A note on $L$-packets and abelian varieties over local fields

Jeffrey D. Achter and Clifton Cunningham

Vol. 273 (2015), No. 2, 395–412

DOI: 10.2140/pjm.2015.273.395

Abstract

A polarized abelian variety $\left(X,\lambda \right)$ of dimension $g$ and good reduction over a local field $K$ determines an admissible representation of ${GSpin}_{2g+1}\left(K\right)$. We show that the restriction of this representation to ${Spin}_{2g+1}\left(K\right)$ is reducible if and only if $X$ is isogenous to its twist by the quadratic unramified extension of $K$. When $g=1$ and $K={\mathbb{Q}}_p$, we recover the well-known fact that the admissible ${GL}_2\left(K\right)$-representation attached to an elliptic curve $E$ with good reduction is reducible upon restriction to ${SL}_2\left(K\right)$ if and only if $E$ has supersingular reduction.

Keywords

abelian varieties, good reduction, local fields, $L$-packets, admissible representations

Mathematical Subject Classification 2010

Primary: 22E50, 11G10

Secondary: 11F70, 14K15, 11S37

Milestones

Received: 21 February 2014

Revised: 25 July 2014

Accepted: 22 August 2014

Published: 23 December 2014