A LOCAL PROOF OF LANGLANDS'S SECOND MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS
Abstract
We give a local proof of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields. The lemma compares the induction expressions attached to compatible characters of distinct intermediate fields of prime degree in a bicyclic extension of prime-square degree. It is one of the main identities used in Langlands's construction of epsilon factors of local Weil representations. The proof builds on the work of Dwork, Lakkis, and Langlands. The main new point in the present paper is the wild dyadic case, i.e. biquadratic extensions over a nonarchimedean local field of residue characteristic two. This completes the proof of the Second Main Lemma, including the equal-characteristic case. Finally, together with the local proof of the First Main Lemma and Langlands's remaining reductions, we obtain a purely local construction of local epsilon factors envisioned by Langlands.
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BibTeX
@misc{ueda2026local,
title = {{A LOCAL PROOF OF LANGLANDS'S SECOND MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS}},
author = {Fukuhiro Ueda},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13960},
eprint = {2609.13960},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05749843v1},
}
RIS
TY - UNPB
TI - A LOCAL PROOF OF LANGLANDS'S SECOND MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS
AU - Fukuhiro Ueda
PY - 2026
DA - 2026/09/15
PB - HAL
DO - 10.48550/arxiv.2609.13960
UR - https://hal.science/hal-05749843v1
AB - We give a local proof of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields. The lemma compares the induction expressions attached to compatible characters of distinct intermediate fields of prime degree in a bicyclic extension of prime-square degree. It is one of the main identities used in Langlands's construction of epsilon factors of local Weil representations. The proof builds on the work of Dwork, Lakkis, and Langlands. The main new point in the present paper is the wild dyadic case, i.e. biquadratic extensions over a nonarchimedean local field of residue characteristic two. This completes the proof of the Second Main Lemma, including the equal-characteristic case. Finally, together with the local proof of the First Main Lemma and Langlands's remaining reductions, we obtain a purely local construction of local epsilon factors envisioned by Langlands.
ER -