934 research outputs found
On a completed generating function of locally harmonic Maass forms
While investigating the Doi-Naganuma lift, Zagier defined integral weight
cusp forms which are naturally defined in terms of binary quadratic forms
of discriminant . It was later determined by Kohnen and Zagier that the
generating function for the is a half-integral weight cusp form. A
natural preimage of under a differential operator at the heart of the
theory of harmonic weak Maass forms was determined by the first two authors and
Kohnen. In this paper, we consider the modularity properties of the generating
function of these preimages. We prove that although the generating function is
not itelf modular, it can be naturally completed to obtain a half-integral
weight modular object
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
On a factorization of Riemann's function with respect to a quadratic field and its computation
Let be a quadratic field, and let its Dedekind zeta function.
In this paper we introduce a factorization of into two functions,
and , defined as partial Euler products of , which lead to
a factorization of Riemann's function into two functions, and
. We prove that these functions satisfy a functional equation which has a
unique solution, and we give series of very fast convergence to them. Moreover,
when the general term of these series at even positive integers is
calculated explicitly in terms of generalized Bernoulli numbers
Modular embeddings of Teichmueller curves
Fuchsian groups with a modular embedding have the richest arithmetic
properties among non-arithmetic Fuchsian groups. But they are very rare, all
known examples being related either to triangle groups or to Teichmueller
curves.
In Part I of this paper we study the arithmetic properties of the modular
embedding and develop from scratch a theory of twisted modular forms for
Fuchsian groups with a modular embedding, proving dimension formulas,
coefficient growth estimates and differential equations.
In Part II we provide a modular proof for an Apery-like integrality statement
for solutions of Picard-Fuchs equations. We illustrate the theory on a worked
example, giving explicit Fourier expansions of twisted modular forms and the
equation of a Teichmueller curve in a Hilbert modular surface.
In Part III we show that genus two Teichmueller curves are cut out in Hilbert
modular surfaces by a product of theta derivatives. We rederive most of the
known properties of those Teichmueller curves from this viewpoint, without
using the theory of flat surfaces. As a consequence we give the modular
embeddings for all genus two Teichmueller curves and prove that the Fourier
developments of their twisted modular forms are algebraic up to one
transcendental scaling constant. Moreover, we prove that Bainbridge's
compactification of Hilbert modular surfaces is toroidal. The strategy to
compactify can be expressed using continued fractions and resembles
Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio
Mat
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