If a is an odd positive integer, then a result of Murty, Murty and Shorey implies that there are at most finitely many positive integers n for which tau(n)=a, where tau(n) is the Ramanujan tau-function. In this talk, I will discuss non-archimidean analogues of this result and show how the machinery of Frey curves and their associated Galois representations can be employed to make such results explicit, at least in certain situations. Much of what I will discuss generalizes readily to the more general situation of coefficients of cuspidal newforms of weight at least 4, under natural arithmetic conditions. This is joint work with Adela Gherga, Vandita Patel and Samir Siksek.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/Fields-2020.pdf