A naive p-adic height on the Jacobians of curves of genus 2
摘要
Consider a genus 2 curve defined over ℚ given by an affine equation of the form y2 = f(x) for some polynomial f of degree 5, and let p be an odd prime. Extending work of Perrin-Riou for elliptic curves, we construct a naive p-adic height function on a finite index subgroup of the Jacobian J of this curve, using the explicit embedding of J in ℙ8 and the associated formal group described by Grant. We use the naive height to construct a global height hp: J(ℚ) → ℚp using a limit construction analogous to Tate’s construction of the Néron–Tate height, and show that it is quadratic. We then compare hp to a p-adic height constructed in a different way by Bianchi and show that they are equal.