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

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

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)
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

References

Brown, F., A multi-variable version of the completed Riemann zeta function and other $L$-functions, https://arxiv.org/abs/1904.00190.Google Scholar
Brown, F., Multiple modular values and the relative completion of the fundamental group of $M_{1,1}$, arXiv:1407.5167.Google Scholar
Brown, F., Depth-graded motivic multiple zeta values, arXiv:1301.3053.Google Scholar
Brown, F., ‘Polylogarithmes multiples uniformes en une variable’, C. R. Math. Acad. Sci. Paris 338(7) (2004), 527–532.CrossRefGoogle Scholar
Brown, F., ‘Zeta elements in depth 3 and the fundamental Lie algebra of the infinitesimal Tate curve’, Forum Math. Sigma 5 (2017), doi:10.1017/fms.2016.29, arXiv:1504.04737.CrossRefGoogle Scholar
Brown, F., ‘A class of non-holomorphic modular forms I’, Res. Math. Sci. 5 (2018), 7, https://doi.org/10.1007/s40687-018-0130-8, (https://arxiv.org/abs/1710.07912).CrossRefGoogle Scholar
Brown, F., ‘A class of non-holomorphic modular forms III: real analytic cusp forms’, Res. Math. Sci. 5 (2018), 34, https://doi.org/10.1007/s40687-018-0151-3, (https://arxiv.org/abs/1710.07912).Google Scholar
Brown, F. and Levin, A., Multiple elliptic polylogarithms, arXiv:1110.6917.Google Scholar
Calaque, D., Enriquez, B. and Etingof, P., ‘Universal KZB equations: the elliptic case’, inAlgebra, Arithmetic, and Geometry: in honour of Yu. I. Manin, Vol. I, Progress in Mathematics, 269 (Springer, Birkhäuser Boston, 2009), 165–266.CrossRefGoogle Scholar
Chen, K. T., ‘Iterated path integrals’, Bull. Amer. Math. Soc. 83 (1977), 831–879.CrossRefGoogle Scholar
Diamantis, N. and O’Sullivan, C., ‘Kernels for products of L-functions’, Algebra Number Theory 7(8) (2013), 1883–1917.CrossRefGoogle Scholar
D’Hoker, E. and Green, M., ‘Identities between modular graph forms’, J. Number Theory 189 (2018), 25–88.CrossRefGoogle Scholar
D’Hoker, E., Green, M. B., Gürdogan, O. and Vanhove, P., Modular graph functions, arXiv:1512.06779.Google Scholar
D’Hoker, E. and Kaidi, J., ‘Hierarchy of modular graph identities’, J. High Energy Phys. (11)051 (2016).CrossRefGoogle Scholar
Eichler, M., ‘Eine Verallgemeinerung der Abelschen Integrale’, Math. Z. 67 (1957), 267–298.CrossRefGoogle Scholar
Enriquez, B., ‘Elliptic associators’, Selecta Math. (N.S.) 20(2) (2014), 491–584.CrossRefGoogle Scholar
Gangl, H., Kaneko, M. and Zagier, D., ‘Double zeta values and modular forms’, inAutomorphic Forms and Zeta Functions (World Sci. Publ., Hackensack, NJ, 2006), 71–106.CrossRefGoogle Scholar
Green, M., Russo, J. and Vanhove, P., ‘Low energy expansion of the four-particle genus-one amplitude in type II superstring theory’, J. High Energy Phys. 2008 (2008), doi:10.1088/1126-6708/2008/02/020.CrossRefGoogle Scholar
Hain, R., ‘Notes on the universal elliptic KZB equation’, Pure Appl. Math. Q. 12(2) (2016), International Press, [arXiv:1309.0580].Google Scholar
Hain, R., Ihara curves, unpublished notes.Google Scholar
doi:10.1017/S1474748018000130Hain, R. and Matsumoto, M., ‘Universal mixed elliptic motives’, J. Inst. Math. Jussieu 19(3) 663–76610.1017/S1474748018000130.CrossRefGoogle Scholar
Hain, R., ‘The Hodge de Rham theory of the relative Malcev completion’, Ann. Sci. Éc. Norm. Supér. 31 (1998), 47–92.CrossRefGoogle Scholar
Hain, R., ‘The Hodge-de Rham theory of modular groups’, inRecent Advances in Hodge Theory, London Mathematical Society Lecture Note Series, 427 (Cambridge University Press, Cambridge, 2016), 422–514.CrossRefGoogle Scholar
Ihara, K., Kaneko, M. and Zagier, D., ‘Derivation and double shuffle relations for multiple zeta values’, Compos. Math. 142 (2006), 307–338.CrossRefGoogle Scholar
Levin, A. and Racinet, G., Towards multiple elliptic polylogarithms, arXiv:math/0703237.Google Scholar
Kontsevich, M. and Zagier, D., ‘Periods’, inMathematics Unlimited- 2001 and Beyond (Springer, Berlin, 2001), 771–808.CrossRefGoogle Scholar
Manin, Y., ‘Periods of parabolic points and p-adic Hecke series’, Math. Sb. (1973), 371–393.CrossRefGoogle Scholar
Manin, Y., ‘Iterated Shimura integrals’, Moscow Math. J. 5 (2005), 869–881.CrossRefGoogle Scholar
Manin, Y., ‘Iterated integrals of modular forms and non-commutative modular symbols’, inAlgebraic Geometry and Number Theory, Progress in Mathematics, 253 (Springer, Birkhäuser Boston, 2006), 565–597.CrossRefGoogle Scholar
Matthes, N., ‘On the algebraic structure of iterated integrals of quasimodular forms’, Algebra Number Theory 11(9) (2017), 2113–2130.CrossRefGoogle Scholar
Nakamura, H., ‘Tangential base points and Eisenstein power series’, inAspects of Galois Theory, London Mathematical Society Lecture Note Series, 256 (Cambridge University Press, Cambridge, 1999), 202–217.Google Scholar
Pollack, A., Relations between derivations arising from modular forms, undergraduate thesis, Duke University, (2009).Google Scholar
Shimura, G., ‘Sur les intégrales attachées aux formes automorphes’, J. Math. Soc. Japan 11 (1959), 291–311.CrossRefGoogle Scholar
Sturm, J., ‘Projections of C ∞ automorphic forms’, Bull. Amer. Math. Soc. (N.S.) 2(3) (1980), 435–439.CrossRefGoogle Scholar
Tsunogai, H., ‘On some derivations of Lie algebras related to Galois representations’, Publ. Res. Inst. Math. Sci. 31 (1995), 113–134.CrossRefGoogle Scholar
Zagier, D., ‘The 1-2-3 of modular forms’, inElliptic Modular Forms and Their Applications, Universitext (Springer, Berlin, 2008), 1–103.Google Scholar
Zerbini, F., ‘Single-valued multiple zeta values in genus 1 superstring amplitudes’, Commun. Number Theory Phys. 10(4) (2016), 703–737.CrossRefGoogle Scholar