A Local Proof of Langlands's First Main Lemma for Local Epsilon Factors over Nonarchimedean Local Fields

Abstract

We give a complete local proof of Langlands's First Main Lemma for local epsilon factors over nonarchimedean local fields. The proof builds on Dwork's $p$-adic calculations and Langlands's formulation of the local method. The present paper reformulates these calculations in a common normalization and extends the argument to equal characteristic. A more systematic treatment of stationary parameters and conductor cases is intended to make the calculations easier to follow. Norm--trace correction formulas yield the equal-characteristic extension. The author is also completing a local proof of Langlands's Second Main Lemma and will verify it in Lean. Together with the present result and Langlands's remaining reductions, that work will give a complete proof of the local construction of local epsilon factors envisioned by Langlands. OpenAI ChatGPT assisted with drafting and wording revisions of this manuscript; the author audited the manuscript, checked the proof, and assumes full responsibility.