General Real Multiplication Values Conjecture
Prove that for any real quadratic irrational \(\rho\) with non-square discriminant \(\Delta=b^2-4ac\), any \(\mathbf{r} \in \mathbb{Q}^2 \setminus \mathbb{Z}^2\), and any \(A \in \Gamma_\mathbf{r}\) fixing \(\rho\), the special value \(\shin^\mathbf{r}_{A}(\rho)\) is an algebraic unit in an abelian Galois extension of \(\mathbb{Q}(\sqrt{\Delta})\), and that for any Galois automorphism \(g\) with \(g(\sqrt{\Delta}) = -\sqrt{\Delta}\), one has \(|g(\shin^\mathbf{r}_{A}(\rho))| = 1\).
References
Conjecture [General Real Multiplication Values Conjecture] Let \rho \in \R such that a\rho2 + b\rho + c = 0 with a,b,c \in \Z and \Delta = b2-4ac is not a square. Let \r \in \Q2 \setminus \Z2 and A \in \Gamma_\r such that A \cdot \rho = \rho. Then: (1) \shin\r_{!A}(\rho) is an algebraic unit in an abelian Galois extension of \Q(\sqrt{\Delta}). (2) If g \in \Gal(\ol{\Q}/\Q) such that g(\sqrt{\Delta}) = -\sqrt{\Delta}, then \abs{g(\shin\r_{!A}(\rho))}=1.
We conjecture that these higher elliptic units are algebraic units which belong to prescribed abelian extensions of the base field where they are evaluated and that they satisfy a Kronecker limit formula which relates the logarithm of their modulus to values of derivatives of partial zeta functions at s = 0 in the base field.