Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T23:29:10.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequences, modular forms and cellular integrals

Published online by Cambridge University Press:  11 October 2018

DERMOT McCARTHY ,
ROBERT OSBURN  and
ARMIN STRAUB
DERMOT McCARTHY
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79410-1042, U.S.A. e-mail: dermot.mccarthy@ttu.edu
ROBERT OSBURN
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland. e-mail: robert.osburn@ucd.ie
ARMIN STRAUB
Affiliation:
Department of Mathematics and Statistics, University of South Alabama. 411 University Blvd N, MSPB 325, Mobile AL 36686, U.S.A. e-mail: straub@southalabama.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 379 - 404
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ahlgren, S. Gaussian hypergeometric series and combinatorial congruences. In Symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math. 4 (Kluwer Academic Publisher, Dordrecht, 2001), pp. 1–12.Google Scholar
[2] Ahlgren, S. The points of a certain fivefold over finite fields and the twelfth power of the eta function. Finite Fields Appl. 8 (2002), 18–33.Google Scholar
[3] Ahlgren, S. and Ono, K. A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518 (2000), 187–212.Google Scholar
[4] Amdeberhan, T. and Tauraso, R. Supercongruences for the Almkvist–Zudilin numbers. Acta Arith. 173 (2016), 255–268.Google Scholar
[5] Almkvist, G., van Enckevort, C., van Straten, D. and Zudilin, W. Tables of Calabi–Yau equations. Preprint (2010), arXiv:math/0507430v2Google Scholar
[6] Aspvall, B. and Liang, F. The dinner table problem. Technical Report STAN-CS-80-829, Computer Science Department, Stanford University, Stanford, California, 1980.Google Scholar
[7] Beukers, F. A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11 (1979), 268–272.Google Scholar
[8] Beukers, F. Some congruences for the Apéry numbers. J. Number Theory 21 (1985), 141–155.Google Scholar
[9] Beukers, F. Another congruence for the Apéry numbers. J. Number Theory 25 (1987), 201–210.Google Scholar
[10] Bogner, C. and Brown, F. Feynman integrals and iterated integrals on moduli spaces of curves of genus zero. Commun. Number Theory Phys. 9 (2015), 189–238.Google Scholar
[11] Broedel, J., Schlotterer, O. and Stieberger, S. Polylogarithms, multiple zeta values and superstring amplitudes. Fortschr. Phys. 61 (2013), 812–870.Google Scholar
[12] Brown, F. Multiple zeta values and periods of moduli spaces $\overline{{\Ncal M}}_{0,n}$. Ann. Sci. Éc. Norm. Supér (4) 42 (2009), 371–489.Google Scholar
[13] Brown, F. Irrationality proofs for zeta values, moduli spaces and dinner parties. Mosc. J. Comb. Number Theory 6 (2016), 102–165.Google Scholar
[14] Brown, F., Carr, S. and Schneps, L. The algebra of cell-zeta values. Compos. Math. 146 (2010), 731–771.Google Scholar
[15] Chan, H. H., Cooper, S. and Sica, F. Congruences satisfied by Apéry-like numbers. Int. J. Number Theory 6 (2010), 89–97.Google Scholar
[16] Cooper, S. Sporadic sequences, modular forms and new series for 1/π. Ramanujan J. 29 (2012), 163–183.Google Scholar
[17] Coster, M. Supercongruences. PhD. thesis, Universiteit Leiden (1988).Google Scholar
[18] Dupont, C. Odd zeta motive and linear forms in odd zeta values. With a joint appendix with Don Zagier. Compos. Math. 154 (2018), 342–379.Google Scholar
[19] Frechette, S., Ono, K. and Papanikolas, M. Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Notices 2004, 3233–3262.Google Scholar
[20] Gessel, I. Some congruences for Apéry numbers. J. Number Theory 14 (1982), 362–368.Google Scholar
[21] Golyshev, V. V. and Zagier, D. Proof of the gamma conjecture for Fano 3-folds of Picard rank 1. Izv. Math. 80 (2016), 24–49.Google Scholar
[22] Goncharov, A. B. and Manin, Y. I. Multiple ζ-motives and moduli spaces $\overline{{\Ncal M}}_{0,n}$. Compos. Math. 140 (2004), 1–14.Google Scholar
[23] Greene, J. Hypergeometric functions over finite fields. Trans. Amer. Math. Soc. 301 (1987), 77–101.Google Scholar
[24] Hardy, G. H. and Wright, E. M. An introduction to the theory of numbers. Fifth edition. (The Clarendon Press, Oxford University Press, New York, 1979).Google Scholar
[25] Kalita, G. and Chetry, A. Congruences for generalised Apéry numbers and Gaussian hypergeometric series. Res. Number Theory 3 (2017), Art. 5, 15 pp.Google Scholar
[26] Koike, M. Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields. Hiroshima Math. J. 25 (1995), 43–52.Google Scholar
[27] Koutschan, C. Advanced Applications of the Holonomic Systems Approach. PhD. thesis. RISC, Johannes Kepler University, Linz, Austria (2009).Google Scholar
[28] Lairez, P. Computing periods of rational integrals. Math. Comp. 85 (2016), 1719–1752.Google Scholar
[29] Loh, P. and Rhodes, R. p-adic and combinatorial properties of modular form coefficients. Int. J. Number Theory 2 (2006), 305–328.Google Scholar
[30] Ono, K. Values of Gaussian hypergeometric series. Trans. Amer. Math. Soc. 350 (1998), 1205–1223.Google Scholar
[31] Osburn, R. and Sahu, B. Congruences via modular forms. Proc. Amer. Math. Soc. 139 (2011), 2375–2381.Google Scholar
[32] Osburn, R. and Sahu, B. Supercongruences for Apéry-like numbers. Adv. in Appl. Math. 47 (2011), 631–638.Google Scholar
[33] Osburn, R. and Sahu, B. A supercongruence for generalised Domb numbers. Funct. Approx. Comment. Math. 48 (2013), 29–36.Google Scholar
[34] Osburn, R., Sahu, B. and Straub, A. Supercongruences for sporadic sequences. Proc. Edinb. Math. Soc. (2) 59 (2016), 503–518.Google Scholar
[35] Osburn, R. and Schneider, C. Gaussian hypergeometric series and supercongruences. Math. Comp. 78 (2009), 275–292.Google Scholar
[36] Osburn, R. and Straub, A. Interpolated sequences and critical $L$-values of modular forms. Preprint (2018), arXiv:1806.05207Google Scholar
[37] Panzer, E. Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals. Comput. Phys. Commun. 188 (2015), 148–166.Google Scholar
[38] Poulet, P. Permutations. L'Intermédiaire des Mathématiciens 26 (1919), 117–121.Google Scholar
[39] Ribet, K. Galois representations attached to eigenforms with Nebentypus. In Modular functions of one variable, V, Lecture Notes in Math. vol. 601 (Springer, 1977), pp. 17–51.Google Scholar
[40] Roberts, D.P., Rodriquez–Villegas, F. and Watkins, M. Hypergeometric motives. “Preprint” (2017).Google Scholar
[41] Schlotterer, O. and Stieberger, S. Motivic multiple zeta values and superstring amplitudes. J. Phys. A 46 (2013), 475401, 37 pp.Google Scholar
[42] Stienstra, J. and Beukers, F. On the Picard–Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271 (1985), 269–304.Google Scholar
[43] Straub, A. Multivariate Apéry numbers and supercongruences of rational functions. Algebra Number Theory 8 (2014), 1985–2008.Google Scholar