Number Theory Seminar

In this talk, we describe a representation of a ramified $p$-adic group $U_5$ which is compactly induced from a 1-dimensional representation of some open compact subgroup. We compute part of Harish-Chandra's expansion that locally expresses its character on $U_5$ as a linear combination of Fourier transforms of nilpotent orbital integrals on Lie $U_5$. The coefficients are in terms of the number of rational points on certain cover of a hyperelliptic curve over the residue field. We also describe roughly how similar representations form an $L$-packet, and how endoscopic transfer, which is an identity between characters of representations (within $L$-packets) of the group and its endoscopic groups, appear as geometric identities obtained via identifying the $H^1$ of different curves and their covers. This is a special case of a general algorithm that is jointly developed with Zhiwei Yun.