Joint PU/IAS Number Theory

We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical “quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is generically non-torsion, we introduce a height-theoretic notion of bigness for families of curves in the moduli space of genus four curves, and give a criterion, via dimensions of modular quotients, for when it holds. We then exhibit two concrete 3- and 4-parameter families (the bi-involutions locus and a CM example) in which the canonical point is provably big, from which we deduce the finiteness of low-height curves and non-torsion at transcendental moduli. Our methods combine adelic Arakelov intersection theory with a generic Betti-rank argument.