Nonanalytic automorphic integrals on the Hecke groups

Pasles, Paul C.

doi:10.4064/aa-90-2-155-171

Cited by 8 publications

(14 citation statements)

References 21 publications

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“…We will take a quite general view, essentially in accordance with [24,25], to obtain a result with broad applicability.…”

Section: Multiplier Systemsmentioning

confidence: 99%

“…[6,21,13,12]), or simply that |υ(S λ )| = 1 (as in [25]), or even that υ(S λ ) = 1 (see [24,26]). A few authors further restrict to α ∈ Z, β = 0, and υ ≡ 1 (e.g.…”

Section: Multiplier Systemsmentioning

confidence: 99%

“…This was shown for the theta group in [24], but since the focus of the current paper will be triangle groups, we present a construction for one of the latter.…”

mentioning

confidence: 99%

“…Several recent papers have also examined modular and automorphic forms and integrals of nonreal weight [24][25][26]; their allied multipliers are among those to be considered in this paper. In particular, we will describe a relation between four parameters associated with multipliers: α (often called the weight or first coweight), β (the coweight or second coweight), λ (the width), and κ = 1/2 + (2πi) −1 log[−υ(S λ )], all of which will be carefully defined in due course.…”

mentioning

confidence: 99%

“…We will obtain like results for the more general scenario: two "coweights" instead of one weight; neither coweight necessarily real; no technical assumptions regarding υ(−I); no bound on the multiplier system υ. The last of these is motivated by the fact that the traditional restriction |υ| ≡ 1 has been relaxed in some recent applications [15,[24][25][26].…”

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confidence: 99%

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Multiplier systems

Pasles

2003

Acta Arith.

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“…We will take a quite general view, essentially in accordance with [24,25], to obtain a result with broad applicability.…”

Section: Multiplier Systemsmentioning

confidence: 99%

“…[6,21,13,12]), or simply that |υ(S λ )| = 1 (as in [25]), or even that υ(S λ ) = 1 (see [24,26]). A few authors further restrict to α ∈ Z, β = 0, and υ ≡ 1 (e.g.…”

Section: Multiplier Systemsmentioning

confidence: 99%

“…This was shown for the theta group in [24], but since the focus of the current paper will be triangle groups, we present a construction for one of the latter.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Multiplier systems

Pasles

2003

Acta Arith.

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A Hecke Correspondence Theorem for Nonanalytic Automorphic Integrals

Pasles

2000

Journal of Number Theory

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A class of non-holomorphic modular forms I

Brown

2018

Res Math Sci

169

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This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms) and to the modular graph functions arising in genus one string perturbation theory.This paper studies examples of real analytic functions on the upper half plane satisfying a modular transformation property of the formfor integers r, s. They do not satisfy a simple condition involving the Laplacian. The raison d'être for this class of functions is two-fold:(1) Holomorphic modular forms f with rational Fourier coefficients correspond to certain pure motives M f over Q. Using iterated integrals, we can construct nonholomorphic modular forms which are associated with iterated extensions of the pure motives M f . Their coefficients are periods. (2) In genus one closed string perturbation theory, one assigns a lattice sum to a graph [16], which defines a real analytic function on the upper half plane invariant under SL 2 (Z).It is an open problem to give a complete description of this class of functions and prove their conjectured properties.In this introductory paper, we describe elementary properties of a class M of modular forms. Within this class are modular iterated integrals, which are analogues of singlevalued polylogarithms, and are obtained by solving a differential equation in M. The basic prototype are real analytic Eisenstein series, defined by E r,s (z) = w! (2πi) w+1 1 2 (m,n) =(0,0)

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Nonanalytic automorphic integrals on the Hecke groups

Cited by 8 publications

References 21 publications

Multiplier systems

Multiplier systems

A Hecke Correspondence Theorem for Nonanalytic Automorphic Integrals

A class of non-holomorphic modular forms I

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