nLab Rankin-Cohen bracket

Overview and definition

Let Γ\Gamma be a congruence subgroup of SL(2,Z)SL(2,\mathbf{Z}) and (Γ)\mathcal{M}(\Gamma) the graded (by the weight) algebra of modular forms with respect to Γ\Gamma. All bidifferential operators which leave that space invariant are linear combinations of Rankin-Cohen brackets [,] n:(f,g)[f,g] n[-,-]_n\colon(f, g)\mapsto [f,g]_n. By definition, nn-th bracket between elements f(Γ) 2kf\in\mathcal{M}(\Gamma)_{2k} and g(Γ) 2lg\in\mathcal{M}(\Gamma)_{2l} is given by the formula

[f,g] n:= r=0 n(1) r(n+2k1nr)(n+2l1r)f (r)g (nr)(Γ) 2k+2l+2n [f,g]_n := \sum_{r=0}^n (-1)^r\binom{n+2k-1}{n-r}\binom{n+2l-1}{r}f^{(r)}g^{(n-r)}\,\in \,\mathcal{M}(\Gamma)_{2k+2l+2n}

where f (r):=(12πiz) rff^{(r)} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial z}\right)^r f. They are directly related to invariant differential operators used to produce new sl(2)sl(2)-invariant bilinear forms from old ones, so called transvectants found by Gordan,

  • P. Gordan, Das Zerfallen der Curven in gerade Linien, Math. Ann., 45 (1894), pp. 411-427 doi

Moscovici and Connes have constructed a sequence of Hopf algebras q\mathcal{H}_q related to geometry of foliations. Hopf algebra q\mathcal{H}_q has deformations which may be given by universal deformation formulas, or in other words, by Drinfeld twists which are power series in formal variable with unit free term. These 2-cocycles have the structure appearing in Rankin-Cohen brackets and are called Rankin-Cohen deformations and are akin in structure to what is in quantum group context known as Jordanian twist?s, coming from the work of Gurevich on (generalized) Jordanian R-matrices, and of Ogievetsky, Coll-Gerstenhaber-Giaquinto and in a closer symmetrized form by Giaquinto and Zhang. An isomorphism between (reduced) Rankin-Cohen deformation and Jordanian deformation has been exhibited by Samsonov.


  • R. A. Rankin, The construction of automorphic forms from the derivatives of a given form, J. Indian Math.Soc., (N.S.) 20 (1956) 103–116
  • H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math.Ann. 217 (1975) 271-285
  • P. Cohen, Yu. Manin, D. Zagier, Automorphic pseudodifferential operators, Algebraic aspects of integrable systems, 17–47, Progr. Nonlinear Differential Equations Appl.26, Birkauser 1997
  • Alain Connes, Henri Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4:1 (2004) 111–130
  • R. Rochberg, X. Tang, Y. Yao, A survey of Rankin-Cohen deformations, Perspectives on Noncommutative Geometry, Fields Inst. Commun. 61:7 (2011) arXiv:0909.4364
  • P. Bieliavsky, X. Tang, Y. Yao, Rankin–Cohen brackets and formal quantization, Adv. Math. 212:1 (2007) 293-314 doi
  • M. Samsonov, Quantization of semi-classical twists and noncommutative geometry, Lett. Math. Phys. 75 (2006) 63–77 doi
  • Y.–J. Choie, W. Eholzer, Rankin–Cohen operators for Jacobi and Siegel forms, J. Number Theory 68 (1998) 160–177

Last revised on August 16, 2021 at 14:54:35. See the history of this page for a list of all contributions to it.