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 epsilonfactors 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. The author has also completed a local proof of Langlands's Second Main Lemma. Together with the present result and Langlands's remaining reductions, this gives a complete proof of the local construction of local epsilon factors envisioned by Langlands.

arXiv commentCompanion to arXiv:2609.13960 and arXiv:2609.30301. The First Main Lemma proved here is used in the Second Main Lemma and its Lean formalization