Session S09 - Number Theory in the Americas

Thursday, July 22, 20:30 ~ 21:00 UTC-3

Values of the Ramanujan-tau function

Mike Bennett

UBC, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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.

Joint work with Adela Gherga (U. Warwick, UK), Vandita Patel (U. Manchester, UK) and Samir Siksek (U. Warwick, UK).

View abstract PDF