Double covers of tori for p=2 and genuine functions
Abstract
In his work on L-embeddings and double covers of tori over local fields, Kaletha associates to a maximal torus S in a connected reductive group a canonical double cover S(F)± and introduces a genuine sign function aS associated to a genuine character of S(F)±. We give two constructions of this genuine sign function over a non-archimedean local field, including residual characteristic $2$. The first uses rootwise lifts to the double cover; the second, in characteristic zero, uses the exponential map. Under explicit ramification conditions, both constructions yield genuine sign functions which, in odd residual characteristic, agree with his construction. We also illustrate the case of PGL2. A brief announcement of the first construction appears in Appendix D of that work.
Related
- On L -embeddings and double covers of tori over local fields
- Higher Schwarzians of Elliptic Double Covers and Eisenstein--Kronecker Functions
- Oka complements in algebraic tori and their logarithmic covers
- Pseudo-spherical representations and types for covers of split tori
- Abelian surfaces of small conductor from genus 3 double covers
BibTeX
@misc{gratien2026double,
title = {{Double covers of tori for p=2 and genuine functions}},
author = {Léo Gratien and Nhat Hoang Le and Kazuki Tokimoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22171},
eprint = {2610.22171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22171},
}
RIS
TY - UNPB
TI - Double covers of tori for p=2 and genuine functions
AU - Léo Gratien
AU - Nhat Hoang Le
AU - Kazuki Tokimoto
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22171
UR - https://arxiv.org/abs/2610.22171
AB - In his work on L-embeddings and double covers of tori over local fields, Kaletha associates to a maximal torus S in a connected reductive group a canonical double cover S(F)_± and introduces a genuine sign function a_S associated to a genuine character of S(F)_±. We give two constructions of this genuine sign function over a non-archimedean local field, including residual characteristic $2$. The first uses rootwise lifts to the double cover; the second, in characteristic zero, uses the exponential map. Under explicit ramification conditions, both constructions yield genuine sign functions which, in odd residual characteristic, agree with his construction. We also illustrate the case of PGL_2. A brief announcement of the first construction appears in Appendix D of that work.
ER -