Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians

We compare three heights attached to a curve of genus two: the weighted height of its Igusa–Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron–Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron–Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\text{inv}}\le h_{\text{GIT}}+\frac{1}{2}\log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron–Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\times E$.