Jacobi symbols on class groups of quadratic forms
Volume 219 / 2025
Acta Arithmetica 219 (2025), 223-247
MSC: Primary 11E16; Secondary 11R11, 11R29, 11R37
DOI: 10.4064/aa240620-11-3
Published online: 17 May 2025
Abstract
Let $H$ be the class group of binary quadratic forms of discriminant $d \lt 0$. For certain $d$, we construct an epimorphism $J \colon H^2\rightarrow \{\pm 1\}$, where $J$ is defined in terms of a Jacobi symbol. We apply $J$, for example, to extend a result of Muskat, Spearman, and Williams concerning values of Legendre symbols of the form $\big(\frac{a+b\sqrt {m}} p\big)$. Moreover, the map $J$ leads to many new evaluations of such symbols.
