非阿基米德局部域上局部ε因子的Langlands第二主引理的一个局部证明

中文摘要 AI 辅助

我们给出了非阿基米德局部域上局部ε因子的Langlands第二主引理的一个局部证明，涵盖混合特征和等特征情形。该引理比较了双循环扩张中两个中间域的特征的局部常数。它是Langlands构造局部Weil表示的ε因子所用的恒等式之一。证明基于Dwork、Langlands和Lakkis的工作。Langlands将第二主引理归功于Dwork，但Dwork未发表完整证明。Lakkis给出了第二主引理的详细局部处理。对于奇素数次数，他的论证证明了所需的等式。然而，在次数为二的情形，他的论证仅证明等式在符号意义下成立。本文的新颖之处在于野二分支情形，即剩余特征为二的非阿基米德局部域上的双二次扩张。对于双二次扩张，第一主引理确定了所需等式的平方，留下可能的符号。我们通过比较Lamprecht公式中出现的有限和来确定该符号。这完成了第二主引理的证明，包括等特征情形。OpenAI ChatGPT在证明的开发中被广泛使用。论证的很大一部分由冗长的局部计算组成，这些计算原则上可通过标准方法完成，但需要大量时间。因此，ChatGPT被用来加速这一技术工作：检查计算并与Dwork、Langlands和Lakkis的论证进行比较，并检测中间版本证明中的不一致之处。作者检查了最终的数学论证并对结果负责。

英文摘要

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.

发表机构

• Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究科）

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