Seminar
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| Location: | SLMath: Online/Virtual |
This seminar will focus on Diophantine problems in a broad sense, with a view towards (but not limited to) interactions between Number Theory and Logic. Particular attention will be given to topics with the potential of further developments in the context of this MSRI scientific program. This will provide an opportunity for researchers to update on new results, techniques and some of the main problems of the field.
To participate in this seminar, please register here: https://www.msri.org/seminars/25206
Nonexistence Of Exceptional Units Via Skolem-Chabauty's Method
To participate in this seminar, please register here: https://www.msri.org/seminars/25206
Abstract:
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.
No Notes/Supplements UploadedNonexistence Of Exceptional Units Via Skolem-Chabauty's Method
| H.264 Video | 25174_28672_8607_Nonexistence_of_Exceptional_Units_via_Skolem-Chabauty's_Method.mp4 |