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Sequences, modular forms and cellular integrals


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.

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[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. 112.
[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), 1833.
[3] Ahlgren, S. and Ono, K. A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518 (2000), 187212.
[4] Amdeberhan, T. and Tauraso, R. Supercongruences for the Almkvist–Zudilin numbers. Acta Arith. 173 (2016), 255268.
[5] Almkvist, G., van Enckevort, C., van Straten, D. and Zudilin, W. Tables of Calabi–Yau equations. Preprint (2010), arXiv:math/0507430v2
[6] Aspvall, B. and Liang, F. The dinner table problem. Technical Report STAN-CS-80-829, Computer Science Department, Stanford University, Stanford, California, 1980.
[7] Beukers, F. A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11 (1979), 268272.
[8] Beukers, F. Some congruences for the Apéry numbers. J. Number Theory 21 (1985), 141155.
[9] Beukers, F. Another congruence for the Apéry numbers. J. Number Theory 25 (1987), 201210.
[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), 189238.
[11] Broedel, J., Schlotterer, O. and Stieberger, S. Polylogarithms, multiple zeta values and superstring amplitudes. Fortschr. Phys. 61 (2013), 812870.
[12] Brown, F. Multiple zeta values and periods of moduli spaces . Ann. Sci. Éc. Norm. Supér (4) 42 (2009), 371489.
[13] Brown, F. Irrationality proofs for zeta values, moduli spaces and dinner parties. Mosc. J. Comb. Number Theory 6 (2016), 102165.
[14] Brown, F., Carr, S. and Schneps, L. The algebra of cell-zeta values. Compos. Math. 146 (2010), 731771.
[15] Chan, H. H., Cooper, S. and Sica, F. Congruences satisfied by Apéry-like numbers. Int. J. Number Theory 6 (2010), 8997.
[16] Cooper, S. Sporadic sequences, modular forms and new series for 1/π. Ramanujan J. 29 (2012), 163183.
[17] Coster, M. Supercongruences. PhD. thesis, Universiteit Leiden (1988).
[18] Dupont, C. Odd zeta motive and linear forms in odd zeta values. With a joint appendix with Don Zagier. Compos. Math. 154 (2018), 342379.
[19] Frechette, S., Ono, K. and Papanikolas, M. Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Notices 2004, 32333262.
[20] Gessel, I. Some congruences for Apéry numbers. J. Number Theory 14 (1982), 362368.
[21] Golyshev, V. V. and Zagier, D. Proof of the gamma conjecture for Fano 3-folds of Picard rank 1. Izv. Math. 80 (2016), 2449.
[22] Goncharov, A. B. and Manin, Y. I. Multiple ζ-motives and moduli spaces . Compos. Math. 140 (2004), 114.
[23] Greene, J. Hypergeometric functions over finite fields. Trans. Amer. Math. Soc. 301 (1987), 77101.
[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).
[25] Kalita, G. and Chetry, A. Congruences for generalised Apéry numbers and Gaussian hypergeometric series. Res. Number Theory 3 (2017), Art. 5, 15 pp.
[26] Koike, M. Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields. Hiroshima Math. J. 25 (1995), 4352.
[27] Koutschan, C. Advanced Applications of the Holonomic Systems Approach. PhD. thesis. RISC, Johannes Kepler University, Linz, Austria (2009).
[28] Lairez, P. Computing periods of rational integrals. Math. Comp. 85 (2016), 17191752.
[29] Loh, P. and Rhodes, R. p-adic and combinatorial properties of modular form coefficients. Int. J. Number Theory 2 (2006), 305328.
[30] Ono, K. Values of Gaussian hypergeometric series. Trans. Amer. Math. Soc. 350 (1998), 12051223.
[31] Osburn, R. and Sahu, B. Congruences via modular forms. Proc. Amer. Math. Soc. 139 (2011), 23752381.
[32] Osburn, R. and Sahu, B. Supercongruences for Apéry-like numbers. Adv. in Appl. Math. 47 (2011), 631638.
[33] Osburn, R. and Sahu, B. A supercongruence for generalised Domb numbers. Funct. Approx. Comment. Math. 48 (2013), 2936.
[34] Osburn, R., Sahu, B. and Straub, A. Supercongruences for sporadic sequences. Proc. Edinb. Math. Soc. (2) 59 (2016), 503518.
[35] Osburn, R. and Schneider, C. Gaussian hypergeometric series and supercongruences. Math. Comp. 78 (2009), 275292.
[36] Osburn, R. and Straub, A. Interpolated sequences and critical $L$-values of modular forms. Preprint (2018), arXiv:1806.05207
[37] Panzer, E. Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals. Comput. Phys. Commun. 188 (2015), 148166.
[38] Poulet, P. Permutations. L'Intermédiaire des Mathématiciens 26 (1919), 117121.
[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. 1751.
[40] Roberts, D.P., Rodriquez–Villegas, F. and Watkins, M. Hypergeometric motives. “Preprint” (2017).
[41] Schlotterer, O. and Stieberger, S. Motivic multiple zeta values and superstring amplitudes. J. Phys. A 46 (2013), 475401, 37 pp.
[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), 269304.
[43] Straub, A. Multivariate Apéry numbers and supercongruences of rational functions. Algebra Number Theory 8 (2014), 19852008.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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