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Shimura Reciprocity for Real Quadratic Fields
Abstract
The purpose of this note is to give conditional explicit formulas for the Galois action on real multiplication values of the Shintani--Faddeev modular cocycle. We assume the order 1 abelian Stark conjecture for real quadratic fields. We conclude this note by presenting our own conjecture about the Galois action. Numerical evidence for this conjecture will be provided in forthcoming work.
Related
- On ℤ2-extensions of real quadratic fields
- An additive problem in connection with some quadratic real fields with class number one
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
- Fermat curves and explicit class field theory for real quadratic number fields
- Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
BibTeX
@misc{dicapua2026shimura,
title = {{Shimura Reciprocity for Real Quadratic Fields}},
author = {Joseph DiCapua},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02419},
eprint = {2610.02419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02419},
}
RIS
TY - UNPB
TI - Shimura Reciprocity for Real Quadratic Fields
AU - Joseph DiCapua
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02419
UR - https://arxiv.org/abs/2610.02419
AB - The purpose of this note is to give conditional explicit formulas for the Galois action on real multiplication values of the Shintani--Faddeev modular cocycle. We assume the order 1 abelian Stark conjecture for real quadratic fields. We conclude this note by presenting our own conjecture about the Galois action. Numerical evidence for this conjecture will be provided in forthcoming work.
ER -