p-adic Eisenstein series, arithmetic holonomicity criteria, and irrationality of the 2-adic $\zeta(5)$

Vesselin Dimitrov (Toronto)

12-Jun-2020, 15:00-16:00 (4 years ago)

Abstract: In this exposition of a joint work in progress with Frank Calegari and Yunqing Tang, I will explain a new arithmetic criterion for a formal function to be holonomic, and how it revives an approach to the arithmetic nature of special values of L-functions. The new consequence to be proved in this talk is the irrationality of the 2-adic version of $\zeta(5)$ (of Kubota-Leopoldt). But I will also draw a parallel to a work of Zudilin, and try to leave some additional open ends where the holonomicity theorem could be useful. The ingredients of the irrationality proof are Calegari's p-adic counterpart of the Apery-Beukers method, which is based on the theory of overconvergent p-adic modular forms (IMRN, 2005) taking its key input from Buzzard's theorem on p-adic analytic continuation (JAMS, 2002), and a Diophantine approximation method of Andre enhanced to a power of the modular curve $X_0(2)$. The overall argument, as we shall discuss, turns out to bear a surprising affinity to a recent solution of the Schinzel-Zassenhaus conjecture on the orbits of Galois around the unit circle.

number theory

Audience: researchers in the topic


London number theory seminar

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