Fundamental Real Multiplication Values Conjecture
Determine whether, for a real quadratic irrational ρ with fundamental discriminant Δ fixed by A ∈ Γ_ρ, the Shintani–Faddeev modular cocycle value shin^ρ_A(ρ) is an algebraic unit in an abelian extension of Q(√Δ) and has unit absolute value under conjugation switching √Δ to −√Δ.
References
\begin{conj}[Fundamental Real Multiplication Values Conjecture]\label{conj:frmvc} Let $\qrt \in \R$ such that $a\qrt2 + b\qrt + c = 0$ with $a,b,c \in \Z$ and $\Delta = b2-4ac$ is a fundamental discriminant. Let $\r \in \Q2 \setminus \Z2$ and $A \in \Gamma_\r$ such that $A \cdot \qrt = \qrt$. Then: \begin{itemize} \item[(1)] $\sfc{\r}{!A}{\qrt}$ is an algebraic unit in an abelian Galois extension of $\Q(\sqrt{\Delta})$. \item[(2)] If $g \in \Gal(\ol{\Q}/\Q)$ such that $g(\sqrt{\Delta}) = -\sqrt{\Delta}$, then $\abs{g(\sfc{\r}{!A}{\qrt})}=1$. \end{itemize} \end{conj}