Period functions for Maass wave forms. I.
Period functions for Maass wave forms. I.
复制标题
马斯波形的周期函数。
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
D. Zagier
中科院分区：
文献类型：
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作者：
John S. Lewis;D. Zagier
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to C which is small as y o infty and satisfies Delta u = lambda u for some lambda in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic Laplacian. These functions give a basis for L_2 on the modular surface GammaH, with the usual trigonometric waveforms on the torus R^2/Z^2, which are also (for this surface) both the Fourier building blocks for L_2 and eigenfunctions of the Laplacian. Although therefore very basic objects, Maass forms nevertheless still remain mysteriously elusive fifty years after their discovery; in particular, no explicit construction exists for any of these functions for the full modular group. The basic information about them (e.g. their existence and the density of the eigenvalues) comes mostly from the Selberg trace formula: the rest is conjectural with support from extensive numerical computations.
