The twisted endoscopy of \(GL(4)\) and \(GL(5)\): Transfer of Shalika germs
Let \(F\) be a \(p\)-adic field of characteristic zero and of odd residual characteristic. Over the field \(F\) the author obtains a necessary statement about a matching of Shalika germs at the identity for the groups \(GL(4)\) and \(GL(5)\): each stable Shalika germ on \({\mathfrak s} o(5)\) is a linear combination of Shalika germs on \({\mathfrak s} p(4)\), and, conversely, each stable Shalika germ on \({\mathfrak s} p(4)\) is a linear combination of Shalika germs on \({\mathfrak s} o(5)\). This matching of stable Shalika germs at the identity implies the transfer of twisted stable Shalika germs from \(GL(4)\) and \(GL(5)\) to the endoscopic groups \(SO(5)\) and \(Sp(4)\), and, coupled with an expected theory of descent, it would imply the transfer of twisted orbital integrals on the groups to stable orbital integrals on the endoscopic groups. In order to obtain the matching the author finds that the stable Shalika germs of the twisted orbital integrals on \(GL(4)\) and \(GL(5)\) and the stable orbital integrals on \(SO(5)\) and \(Sp(4)\) are expressed by the number of points on a family of elliptic curves, and then he obtains the relations between the families of elliptic curves, that implies the matching. These calculations give a complete formula for the Shalika germs on \(Sp(4)\), especially, the analysis of two-regular conjugacy class of \(Sp(4)\) gives a proof of the transfer of orbital integrals on \(Sp(4)\) to its endoscopic groups.
scientific article; zbMATH DE number 727837
scientific article