Apry extensions

The Ap & eacute;ry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau-Ginzburg (LG) models - and thus, in particular, as periods. We also construct an Ap & eacute;ry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard- Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the "elementary" Ap & eacute;ry numbers in terms of regulators of higher cycles (i.e., algebraic K-theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of K3 surfaces, and the distinction between multiples of zeta(2) and zeta(3) (or (2 pi i)(3)) translates ultimately into one between algebraic K-1 and K-3 of the family.