Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor doubling.

Coleman–Gross height pairings and the p-adic sigma function

We give a direct proof that the Mazur–Tate and Coleman–Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman–Gross height to the case of divisors with non-disjoint support and, doing some