Modern analysis, cuspforms
نویسنده
چکیده
We prove that there is an orthonormal basis for square-integrable (waveform) cuspforms on SL2(Z)\H. First, we reconsider the well-known fact that the Hilbert space L(T) on the circle T = R/2πZ has an orthogonal Hilbert-space basis of exponentials e with n ∈ Z, using ideas relevant to situations lacking analogues of Dirichlet or Fejer kernels. These exponentials are eigenfunctions for the Laplacian ∆ = d/dx, so it would suffice to show that L(T) has an orthogonal basis of eigenfunctions for ∆. Two technical issues must be overcome: that ∆ does not map L(T) to itself, and that there is no guarantee that infinitedimensional Hilbert spaces have Hilbert-space bases of eigenfunctions for a given linear operator. [1]
منابع مشابه
On Atkin and Swinnerton-dyer Congruence Relations (3)
In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long, Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied special families of cuspforms for noncongruence arithmetic subgroups. It was found that the Fourier coefficients of these modular forms at infinity satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the p-adic anal...
متن کاملAutomorphic Representations and L-functions
• Decomposition by central characters • Square-integrable cuspforms • Smoothness of cuspforms • Eigen-cuspforms and automorphic representation • Dirichlet series versus zeta and L-functions • L-functions defined via local data • Factoring unitary representations of adele groups • Spherical representations and Satake parameters • Local data, L-groups, higher L-functions References and historical...
متن کاملDecomposition and estimates for cuspforms
The discrete decomposition with finite multiplicities of square-integrable cuspforms, and concommitant estimates, are foundational. Lie-group and symmetric-space versions of these results were known [Selberg 1956], [Gelfand-Fomin 1952] in the mid 1950’s. The general assertion was made in [GelfandGraev 1962] and [Gelfand-PS 1963], the latter observing that an adelic formulation can proceed in th...
متن کاملColin de Verdière’s meromorphic continuation of Eisenstein series
1. Harmonic analysis on H 2. Meromorphic continuation up to the critical line 3. Sobolev inequality/imbedding 4. Eventually-vanishing constant terms 5. Compactness of Sob(Γ\H)a → L(Γ\H) 6. Discreteness of cuspforms 7. Meromorphic continuation beyond the critical line 8. Discrete decomposition of truncated Eisenstein series 9. Appendix: Friedrichs extensions 10. Appendix: simplest Maass-Selberg ...
متن کاملThe First Negative Hecke Eigenvalue of Genus
In this short note we extend results of Kohnen and Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level N > 1. We also discuss the signs of Hecke eigenvalues of CAP forms.
متن کامل