Singular moduli for real quadratic fields: a rigid analytic approach. (English) Zbl 1486.11137

Authors’ abstract: A rigid meromorphic cocycle is a class in the first cohomology of the discrete group \(\Gamma :=\mathrm{SL}_2(\mathbb{Z}[1/p])\) with values in the multiplicative group of nonzero rigid meromorphic functions on the \(p\)-adic upper half-plane \(\mathcal{H}_p:=\mathbb{P}_1(\mathbb{C}_p)-\mathbb{P}_1(\mathbb{Q}_p)\). Such a class can be evaluated at the real quadratic irrationalities in \(\mathcal{H}_p\), which are referred to as “RM points.” Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of \(\Gamma\)-orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the values of modular functions on \(\mathrm{SL}_2(\mathbb{Z})\backslash\mathcal{H}\) at complex multiplication (CM) points: in particular, they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high \(p\)-adic accuracy leads to convincing numerical evidence for the algebraicity and factorization of the resulting singular moduli for real quadratic fields.


11R37 Class field theory
11G15 Complex multiplication and moduli of abelian varieties
Full Text: DOI Euclid


[1] A. Ash, Parabolic cohomology of arithmetic subgroups of \({\text{SL}}(2,\mathbb{Z})\) with coefficients in the field of rational functions on the Riemann sphere, Amer. J. Math. 111 (1989) no. 1, 35-51. · Zbl 0664.10014
[2] Y. Choie and D. B. Zagier, “Rational period functions for \({\text{PSL}}(2,\mathbb{Z})\)” in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math. 143, Amer. Math. Soc., Providence, 1993, 89-108. · Zbl 0790.11044
[3] H. Darmon, Integration on \(\mathcal{H}_p\times \mathcal{H}\) and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589-639. · Zbl 1035.11027
[4] H. Darmon, The \(p\)-adic uniformisation of modular curves by \(p\)-arithmetic groups, RIMS Kôkyûroku Bessatsu 2120 (2019), 52-61.
[5] H. Darmon and S. Dasgupta, Elliptic units for real quadratic fields, Ann. of Math. (2) 163 (2006), no. 1, 301-346. · Zbl 1130.11030 · doi:10.4007/annals.2006.163.301
[6] H. Darmon and R. Pollack, Efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), no. 1, 319-354. · Zbl 1157.11028 · doi:10.1007/BF02771789
[7] H. Darmon and J. Vonk, A real quadratic Borcherds lift, in preparation.
[8] S. Dasgupta, Stark-Heegner points on modular Jacobians, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 3, 427-469. · Zbl 1173.11334 · doi:10.1016/j.ansens.2005.03.002
[9] S. Dasgupta, H. Darmon, and R. Pollack, Hilbert modular forms and the Gross-Stark conjecture, Ann. of Math. (2) 174 (2011), no. 1, 439-484. · Zbl 1250.11099 · doi:10.4007/annals.2011.174.1.12
[10] S. Dasgupta and M. Kakde, in preparation.
[11] S. Dasgupta, M. Kakde, and K. Ventullo, On the Gross-Stark conjecture, Ann. of Math. (2) 188 (2018), no. 3, 833-870. · Zbl 1416.11160 · doi:10.4007/annals.2018.188.3.3
[12] W. Duke, O. Imamo?lu, and A. Tóth, Modular cocycles and linking numbers, Duke Math. J. 166 (2017), no. 6, 1179-1210. · Zbl 1408.57005 · doi:10.1215/00127094-3793032
[13] L. Gerritzen and M. van der Put, Schottky Groups and Mumford Curves, Lecture Notes in Math. 817, Springer, Berlin, 1980. · Zbl 0442.14009
[14] E. Gethner, “Rational period functions with irrational poles are not Hecke eigenfunctions” in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math. 143, Amer. Math. Soc., Providence, 1993, 371-383. · Zbl 0795.11023
[15] B. H. Gross and D. B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191-220. · Zbl 0545.10015
[16] M. I. Knopp, Rational period functions of the modular group, with an appendix by G. Grinstein, Duke Math. J. 45 (1978), no. 1, 47-62. · Zbl 0374.10014 · doi:10.1215/S0012-7094-78-04504-0
[17] J. Rickards, Intersections of closed geodesics on Shimura curves, Ph.D. dissertation, McGill University, 2020.
[18] P. Schneider and J. Teitelbaum, \(p\)-adic boundary values, Astérisque 278 (2002), 51-125. · Zbl 1051.14024
[19] J.-P. Serre, Trees, Springer, New York, 1980. · Zbl 0548.20018
[20] M. Van der Put, Discrete groups, Mumford curves and theta functions, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 3, 399-438. · Zbl 0789.14020 · doi:10.5802/afst.754
[21] D. B. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. · Zbl 0283.12004 · doi:10.1007/BF01343950
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.