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Gamma functions, monodromy and Apéry constants

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KAH - K-theory, algebraic cycles and motivic homotopy theory

In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce a sequence of Apéry constants associated to an ordinary linear differential operator with a choice of two singular points and a path between them. Their setting also involves assumptions on the local monodromies around the two points (maximally unipotent and reflection point respectively). In particular, these assumptions are satisfied in the situation of Apéry's proof of irrationality of zeta(3), and in this case Golyshev and Zagier discover that (numerically, with high precision) the higher constants in the sequence seem to be rational linear combinations of weighted products of zeta and multiple zeta values. In the joint work with Spencer Bloch we show that, quite generally, the generating series of Apéry constants is a Mellin transform of a solution of the adjoint differential operator. This peculiar property explains why Apéry constants of geometric differential operators are periods, which seems to be the first step in the study of their motivic nature.




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