Period functions for Maass wave forms. I.

By John S. Lewis, Don Zagier. 2001

211 citations recorded by the source corpus.

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 \to \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 Gamma\H, 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.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2b4ff7cb4dbb668a5930a1f6bab306bc4185847c

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Citations in the corpus

Quantum modular forms (2010, background): “The fact that it is C∞ also at x = −1/2 follows by looking more closely at the integral and using that u is a cusp form, as was done in [13] for the period functions of Maass forms on the full modular group.”
Double Zeta values and modular forms (2006, background): “Remark. The operator L = 1 T T 0 plays a key role in the discovery by J. Lewis that there are holomorphic functions annihilated by this operator which have the same relation to the so-called Maass wave forms as period polynomials have to holomorphic modular forms ([10], [ 11 ]).”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, methodology): “This is reminiscent of the properties of similar functions discussed in [83], in relation to the Eichler integrals of Maass waveforms.”
Period functions and cotangent sums (2011, background): “MR 2004c:11159 Zbl 1117.11305 [Lewis and Zagier 2001] J. Lewis and D. Zagier, “Period functions for Maass wave forms, I”, Ann. of Math.”
The notion of dimension in geometry and algebra (2005, background): “In the following we will briefly describe recent work of D. Mayer ([May]), J. Lewis, and D. Zagier ([ LZ1 ], [LZ2]), giving a very beautiful description of the spectrum of values of s for which an L–invariant density exists.”