Lectures on Modular Functions of One Complex Variable

Maaß, Hans; Lal, Sunder

doi:10.1007/978-3-662-02380-8

Cited by 88 publications

(97 citation statements)

References 2 publications

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“…Following [20], take arg(cz +d) ∈ (−π, π] for Im z > 0 and c, d ∈ R (c and d not both zero), and arg w ∈ [−π, π) otherwise. This ensures usefully that arg(cz +d) = − arg(cz +d) and log(cz +d)+log(cz +d) ∈ R, thus simplifying many calculations.…”

Section: Multiplier Systemsmentioning

confidence: 99%

Multiplier systems

Pasles

2003

Acta Arith.

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Section: Multiplier Systemsmentioning

confidence: 99%

Multiplier systems

Pasles

2003

Acta Arith.

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“…If one of the mapping class generators g α modifies K(i) very little, where K(i) is a classical solution which lies within the packet a(k), then by applying this generator again and again one obtains a sequence of contributions from K (jn) , with g (jn) = (g α ) n g(i). The series (3.22) may be exactly summable in this case; in the example of the torus the result should be related to the Maass forms [16].…”

Section: The Stationary Phase Theorem and The Sum Over Mapping Classmentioning

confidence: 99%

Canonical quantization of (2+1)-dimensional gravity

Waelbroeck

1994

Phys. Rev. D

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We consider the quantum dynamics of both open and closed two-dimensional universes with "wormholes" and particles. The wave function is given as a sum of freely propagating amplitudes, emitted from a network of mapping class images of the initial state. Interference between these amplitudes gives non-trivial scattering effects, formally analogous to the optical diffraction by a multidimensional grating; the "bright lines" correspond to the most probable geometries.

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“…The space M ! is equipped with differential operators ∂, ∂ closely related to Maass' raising and lowering operators [17], and a Laplacian . In [2], we defined a subspace MI !…”

Section: Introductionmentioning

confidence: 99%

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Brown

2018

Res Math Sci

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We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit 'weak harmonic lifts' for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n 1−k (a n + ρa n ) for n = 0, where ρ is the normalised permanent of the period matrix of the corresponding motive, and a n , a n are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

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Lectures on Modular Functions of One Complex Variable

Cited by 88 publications

References 2 publications

Multiplier systems

Multiplier systems

Canonical quantization of (2+1)-dimensional gravity

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

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