NON-ABELIAN BASE CHANGE FOR TOTALLY REAL FIELDS
NON-ABELIAN BASE CHANGE FOR TOTALLY REAL FIELDS
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完全实数域的非阿贝尔基数变化
DOI：
10.2140/pjm.1997.181.189
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发表时间：
1997
影响因子：
0.6
通讯作者：
Y. Maeda
中科院分区：
文献类型：
--
作者：
H. Hida;Y. Maeda
设F是一个数域。F的Adele环FA的GL 2（FA）上的上同调Hecke本征尖点形式f称为f的基变换，如果L（s，f）= L（s，ρF），其中ρF = ρ Gal（Q/F）.当f存在时，我们称f为“可提升到F”或“F -可提升”。当F是全真实的时，Hilbert尖点型是上同调的当且仅当它在F的每一个阿基米德位置上是权k ≥ 2的全纯的.由于基变式f的定义是用纯代数的形式给出的，所以在用解析方法刻画的上同调形式空间中求基变式f是一个非平凡的问题。这个问题首先由Doi和Naganuma [DN]和[N]对二次扩张F/Q进行了研究，后来Langlands [L]证明了f的存在性对于相对素循环的情况（因此最终覆盖了所有可解的情况）。
Let F be a number field. A cohomological Hecke eigen cusp form f on GL2(FA) for the adele ring FA of F is called a base change of f if L(s, f) = L(s, ρF ) for ρF = ρ ∣∣ Gal(Q/F ). When f exists, we call f “liftable to F” or “F -liftable”. When F is totally real, a Hilbert cusp form is cohomological if and only if it is holomorphic of weight k ≥ 2 at every archimedean place of F . Since the definition of the base change f is given in purely algebraic terms, it is a non-trivial problem to find the base-change form f in the space of cohomological forms characterized by analytic means. This problem was first studied by Doi and Naganuma [DN] and [N] for quadratic extensions F/Q, and the existence of f was later proved by Langlands [L] for relative prime cyclic cases (so eventually covering all solvable cases).