no. 5
A modular supercongruence for 6 F 5 : An Apéry-like story
[Une supercongruence modulaire pour 6 F 5  : un conte à la Apéry]
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1987-2004.

On démontre une supercongruence modulo p 3 entre le p-ième coefficient de Fourier d’une forme modulaire de poids 6 et une série hypergéométrique 6 F 5 tronquée. Les nouveaux ingrédients de la preuve sont la comparaison de deux approximations rationnelles de ζ(3) pour produire des identités non triviales entre sommes harmoniques, et la réduction des congruences qui en résultent entre des sommes via une congruence qui relie les nombres d’Apéry á une autre suite du type de celle d’Apéry.

We prove a supercongruence modulo p 3 between the pth Fourier coefficient of a weight 6 modular form and a truncated 6 F 5 -hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to ζ(3) to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence relating the Apéry numbers to another Apéry-like sequence.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3201
Classification : 11B65,  33C20,  33F10
Mots clés : supercongruence, nombres d’Apéry, nombres de type Apéry, fonction hypergéométrique 
@article{AIF_2018__68_5_1987_0,
     author = {Osburn, Robert and Straub, Armin and Zudilin, Wadim},
     title = {A modular supercongruence for $_6F_5$: {An} {Ap\'ery-like~story}},
     journal = {Annales de l'Institut Fourier},
     pages = {1987--2004},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     doi = {10.5802/aif.3201},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3201/}
}
TY  - JOUR
AU  - Osburn, Robert
AU  - Straub, Armin
AU  - Zudilin, Wadim
TI  - A modular supercongruence for $_6F_5$: An Apéry-like story
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 2018///
SP  - 1987
EP  - 2004
VL  - 68
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3201/
UR  - https://doi.org/10.5802/aif.3201
DO  - 10.5802/aif.3201
LA  - en
ID  - AIF_2018__68_5_1987_0
ER  - 
Osburn, Robert; Straub, Armin; Zudilin, Wadim. A modular supercongruence for $_6F_5$: An Apéry-like story. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1987-2004. doi : 10.5802/aif.3201. https://aif.centre-mersenne.org/articles/10.5802/aif.3201/

[1] Ahlgren, Scott; Ono, Ken A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., Tome 518 (2000), pp. 187-212 | Article | MR 1739404 | Zbl 0940.33002

[2] Apéry, Roger Irrationalité de ζ(2) et ζ(3) (Astérisque) Tome 61, Société Mathématique de France, 1979, pp. 11-13 | Zbl 0401.10049

[3] Bailey, Wilfrid Norman Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Tome 32, Stechert-Hafner, 1964, v+108 pages | MR 0185155

[4] Beukers, Frits Another congruence for the Apéry numbers, J. Number Theory, Tome 25 (1987) no. 2, pp. 201-210 | Article | MR 873877 | Zbl 0614.10011

[5] Beukers, Frits Irrationality proofs using modular forms, Journées arithmétiques de Besançon (Besançon, 1985) (Astérisque) Tome 147-148, Société Mathématique de France, 1987, pp. 271-283 | MR 891433 | Zbl 0613.10031

[6] Chu, Wenchang; De Donno, Livia Hypergeometric series and harmonic number identities, Adv. Appl. Math., Tome 34 (2005) no. 1, pp. 123-137 | Article | MR 2102278 | Zbl 1062.05017

[7] Cooper, Shaun Sporadic sequences, modular forms and new series for 1/π, Ramanujan J., Tome 29 (2012) no. 1-3, pp. 163-183 | Article | MR 2994096 | Zbl 1336.11031

[8] Frechette, Sharon; Ono, Ken; Papanikolas, Matthew Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., Tome 2004 (2004) no. 60, pp. 3233-3262 | Article | MR 2096220 | Zbl 1088.11029

[9] Fuselier, Jenny G.; Long, Ling; Ramakrishna, Ravi; Swisher, Holly; Tu, Fang-Ting Hypergeometric functions over finite fields (2015) (http://arxiv.org/abs/1510.02575)

[10] Fuselier, Jenny G.; McCarthy, Dermot Hypergeometric type identities in the p-adic setting and modular forms, Proc. Am. Math. Soc., Tome 144 (2016) no. 4, pp. 1493-1508 | Article | MR 3451227 | Zbl 06549217

[11] Greene, John Hypergeometric functions over finite fields, Trans. Am. Math. Soc., Tome 301 (1987) no. 1, pp. 77-101 | Article | MR 879564 | Zbl 0629.12017

[12] van Hamme, L. Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 1996) (Lecture Notes in Pure and Appl. Math.) Tome 192, Dekker, 1997, pp. 223-236 | MR 1459212 | Zbl 0895.11051

