Class invariants by Siegel resolvents and the modularity of their Galois traces

被引:0
作者
Jung, Ho Yun [1 ]
Lee, Yoonjin [2 ]
机构
[1] Dankook Univ, Dept Math, Cheonan Si 31116, Chungnam, South Korea
[2] Ewha Womans Univ, Dept Math, Seoul 03760, South Korea
基金
新加坡国家研究基金会;
关键词
Modular forms; Modular traces; Galois traces; Siegel functions; Class field theory;
D O I
10.1007/s11139-020-00382-8
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level.
引用
收藏
页码:135 / 152
页数:18
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