Construction of half integral weight Siegel modular forms of Sp (2, R) from automorphic forms of the compact twist Sp (2).
Construction of half integral weight Siegel modular forms of Sp (2, R) from automorphic forms of the compact twist Sp (2).
复制标题
从紧扭曲 Sp (2) 的自同构形式构造 Sp (2, R) 的半积分权西格尔模形式。
DOI：
10.1515/crll.1985.359.188
复制
发表时间：
1985
期刊：
影响因子：
--
通讯作者：
T. Ibukiyama
中科院分区：
文献类型：
--
作者：
T. Ibukiyama
Sp（2，IR）（矩阵大小为4）的一般辛群Sp（2，IR）与其紧扭群Sp（2）= Sp（2，C）n U（4）（U（4）：大小为4的酉群）的关系。我们的主要观点是这个构造保持了L函数。众所周知，我们有5/？(2)/{±1} = 5（5），（SO（5），Sp（2，P））构成Howe [9]定义的一个对偶约化对，所以这种构造是自然期望的。实际上，我们可以省略Sp（2）而只给出SO（5）的公式，但我们没有这样做。例如，我们在Sp（2）上而不是在SO（5）上表述Hecke理论。这是因为我们有以下动机。Ihara [13]或Langlands [20]证明了Sp（n）与Sp（n，IR）的自守形式之间存在着某种良好的对应关系。当n = 1时，这是Eichler的经典定理。对于n = 2，已知这些形式之间的一些例子和一些良好的量纲关系（参见。[8]，[10]，[11]，[12]）。证明这种猜想的唯一方法似乎是迹公式。它至少在维度关系方面工作得很好（同上）。cit.）。但更直接的对应关系，如果存在的话，也会非常有趣。这里，不是从Sp（2）到5/7（2，P），
Sp(2,IR) of the usual symplectic group Sp(2,IR) (matrix size four) from those of its compact twist Sp(2) = Sp(2, C) n U(4) (U(4): the unitary group of size four). Our main point is that this construction preserves L functions. As well known, we have 5/?(2)/{±1} = 5 (5), and (SO(5), Sp(2, P)) form a dual reductive pair defmed by Howe [9], so such construction is naturally expected. Actually, one could omit Sp(2) and give a formulation only on SO (5), but we did not do so. For example, we formulate Hecke theory on Sp (2), and not on SO (5). This is because we have the following motivation. By Ihara [13] or Langlands [20], it has been conjectured that there should exist some good correspondence between automorphic forms of Sp (n) and Sp(n,IR). When n = l, this is Eichler's classical theorem. For « = 2, some examples and some good dimensional relations between these forms have been known (cf. [8], [10], [11], [12]). The only method to prove such conjecture seems to be the trace formula. It has worked well at least for dimensional relations (loc. cit.). But a more direct correspondence, if it exists, would be also very interesting. Here, instead of passing from Sp(2) to 5/7 (2, P),