Collapsing Fifty Dilogarithm Arguments to Five Terms over $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$

Abstract

We give a one-parameter functional equation for the real Rogers dilogarithm, with arguments in $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$, and prove it by an explicit array of ten instances of Rogers' five-term relation: the fifty arguments so contributed cancel down to the five of the identity, which admit no shorter relation among themselves. The equation comes from a pair of integrals whose equality is elementary, and we show that the underlying integrand is essentially forced. Specialising the parameter gives identities over $\mathbb{Q}(\sqrt{33})$ and $\mathbb{Q}(\sqrt{17})$, a relation between $\mathbb{Q}(\sqrt{13})$, $\mathbb{Q}(\sqrt{3})$, and $\operatorname{Cl}_2(π/3)$ with a new analogue for Catalan's constant, and a pair of dilogarithm ladders of quartic base over $\mathbb{Q}(\sqrt{33})$. The base equations of these ladders, and of four conjectural ones located by integer-relation search, are irreducible quartics with four real roots. All previously recorded ladders of degree four known to us have base equations with two real roots, so these appear to be the first totally real ladders of degree exceeding three.