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On a conjecture of Hecke concerning elementary class number formulas

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Abstract

Let K be a totally real algebraic number field of class number hK and L a totally imaginary quadratic extension of K of class number hL. Hecke conjectured that there exists an “elementary” formula for the first factor hL/hK of the class number of L. The paper develops a theory which allows computation of hL/hK in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, Maryland

    Larry Joel Goldstein

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  1. Larry Joel Goldstein
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Research supported by NSF Grant GP 20538

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Goldstein, L.J. On a conjecture of Hecke concerning elementary class number formulas. Manuscripta Math 9, 245–305 (1973). https://doi.org/10.1007/BF01303855

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