Kronecker’s first limit formula, revisited

  • W. Duke
  • Ö. Imamoḡlu
  • Á. Tóth
Part of the following topical collections:
  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday


We give some new applications of Kronecker’s first limit formula to real quadratic fields. In particular, we give a surprising geometrical relationship between the CM points associated with two imaginary quadratic fields with discriminants d and \(d^{\prime }\) and certain winding number functions coming from the closed geodesics associated with the real quadratic field of discriminant \(d^{\prime }d\).


Author's contributions


Duke and Tóth are grateful to FIM of ETH Zürich for its generous continued support of our joint research. Duke’s research on this paper was supported by NSF Grant DMS 1701638, the Simons Foundation and the Mathematisches Forschungsinstitut Oberwolfach. Á. Tóth is supported by NKFIH (National Research, Development and Innovation Office) Grant ERC_HU_15 118946.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Authors and Affiliations

  1. 1.Mathematics DepartmentUCLALos AngelesUSA
  2. 2.Mathematics DepartmentETHZurichSwitzerland
  3. 3.Eotvos Lorand UniversityBudapestHungary

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