[13] Kibelbek, Jonas; Long, Ling; Moss, Kevin; Sheller, Benjamin; Yuan, Hao Supercongruences and complex multiplication, J. Number Theory, Tome 164 (2016), pp. 166-178 | Article | MR 3474383 | Zbl 1334.33020

[14] Kilbourn, Timothy An extension of the Apéry number supercongruence, Acta Arith., Tome 123 (2006) no. 4, pp. 335-348 | Article | MR 2262248 | Zbl 1170.11008

[15] Krattenthaler, Christian; Rivoal, Tanguy Hypergéométrie et fonction zêta de Riemann, Mem. Am. Math. Soc., Tome 186 (2007) no. 875, x+87 pages | Article | MR 2295224 | Zbl 1113.11039

[16] McCarthy, Dermot Binomial coefficient-harmonic sum identities associated to supercongruences, Integers, Tome 11 (2011), A37 (Art A37, 8 p.) | Article | MR 2798613 | Zbl 1234.05039

[17] McCarthy, Dermot Extending Gaussian hypergeometric series to the p-adic setting, Int. J. Number Theory, Tome 8 (2012) no. 7, pp. 1581-1612 | Article | MR 2968943 | Zbl 1253.33024

[18] McCarthy, Dermot On a supercongruence conjecture of Rodriguez-Villegas, Proc. Am. Math. Soc., Tome 140 (2012) no. 7, pp. 2241-2254 | Article | MR 2898688 | Zbl 1354.11030

[19] Nesterenko Some remarks on ζ(3), Mat. Zametki, Tome 59 (1996) no. 6, pp. 865-880 | Article | MR 1445472 | Zbl 0888.11028

[20] Osburn, Robert; Schneider, Carsten Gaussian hypergeometric series and supercongruences, Math. Comput., Tome 78 (2009) no. 265, pp. 275-292 | Article | MR 2448707 | Zbl 1209.11049

[21] Osburn, Robert; Zudilin, Wadim On the (K.2) supercongruence of Van Hamme, J. Math. Anal. Appl., Tome 433 (2016) no. 1, pp. 706-711 | Article | MR 3388817 | Zbl 06485394

[22] Paule, Peter; Schneider, Carsten Computer proofs of a new family of harmonic number identities, Adv. Appl. Math., Tome 31 (2003) no. 2, pp. 359-378 | Article | MR 2001619 | Zbl 1039.11007

[23] Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron A=B, Peters, 1996, xii+212 pages (With a foreword by Donald E. Knuth, With a separately available computer disk) | MR 1379802 | Zbl 0848.05002

[24] van der Poorten, Alfred A proof that Euler missed: Apéry’s proof of the irrationality of ζ(3), Math. Intell., Tome 1 (1979) no. 4, pp. 195-203 | Article | MR 547748 | Zbl 0409.10028

[25] Rivoal, Tanguy Propriétés diophantinnes des valeurs de la fonction zêta de Riemann aux entiers impairs (2001) (Ph. D. Thesis)

[26] Rodriguez-Villegas, Fernando Hypergeometric families of Calabi–Yau manifolds, Calabi–Yau varieties and mirror symmetry (Toronto, ON, 2001) (Fields Inst. Commun.) Tome 38, American Mathematical Society, 2003, pp. 223-231 | MR 2019156 | Zbl 1062.11038

[27] Schneider, Carsten Symbolic summation assists combinatorics, Sémin. Lothar. Comb., Tome 56 (2007), B56b http://www.mat.univie.ac.at/slc/wpapers/s56schneider.html (Art. B56b, 36 p.) | Zbl 1188.05001

[28] Sloane, Neil J. A. The On-Line Encyclopedia of Integer Sequences, 2017 (published electronically at http://oeis.org)

[29] Swisher, Holly On the supercongruence conjectures of van Hamme, Res. Math. Sci., Tome 2 (2015), 18 (Art. 18, 21 p.) | Article | MR 3411813 | Zbl 1337.33005

[30] Zagier, Don Integral solutions of Apéry-like recurrence equations, Groups and symmetries (CRM Proc. Lecture Notes) Tome 47, American Mathematical Society, 2009, pp. 349-366 | MR 2500571 | Zbl 1244.11042

[31] Zudilin, Wadim Apéry’s theorem. Thirty years after, Int. J. Math. Comput. Sci., Tome 4 (2009) no. 1, pp. 9-19 | MR 2598496 | Zbl 1223.11089

[32] Zudilin, Wadim A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., Tome 89 (2014) no. 1, pp. 125-131 | Article | MR 3163010 | Zbl 1334.33022

[33] Zudilin, Wadim Hypergeometric heritage of W. N. Bailey. With an appendix: Bailey’s letters to F. Dyson (2016) (http://arxiv.org/abs/1611.08806)

Cité par Sources :