Abstract
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.
Highlights
In the companion Part I [1] to this paper we introduced the Laplace equations (∆ − s(s − 1))F+m(,ks) = EmEk, (∆ − s(s 1))F−m(,ks) =(∇Em)(∇Ek) − (∇Ek)(∇Em) 2(Im τ )2(1.1a) (1.1b) with integers s ≥ 2 and 2 ≤ m ≤ k, and where Ek are non-holomorphic Eisenstein series (Im τ )k Ek = πk |mτ + n|2k . (m,n)=(0,0)
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper
The source term of the Laplace equation is a product of two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of singlevalued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms
Summary
The latter were fixed from certain requirements on the desired solutions concerning their Cauchy-Riemann derivatives and asymptotics at the cusp, see Part I for further details This procedure was tailored towards solving the Laplace system in terms of the building blocks βsv of MGFs and does not guarantee that the resulting expression is modular invariant. Perhaps more interestingly, due to the presence of iterated integrals of holomorphic cusp forms we find that the exponentially suppressed terms of the form qnq0 (and their complex conjugates q0qn) with n > 0 are multiplied by rich Laurent polynomials in y: their coefficients are either rationals, or Q-multiples of single odd zeta values or surprisingly rationals (or more general number-field extensions of Q) times special ratios of completed L-values associated to whichever cusp form is at play These results allow us to make novel predictions regarding the non-zero Fourier-mode decomposition of the Poincaré series (1.5). An ancillary file that accompanies the arXiv submission and the supplementary material of the journal publication of this work contains many examples and explicit expressions related to the functions F±m(,ks)
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