An exceptional (S-)unit is a unit x in a ring of (S-)integers of a number field K such that 1-x is also an (S-)unit. For fixed K and S, the set of exceptional S-units is finite by work of Siegel from the early 1900s. In the hundred years since, exceptional S-units have found wide-ranging applications, including to enumerating elliptic curves with good reduction outside a fixed set of primes and to proving "asymptotic" versions of Fermat's last theorem.
In this talk, we give an elementary p-adic proof of a new nonexistence result on exceptional units: there are no exceptional units in number fields of degree prime to 3 where 3 splits completely. We will also explain the geometric inspiration for the proof -- a version of Skolem-Chabauty's method for finding integral points on curves. Time permitting, we will discuss an application to periodic points of odd order in arithmetic dynamics.