On representations of finite groups in the space of modular forms of half-integral weight
On representations of finite groups in the space of modular forms of half-integral weight
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半积分权模形式空间中有限群的表示
DOI：
10.3792/pjaa.70.198
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
Masaru Ueda
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文献类型：
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作者：
Masaru Ueda
导论.设p是素数，k是整数。在1940年，赫克研究了一个代表性的SL 2（Z/pZ），这是实现空间的模块形式的水平p和重量/c，并取得了美丽的结果。本文研究了在4p层权k + 1/2的尖点型空间中实现的一个类似表示rk+1/2。特别是，我们详细研究的子表示Ps生成的Hecke共同特征值f的水平4p（“新形式”）。这样我们就得到了与积分权情形完全不同的结果。例如，ps总是不可约的，如果f是Neben型的，那么p是“剩余的”还是“非剩余的”（参见：下面（1.2））由Atkin-Lehner对合W（p）确定（参见定理（4.1）的细节）。最后，我们注意到Q（v/p）的类数也出现在我们的结果中，就像Hecke的经典工作一样（参见：注（4.2））。0个字母。在本文中，我们始终保持[4]中的符号。特别地，我们使用以下通用符号。令/c表示正整数，p表示奇素数。如果z C和x C，我们将zxexp（x，log（z））与log（z）-log（lz[）+v/1 arg（z），arg（z）由7 r < arg（z）_< zr确定。我们也把e（z）exp（2 zr/I z）。（a B）/，o（4）和设gp为复上半平面。对于7 c d
Introduction. Let p be a prime and k an integer. In 1940, Hecke studied a representation of SL2(Z/pZ) which is realized in the space of modular forms of level p and of weight/c and obtained beautiful results. In this paper, we study a similar representation rk+l/2 which is realized in the space of cusp forms of level 4p and of weight k + 1/2. In particular, we study in detail the subrepresentation Ps generated by Hecke common eigenforrn f of level 4p ("newform"). Then we have some completely different facts from the results in the case of integral weight. For example, ps is always irreducible, and if f is of Neben-type, whether p is "residual" or "non-residual" (cf. below (1.2)) is determined by the Atkin-Lehner involution W(p) (cf. Theorem (4.1) for the details). Finally, we remark that the class number of Q(v/p) also occurs in our results as in the classical work of Hecke (cf. Remark(4.2)). 0 Preliminaries. Throughout this paper, we keep to the notation in [4]. In particular, we use the following general notation. Let /c denote a positive integer and p an odd prime number. If z C and x C, we put zxexp(x, log(z)) with log(z) --log(lz[) +v/1 arg(z), arg(z) being determined by 7r < arg(z) _< zr. Also we put e(z) exp (2zr/I z). (a b) /,o(4) and Let gp be the complex upper half plane. For 7 c d