Approximations and complex multiplication according to Ramanujan

By D. Chudnovsky, G. Chudnovsky. 2000

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Polynomial evaluation and interpolation on special sets of points (2005, unknown): “…occurs during computations of matrix Pad(cid:19)e-type approxi-mants (Beckermann and Labahn, 1992, 1994; Thom(cid:19)e, 2001, 2002; Giorgi et al., 2003), recurrences with polynomial coe(cid:14)cients (Chudnovsky and Chudnovsky, 1988; Bostan et al., 2004) and linear di(cid:11)erential operators.”
Eisenstein series and Ramanujan-type series for 1/π (2010, background): “Chudnovsky [27] independently proved several of Ramanujan’s series representations for 1/π and established new ones as well.”
Open Conjectures on Congruences (2009, background): “p with ( p 43 ) = 1 there are unique positive integers x and y such that 4p = x2 + 43y2. The conjectured congruence modulo p has been conﬁrmed by Z. H. Sun [Su3]. D. V. Chudnovsky and G. V. Chunovsky [CC] showed that X∞ k=0 5418k + 263 (−960)3k 6k 3k 3k k,k,k = 640 √ 15 3π , and Zudilin [Zu] sugg”
Hypergeometric Functions over Finite Fields (2015, background): “Later in the 1980’s, Borwein-Borwein [19] and Chudnovsky-Chudnovsky [21] proved these formulas using essentially the theory of elliptic curves with complex multiplication.”