nach oben

Erschienen in:

2019 | OriginalPaper | Buchkapitel

Eta Quotients and Rademacher Sums

verfasst von : Kevin Acres, David Broadhurst

Erschienen in: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Eta quotients on \(\varGamma _0(6)\) yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome q. Atkin–Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for \(\varGamma _0(6)\) and its generalization is found for all levels with genus 0, namely for \(N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25\). There are elliptic obstructions at \(N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,\) with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases. We show how to handle the levels \(N=22, 23, 26, 28, 29, 31, 37, 50\), with genus 2, and the levels \(N=30,33,34,35,39,40,41,43,45,48,64\), with genus 3. We also solve examples with genera 4, 5, 6, 7, 8, 13.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nächstes Kapitel On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.-S. Radu, C. Schneider, Iterated elliptic and hypergeometric integrals for Feynman diagrams. J. Math. Phys. 59(6), 062305 (2018), arXiv:1706.01299MathSciNetCrossRef

D.H. Bailey, J.M. Borwein, D. Broadhurst, M.L. Glasser, Elliptic integral evaluations of Bessel moments. J. Phys. A 41, 205203 (2008), arXiv:0801.0891

F. Beukers, Irrationality proofs using modular forms. Journées arithmétiques de Besançon, Astérisque 147–148, 271–283 (1987)MathSciNetMATH

S. Bloch, P. Vanhove, The elliptic dilogarithm for the sunset graph. J. Number Theory 148, 328–364 (2015), arXiv:1309.5865MathSciNetCrossRef

S. Bloch, M. Kerr, P. Vanhove, A Feynman integral via higher normal functions. Compos. Math. 151, 2329–2375 (2015), arXiv:1406.2664MathSciNetCrossRef

C. Bogner, A. Schweitzer, S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral. Nucl. Phys. B 922, 528–550 (2017), arXiv:1705.08952MathSciNetCrossRef

D. Broadhurst, Multiple zeta values and modular forms in quantum field theory, in Computer Algebra in Quantum Field Theory. Texts and Monographs in Symbolic Computation, ed. by C. Schneider, J. Blümlein (Springer, Vienna, 2013), pp. 33–73MATH

D. Broadhurst, Feynman integrals, L-series and Kloosterman moments, Commun. Number Theory Phys. 10, 527–569 (2016), arXiv:1604.03057MathSciNetCrossRef

D. Broadhurst, A. Mellit, Perturbative quantum field theory informs algebraic geometry, in Loops and Legs in Quantum Field Theory, PoS (LL2016) 079 (2016)

10.

D. Broadhurst, O. Schnetz, Algebraic geometry informs perturbative quantum field theory, in Loops and Legs in Quantum Field Theory, PoS (LL2014) 078 (2014)

11.

D.J. Broadhurst, The master two-loop diagram with masses. Z. Phys. C 47, 115–124 (1990)CrossRef

12.

D.J. Broadhurst, J. Fleischer, O.V. Tarasov, Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension. Z. Phys. C 60, 287–301 (1993), arXiv:hep-ph/9304303

13.

F. Brown, A class of non-holomorphic modular forms III: real analytic cusp forms for \(SL_2(Z)\), arXiv:1710.07912

14.

F. Brown, O. Schnetz, A K3 in \(\phi ^4\). Duke Math. J. 161, 1817–1862 (2012), arXiv:1006.4064MathSciNetCrossRef

15.

H.H. Chan, W. Zudilin, New representations for Apéry-like sequences. Mathematika 56, 107–117 (2010)CrossRef

16.

H. Cohen, Tutorial for modular forms in Pari/GP (2018), http://pari.math.u-bordeaux.fr/pub/pari/manuals/2.10.0/tutorial-mf.pdf

17.

J.F.R. Duncan, M.J. Griffin, K. Ono, Moonshine. Res. Math. Sci. 2, 11 (2015), arXiv:1411.6571

18.

M. Eichler, D. Zagier, The Theory of Jacobi Forms. Progress in Mathematics, vol. 55 (Birkhäuser, Boston, 1985)CrossRef

19.

N. Elkies, The automorphism group of the modular curve \(X_0(63)\). Compos. Math. 74, 203–208 (1990)

20.

G.S. Joyce, On the simple cubic lattice Green function. Philos. Trans. R. Soc. Math. Phys. Sci. 273, 583–610 (1973)MathSciNetCrossRef

21.

P. Kleban, D. Zagier, Crossing probabilities and modular forms. J. Stat. Phys. 113, 431–454 (2003)MathSciNetCrossRef

22.

M.I. Knopp, Rademacher on \(J(\tau )\), Poincaré series of nonpositive weights and the Eichler cohomology. Not. Am. Math. Soc. 37, 385–393 (1990)

23.

S. Laporta, High-precision calculation of the 4-loop contribution to the electron \(g-2\) in QED. Phys. Lett. B 772, 232–238 (2017), arXiv:1704.06996

24.

R.S. Maier, On rationally parametrized modular equations. J. Ramanujan Math. Soc. 24, 1–73 (2009), arXiv:math/0611041

25.

G. Martin, Dimensions of the spaces of cusp forms and newforms on \(\Gamma _0(N)\) and \(\Gamma _1(N)\). J. Number Theory 112, 298–331 (2005), arXiv:math/0306128

26.

H. Petersson, Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58, 169–215 (1932)MathSciNetCrossRef

27.

H. Rademacher, The Fourier coefficients of the modular invariant \(J(\tau )\). Am. J. Math. 60, 501–512 (1938)

28.

H. Rademacher, The Fourier series and the functional equation of the absolute modular invariant \(J(\tau )\). Am. J. Math. 61, 237–248 (1939)

29.

H. Rademacher, On the expansion of the partition function in a series. Ann. Math. 44, 416–422 (1943)MathSciNetCrossRef

30.

A. Sabry, Fourth order spectral functions for the electron propagator. Nucl. Phys. 33, 401–430 (1962)MathSciNetCrossRef

31.

N.-P. Skoruppa, D. Zagier, Jacobi forms and a certain space of modular forms. Invent. Math. 94(1988), 113–146 (1988)MathSciNetCrossRef

32.

Y. Yang, Transformation formulas for generalized Dedekind eta functions. Bull. Lond. Math. Soc. 36, 671–682 (2004)MathSciNetCrossRef

33.

Y. Yang, Defining equations of modular curves. Adv. Math. 204, 481–508 (2006)MathSciNetCrossRef

34.

Y. Zhou, Hilbert transforms and sum rules of Bessel moments. Ramanujan J. (2017). https://doi.org/10.1007/s11139-017-9945-y, arXiv:1706.01068MathSciNetCrossRef

35.

Y. Zhou, Wick rotations, Eichler, integrals, and multi-loop Feynman diagrams. Commun. Number Theory Phys. 12, 127–192 (2018), arXiv:1706.08308MathSciNetCrossRef

36.

Y. Zhou, Wronskian, factorizations and Broadhurst-Mellit determinant formulae. Commun. Number Theory Phys. 12, 355–407 (2018), arXiv:1711.01829MathSciNetCrossRef

37.

Y. Zhou, On Laporta’s 4-loop sunrise formulae, arXiv:1801.02182

38.

Y. Zhou, Some algebraic and arithmetic properties of Feynman diagrams, to appear in this volume, arXiv:1801.05555

Titel: Eta QuotientsEta quotients and Rademacher SumsRademacher sums
verfasst von: Kevin Acres
David Broadhurst
Verlag: Springer International Publishing
Buch: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory
Print ISBN: 978-3-030-04479-4

Electronic ISBN: 978-3-030-04480-0

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-04480-0_1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"