On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura's reciprocity law

Composition law and complex multiplication

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let Q(N) (d(K)) be the set of primitive positive definite binary quadratic forms of discriminant d(K) whose leading coefficients are relatively prime to N. We adopt an equivalence relation similar to(n) on Q(N) (dK) so that the set of equivalence classes Q(N)(d(K))/ similar to(n) can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of Q(N)(d(K))/ similar to(n) onto Gal(K-n/K) in terms of Fricke invariants, where K-n denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. (C) 2019 Elsevier Inc. All rights reserved

Modularity of Galois traces of ray class invariants

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level N≥ 2 and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision). © 2020, Springer Science+Business Media, LLC, part of Springer Nature

지겔 함수에 의한 복소이차체 상의 방사유체 불변생성자와 정규기저

학위논문(박사) - 한국과학기술원 : 수리과학과, 2011.8, [ iii, 57 p. ]한국과학기술원 : 수리과학과