The arithmetic and topology of differential equations

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

“…is rationally proportional to L(f, 2)/π 2 , where L(f, s) denotes the L-function of the modular form. Furthermore, they prove [34] that…”

“…On the other hand, the Apéry sequence and the modular parametrization of its generating series ∞ n=0 A(n)z n gives one a natural way to construct the right-hand side of (3) (namely, the eigenform (2) whose Fourier coefficients show up) modulo p. This construction is performed in [3] and nicely explained in a certain generality in [33]. More recently, V. Golyshev and D. Zagier [34,Section 7] show that the p-adic interpolation of the coefficients a(p) of the newform f (τ ) = η(2τ ) 4 η(4τ ) 4 is part of a much more general picture that, in particular, predicts that…”

“…One of the principal results in [1] is a summation formula for Greene's hypergeometric function, which serves as a finite-field analogue of the classical hypergeometric series given in (4). Curiously enough, R. Evans in his review [7] of [1] mentions that no summation formula is known for this 4 F 3 -value in (4); the evaluation (4) established in [23,34] thus fills in this gap in the hypergeometric literature.…”

Picard–Fuchs equations for Shimura curves over Q

“…As t varies in t(U ), we choose a continuous family of paths γ t ∈ H 1 (A t , Z) and holomorphic differential forms ω t on the abelian surfaces A t varying algebraically in t. Let p(t) = γt ω t . The Picard-Fuchs equation is a differential equation Φ p in the differential operator D which has p(t) as a solution, we refer the reader to [13] for details. Theorem 4.1.…”

“…This is a hypergeometric differential equation, and its monodromy group is isomorphic to the congruence group Γ (2). Moreover, if we choose t = λ(z), the Legendre modular function, then it can be shown that this differential equation has ϑ 2 3 (z) as a solution, where ϑ 3 (z) is a particular theta function [13].…”