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Moduli of abelian varieties and p-divisible groups
Lecture notes at a conference on Arithmetic Geometry, Goettingen, July/August
2006: Density of ordinary Hecke orbits and a conjecture by Grothendieck on
deformations of p-divisible groups.Comment: 92 page
Period functions for Maass wave forms. I
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.Comment: 68 pages, published versio
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