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This lecture titled "On the Arithmetic of Genus 4 Curves" by Jiahui Gao from Princeton University explores a novel arithmetic invariant of non-hyperelliptic genus-4 curves. Learn about the canonical "quadratic" point on the Jacobian, defined by two natural degree-2 maps to projective lines. The talk builds on Xue's result that this point is generically non-torsion, introducing a height-theoretic notion of bigness for families of curves in the moduli space of genus four curves, along with criteria for when it applies. Discover two concrete examples—the bi-involutions locus and a CM example—where the canonical point is provably big, leading to conclusions about finiteness of low-height curves and non-torsion at transcendental moduli. The presentation combines adelic Arakelov intersection theory with a generic Betti-rank argument. This 59-minute Joint PU/IAS Number Theory lecture will be held at Simonyi 101 with remote access available on May 15, 2025, at 3:30pm.
