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 modularity. 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 becomes 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.
Bibliographical noteFunding Information:
H. Y. Jung was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01073055) and by the research fund of Dankook university in 2020, and Y. Lee is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574) and also by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177).
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- Class field theory
- Galois traces
- Modular forms
- Modular traces
- Siegel functions