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Author Hilgert, J. ♦ Mayer, D. ♦ Movasati, H.
Source CiteSeerX
Content type Text
File Format PDF
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Old Solution ♦ New Form ♦ Special Eigenfunctions ♦ Hecke Operator ♦ Surprising Relation ♦ Modular Group Psl ♦ Modular Form ♦ Way Linear Operator ♦ Period Function ♦ Modular Group ♦ Eichler-manin-shimura Correspondence ♦ Old Eigenfunctions ♦ Group Psl ♦ Period Polynomial ♦ Transfer Operator ♦ Lewis Equation ♦ Atkin-lehner Theory
Description Of PSL(2, Z). Math. Proc. Of the Cambridge Philosophical Society
In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group PSL(2, Z). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓1, which are also solutions of the Lewis equations for the groups Γ0(n) and which are determined by eigenfunctions of the transfer operator for the modular group PSL(2, Z). In the language of the Atkin-Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group Γ0(n) determine for any n a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group. Our construction gives in this way linear operators in the space of period functions for the group PSL(2, Z). Indeed these operators are just the Hecke operators for the period functions of the modular group derived previously by Zagier and Mühlenbruch using the Eichler-Manin-Shimura correspondence between period polynomials and modular forms for the modular group. 1
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