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On a conjecture of Hecke concerning elementary class number formulas. (English) Zbl 0259.12006
Author’s summary: Let \(K\) be a totally real algebraic number field of class number \(h_K\) and \(L\) a totally imaginary quadratic extension of \(K\) of class number \(h_L\). Hecke conjectured that there exists an “elementary” formula for the first factor \(h_L/h_K\) of the class number of \(L\). The paper develops a theory which allows computation of \(h_L/h_K\) 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 \(K\).

MSC:
11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
11R80 Totally real fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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