Number Theory Seminar

Speaker: Peter Vang Uttenthal

Title: Level-raising of even representations and spin conditions

Abstract: The arithmetic data in the Langlands program conjectured to parameterize Maass wave forms consist of even two-dimensional complex Galois representations. However, besides being much rarer than odd representations, we know almost nothing about the even parity. For instance, Ribet’s work on level-raising of odd representations relies on geometric and automorphic techniques that do not carry over to the even case.

Level-raising of even representations, on the other hand, can be carried out with techniques from global class field theory over totally real fields. From this perspective, level-raising can be reduced to spin conditions (splitting conditions on primes in number fields that depend on the prime). Spin conditions show up naturally in several areas of number theory and give rise to statistical questions beyond the scope of Chebotarev’s theorem. Instead, new analytic results on the equidistribution of spin symbols are opening up promising lines of work.