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 normalized one-dimensional local epsilon factors over every nonarchimedean local field, in both mixed and equal characteristic. The proof reconstructs and completes the local method initiated by Dwork and organized by Langlands; its decisive p-adic calculations and guiding ramification-theoretic and stationary-phase ideas are due to them. The present paper unifies their normalizations and supplies omitted endpoint and non-stable wild calculations. Exact 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 epsilon factors envisioned by Langlands, without global functional equations. Open AI ChatGPT assisted the drafting and English-language editing of this manuscript; the author audited the complete text, checked the proof against the completed Lean formalization, and assumes full responsibility.
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BibTeX
@misc{ueda2026local,
title = {{A LOCAL PROOF OF LANGLANDS'S FIRST MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS}},
author = {Fukuhiro Ueda},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2610.10583},
eprint = {2610.10583},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05735672v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A LOCAL PROOF OF LANGLANDS'S FIRST MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS
AU - Fukuhiro Ueda
PY - 2026
DA - 2026/09/02
PB - HAL
DO - 10.48550/arxiv.2610.10583
UR - https://hal.science/hal-05735672v1
AB - We give a complete local proof of Langlands’s First Main Lemma for normalized one-dimensional local epsilon factors over every nonarchimedean local field, in both mixed and equal characteristic. The proof reconstructs and completes the local method initiated by Dwork and organized by Langlands; its decisive p-adic calculations and guiding ramification-theoretic and stationary-phase ideas are due to them. The present paper unifies their normalizations and supplies omitted endpoint and non-stable wild calculations. Exact 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 epsilon factors envisioned by Langlands, without global functional equations. Open AI ChatGPT assisted the drafting and English-language editing of this manuscript; the author audited the complete text, checked the proof against the completed Lean formalization, and assumes full responsibility.
ER -