trans. Amer. Math. Soc. 274 (1982), 573–606.CrossRefGoogle Scholar
Schofer, J., Borcherds forms and generalizations of singular moduli, J. reine angew. Math. 629 (2009), 1–36.Google Scholar
Scholz, A., Über die Lösbarkeit der Gleichung t 2 - Du 2 = -4, Math. Z. 39 (1934), 95–111.Google Scholar
Shimura, G., Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46 (Princeton University Press, Princeton, NJ, 1998).Google Scholar
Stark, H. M., L-functions at s = 1. I–IV, Adv. Math. 7 (1971), 301–343; 17 (1975), 60–92; 22 (1976), 64–84; 35 (1980), 197–235.Google Scholar
Stark, H. M., The genus theory of number fields, Comm. Pure Appl. Math. XXIX (1976), 805–811.CrossRefGoogle Scholar
Stark, H. M., Class fields and modular forms of weight one, in Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Mathematics, vol. 601 (Springer, Berlin, 1977), 277–287.Google Scholar
Stark, H. M., Class fields for real quadratic fields and L-series at 1, in Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, 1975) (Academic Press, 1977), 355–375.Google Scholar
Stein, W. A. et al. , Sage mathematics software (version 5.0.1), The Sage Development Team, 2012, http://www.sagemath.org.Google Scholar
Sturm, J., Projection of C ∞ automorphic forms, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 435–439.Google Scholar
Tsuyumine, S., On Shimura lifting of modular forms, Tsukuba J. Math. 23 (1999), 465–483.CrossRefGoogle Scholar
Ueda, M., On twisting operators and newforms of half-integral weight, Nagoya Math. J. 131 (1993), 135–205.CrossRefGoogle Scholar
van Asch, A. G., Modular forms of half integral weight, some explicit arithmetic, Math. Ann. 262 (1983), 77–89.Google Scholar
Viazovska, M., Petersson inner products of weight one modular forms, Preprint (2012),arXiv:1211.4715.Google Scholar
Washington, L., Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, second edition (Springer, New York, 1997).Google Scholar
Zagier, D., Ramanujan’s mock theta functions and their applications (after Zwegers and Ono–Bringmann), in Seminaire Bourbaki, vol. 2007/2008, Astérisque, vol. 326 (Société Mathématique de France, 2009); Exp. No. 986, vii–viii, 143–164.Google Scholar
Zwegers, S. P., Mock theta functions, PhD thesis, Utrecht (2002), ISBN 90-393-3155-3.Google Scholar