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, in mixed and equal characteristic. The lemma compares local constants of characters of two intermediate fields in a bicyclic extension. It is one of the identities used in Langlands's construction of epsilon factors of local Weil representations. The proof builds on the work of Dwork, Langlands, and Lakkis. Langlands attributes the Second Main Lemma to Dwork, but Dwork did not publish a complete proof. Lakkis gave a detailed local treatment of the Second Main Lemma. For odd prime degree, his argument proves the required equality. In degree two, however, it proves only that the equality holds up to sign. The new point in the present paper is the wild dyadic case, i.e. biquadratic extensions over a nonarchimedean local field of residue characteristic two. For a biquadratic extension the First Main Lemma determines the square of the required equality, leaving a possible sign. We determine this sign by comparing the finite sums occurring in Lamprecht's formula. This completes the proof of the Second Main Lemma, including the equal-characteristic case. OpenAI ChatGPT was used extensively in the development of this proof. A substantial part of the argument consists of long local calculations, which are in principle accessible by standard methods, but would have required a very large amount of time. ChatGPT was therefore used to accelerate this technical work: to check calculations and compare them with the arguments of Dwork, Langlands, and Lakkis, and detect inconsistencies in intermediate versions of the proof. The author checked the final mathematical arguments and assumes responsibility for the results.