Appears in collection : A Conference in Arithmetic Algebraic Geometry in Memory of Jan Nekovář

In this talk, we will discuss various irrationality and linear independence problems including the irrationality of 2-adic zeta value at 5. The proofs use an arithmetic holonomicity theorem, the special case of which was used in the proof of the unbounded denominators conjecture; arithmetic holonomicity theorems have also been studied in recent work of Bost and Charles. This is joint work in progress with Frank Calegari and Vesselin Dimitrov.