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A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS

Part of: Quantum field theory; related classical field theories Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  28 May 2020

FRANCIS BROWN Open the ORCID record for FRANCIS BROWN [Opens in a new window]
FRANCIS BROWN*
Affiliation:
Mathematical Institute, University of Oxford; francis.brown@all-souls.ox.ac.uk

Abstract

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We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

MSC classification

Primary: 11F03: Modular and automorphic functions 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 81T30: String and superstring theories; other extended objects (e.g., branes)

Information

Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e31
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2020

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