A short proof for the relation between Weil indices and 𝜖-factors

ABSTRACT

Let F be a finite extension of ℚ_p and let ψ be a non-trivial character of F. For a∈F* let γ(a,ψ) be the normalized Weil index splitting the Hilbert symbol. In this short note we give a simple proof for the relation

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{\gamma (a,\psi )=𝜖(1∕2,{\eta }_a,\overline{\psi })}\end{array}}$

where η_a is the quadratic character of F* whose kernel is $N(F\left(\sqrt{a}\right))$ and where 𝜖(⋅,⋅,⋅) is the epsilon factor appearing in Tate’s thesis.

KEYWORDS:

• Epsilon factor
• Hilbert symbol
• local factors
• Weil index

2010 Mathematics Subject Classification:

• 11F27

Acknowledgments

We would like to thank Wee Teck Gan for providing useful information on the subject matter.