Abstract:
It is well-known that in the Apery's proof of irrationality of $\zeta(3)$ a
key role is played by some linear recurrence with polynomial coefficients
admitting a solution in positive integers. The Apery's proof has a reputation
of somewhat mysterious, for this recurrence appears in it without any
motivation at all, and the proof itself cannot be easily generalised.
Following Golyshev, we will discuss how the Apery recurrence arises from the
geometry of the Fano threefold $V_12$. We will also introduce the Apery
constants for any smooth Fano variety and discuss their connections with
mirror symmetry.
