A Supercongruence Motivated by the Legendre Family of Elliptic Curves
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 620-624.

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A new supercongruence associated with a Gaussian hypergeometric series, as well as one of Mortenson's supercongruences, are established with new congruence relations and the Legendre transforms of certain sequences.
Keywords: elliptic curve, ramified double cover, finite field, supercongruence
Mots-clés : Hasse invariant, Legendre transform.
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Heng Huat Chan; Ling Long; W. Zudilin. A Supercongruence Motivated by the Legendre Family of Elliptic Curves. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 620-624. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a12/

[1] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math., 106, Springer-Verlag, New York, 1986 | MR | Zbl

[2] C. H. Clemens, A Scrapbook of Complex Curve Theory, Univ. Ser. Math., Plenum Press, New York, 1980 | MR | Zbl

[3] R. Osburn, C. Schneider, “Gaussian hypergeometric series and supercongruences”, Math. Comp., 78:265 (2009), 275–292 | MR

[4] W. Zudilin, “Ramanujan-type supercongruences”, J. Number Theory, 129:8 (2009), 1848–1857 | DOI | MR | Zbl

[5] S. Ahlgren, K. Ono, “A Gaussian hypergeometric series evaluation and Apéry number congruences”, J. Reine Angew. Math., 518 (2000), 187–212 | MR | Zbl

[6] D. McCarthy, R. Osburn, “A $p$-adic analogue of a formula of Ramanujan”, Arch. Math. (Basel), 91:6 (2008), 492–504 | MR | Zbl

[7] E. Mortenson, “A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function”, J. Number Theory, 99:1 (2003), 139–147 | DOI | MR | Zbl

[8] E. Mortenson, “Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs”, Trans. Amer. Math. Soc., 355:3 (2003), 987–1007 | DOI | MR | Zbl