Vesselin Dimitrov: The next case after Apéry on mixed Tate periods (NTWS 202)

Abstract: I will introduce a method, joint with Frank Calegari and Yunqing Tang, for proving linear independence results and effective bad approximability measures. It is an outgrowth of our previous joint work on the so-called "unbounded denominators conjecture," which was in some sense an application of transcendental number theory to modular forms theory, with the key step being to prove sufficiently sharp $\mathbb{Q}(x)$-linear dimension bounds on certain spaces of algebraic functions. This time, we step into the wilder realm of G-functions with infinite monodromy, and devise holonomy bounds fine enough to prove the linear independence of two certain Dirichlet L-function values, a result that, in the realm of mixed Tate periods, can be considered as the next-simplest case after Apery's proof of the irrationality of $\zeta(3)$ (excluding the cases that reduce to the Hermite--Lindemann theorem or the Gelfond--Baker theorem on linear forms in logarithms). One key input turns out to be the classical Shidlovsky lemma on functional bad approximability, the point Siegel missed for three decades to complete his theory of algebraic relations among special values of E-functions.
This is all a joint work with Frank Calegari and Yunqing Tang.

Original air date:
Thursday, March 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 8, 2024 (12am CST, 3am AEDT, 5am NZDT)