Adelic framed form class groups and explicit class field theory
Abstract
Let D be a negative discriminant, and let K=ℚ(√D). Let 𝒬(D) denote the set of primitive positive definite binary quadratic forms over ℤ of discriminant D. We introduce the set of adelic framed forms 𝒬̂(D)= {(Q, γ)∈ 𝒬(D)×SL2(ℤ̂) | Q(γ1 0)∈ ℤ̂×} and its orbit space Ĉ(D) under the natural action of SL2(ℤ). We define an explicit adelic analogue of the Gauss-Dirichlet composition law on Ĉ(D) and endow Ĉ(D) with the quotient topology induced by the subspace topology on 𝒬̂(D) inherited from the product topology on 𝒬(D)×SL2(ℤ̂), where 𝒬(D) is discrete and SL2(ℤ̂) has its profinite topology. We then prove that there is an isomorphism of topological groups Ĉ(D)≃Gal(Kab(t1/∞)/K(t)), where the Galois group is endowed with the Krull topology, t is a positive transcendental real number, and t1/∞={√[N]t | N≥1}. Moreover, we identify an explicitly defined subgroup of Ĉ(D) with Gal(Kab/K) and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of Ĉ(D) uniquely determines the imaginary quadratic field K.
Related
- Fermat curves and explicit class field theory for real quadratic number fields
- Explicit bounds for the number of solutions to Roth-type inequalities over adelic curves
- Construction of Galois modules describing ideal class groups and applications to Iwasawa theory
- The local structure of refined class groups in cyclotomic Iwasawa theory
- Multi-framed real monopole Floer theory
BibTeX
@misc{koo2026adelic,
title = {{Adelic framed form class groups and explicit class field theory}},
author = {Ja Kyung Koo and Dong Hwa Shin and Dong Sung Yoon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.04873},
eprint = {2608.04873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adelic framed form class groups and explicit class field theory
AU - Ja Kyung Koo
AU - Dong Hwa Shin
AU - Dong Sung Yoon
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.04873
UR - https://arxiv.org/abs/2608.04873
AB - Let D be a negative discriminant, and let K=ℚ(√D). Let 𝒬(D) denote the set of primitive positive definite binary quadratic forms over ℤ of discriminant D. We introduce the set of adelic framed forms 𝒬̂(D)= {(Q, γ)∈ 𝒬(D)×SL_2(ℤ̂) | Q(γ1 0)∈ ℤ̂^×} and its orbit space Ĉ(D) under the natural action of SL_2(ℤ). We define an explicit adelic analogue of the Gauss-Dirichlet composition law on Ĉ(D) and endow Ĉ(D) with the quotient topology induced by the subspace topology on 𝒬̂(D) inherited from the product topology on 𝒬(D)×SL_2(ℤ̂), where 𝒬(D) is discrete and SL_2(ℤ̂) has its profinite topology. We then prove that there is an isomorphism of topological groups Ĉ(D)≃Gal(K^ab(t^1/∞)/K(t)), where the Galois group is endowed with the Krull topology, t is a positive transcendental real number, and t^1/∞={√[N]t | N≥1}. Moreover, we identify an explicitly defined subgroup of Ĉ(D) with Gal(K^ab/K) and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of Ĉ(D) uniquely determines the imaginary quadratic field K.
ER -