A Hecke Correspondence Theorem for Nonanalytic Automorphic Integrals

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)

show abstract

“…A more general class of functions was considered in [27]. A function in M r,s can be written explicitly, for some N ∈ N, in the form…”

Section: First Definitionsmentioning

confidence: 99%

“…(3.7) and Lemma 2.6 except for a single constant term of the form αL −w , where α is typically transcendental. This missing constant (when w > 0) can be determined from the others by analytic continuation using an L-function [27].…”

Section: L-functions and Constant Termsmentioning

confidence: 99%

A class of non-holomorphic modular forms I

Brown

2018

Res Math Sci

169

View full text Add to dashboard Cite

show abstract

“…An important special case of f , which appears in [10], is the Maass nonanalytic Eisenstein series (this is the case λ = 1, α − β ∈ 2Z, υ j ≡ 1 and ν j = 0 ∀j); it suggests a possible generalization of the so-called Riemann-Hecke-Bochner correspondence for nonanalytic automorphic integrals (see [13]). Any space which contains the Maass series and is closed under certain commonly used weight-changing operators (see [12]) should allow Fourier coefficients so that a typical term has the form…”

mentioning

confidence: 99%

A generalization of the Lipschitz summation formula and some applications

Pasles

Pribitkin

2001

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms. Lipschitz's formulaAs originally conceived, the Lipschitz Summation Formula (henceforth, LSF ) gives a Fourier expansion for certain functions which arise in the theory of modular forms. More explicitly, it states that if µ ∈ Z and Re α > 1 or µ ∈ R \ Z and Re α > 0, then . In number theory, the LSF is a useful tool in solving the problem of representations of a given integer as a sum of squares [2]; this is the starting point for its connection to the theory of modular forms. Later on, H. Maass considers an analogous series, motivated by his quest for the Fourier coefficients of nonanalytic modular forms. Specifically, he looks atwhere α and β are complex numbers such that Re (α + β) > 1; this requires a more complicated LSF (see [10, pp. 209-211] for a statement and proof of the formula), which was also used by Siegel [17]. Both versions of Lipschitz's formula are incorporated in the following, due to John Hawkins, which first appears in [4]:

show abstract