AI 中文总结
本文研究数域二次扩张上酉群的剩余艾森斯坦上同调，识别艾森斯坦级数的上同调相关极点并证明残数留存，以F=Q(∛2)上秩3酉群为例显式构造非零类，方法可推广。
AI 中文摘要
我们研究任意数域二次扩张E/F对应的任意酉群U(V)的剩余艾森斯坦上同调，重点关注U(V)的极大抛物F-子群的贡献，我们识别了艾森斯坦级数上与上同调相关的极点，并证明由此得到的所有残数在显式次数下均作为自守上同调中的非平凡类留存。为说明涉及的现象范围，我们详细研究了F=Q(∛2)上F-秩为3的酉群的情形，对该酉群，我们显式构造了尖点自守表示，其满足主定理的所有假设，因此为该酉群显式构造了非零的剩余艾森斯坦上同调类。不过，该构造所用方法具有范式性，即通过基变换可推广到其他基域F上的其他酉群。
英文摘要
We investigate the residual Eisenstein cohomology of an arbitrary unitary group $U(V)$ attached to an arbitrary quadratic extension of number fields $E/F$. Our focus lies on the contribution of the maximal parabolic $F$-subgroups of $U(V)$, for which we identify the cohomologically relevant poles of Eisenstein series and prove that the resulting residues all survive as non-trivial classes in automorphic cohomology in an explicit degree. To illustrate the range of phenomena involved, we study in detail the case of a unitary group over $F=\mathbb{Q}(\sqrt[3]{2})$ of $F$-rank $3$, for which we explicitly construct cuspidal automorphic representations, which satisfy all the assumptions of our main theorem and hence explicitly construct non-zero residual Eisenstein cohomology classes for this unitary group. The methods used in this construction are paradigmatic however, i.e., generalize to other unitary groups over other ground fields $F$ by the use of base change.
CommentsAdded a reference to [AGIKMS26], to make our use of base change more self-contained, and a reference to two missing funding agencies
