Values of the Ramanujan tau-function

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).