Résume | Consider the hypergeometric series F(t):=F(1/2,1/2,1|t). For any positive integer m we define F_m(t) as the truncation at tm, i.e we drop all terms in F(t) of degree ≥ m. Let p be an odd prime and z_0 a p-adic integer≠0,1. Then Dwork found that if F_p(z_0) is a unit in ℤ_p, the quotient F_{p^s}(z_0)/F_{p^{s−1}}(z0) converges p-adically to (−1)(p−1)/2 times the zero of the ζ-function of the elliptic curve
y2≡x(x−1)(x−z0) mod p
with p-adic valuation 1. There exist many far reaching generalizations. In two recent papers, Dwork-crystals I,II (arXiv:1903.11155, arXiv:1907.10390) Masha Vlasenko and I have developed an elementary framework which explains many of these phenomena. In this lectures I would like to present some of the ideas. |