Seminario Teoría de Números
Local Kummer theory for Drinfeld modules
Ponente: Maxim Mornev (EPFL, Lausanne)Fecha: jueves 19 de septiembre de 2024 - 11:00Lugar: Aula 420, Módulo 17, Departamento de Matemáticas, UAM
Resumen:
This is joint work with Richard Pink.
Let \(\varphi\) be a Drinfeld \(A\)-module of finite residual characteristic \(\bar{\mathfrak{p}}\) over a local field \(K\). We study the action of the inertia group of \(K\) on a modified adelic Tate module \(T^\circ_{\textup{ad}}(\varphi) \) which differs from the usual adelic Tate module only at the \(\bar{\mathfrak{p}}\)-primary component. After replacing \(K\) by a finite extension we can assume that \(\varphi\) is the analytic quotient of a Drinfeld module \(\psi\) of good reduction by a lattice \(M\subset K\). The image of inertia acting on \(T^\circ_{\textup{ad}}(\varphi) \) is then naturally a subgroup of \(\operatorname{Hom}_A(M,\,T^\circ_{\textup{ad}}(\varphi)) \).
This subgroup is described by a canonical local Kummer pairing that is the main subject of our research. In particular we give an effective formula for the image of inertia up to finite index, and obtain a necessary and sufficient condition for this image to be open. We also determine the image of the ramification filtration.
The talk is aimed at a general audience of number theorists and arithmetic geometers. No preliminary knowledge of Drinfeld modules or Kummer theory will be assumed.
