2023/02/07 by Bianchi, Francesca
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2302.03454
Let C be a genus 2 hyperelliptic curve over a number field K, with a Weierstrass point ∞ at infinity, let J be its Jacobian, let Θ be the theta divisor with respect to ∞, and let p be any prime number. We give an explicit construction of a p-adic height hp\colon J(ℚ)→ ℚp by means of p-adic analogues of Néron functions of divisor 2Θ. We define such Néron functions using division polynomials and a generalisation of Blakestad's p-adic sigma function on the formal group of J. We prove that our p-adic Néron function λv at a non-archimedean place v of K is the image, under a suitable trace map, of a symmetric v-adic Green function of divisor Θ à la Colmez. We use this to relate λv and hp to local and global extended Coleman-Gross (and hence Nekovář) p-adic height pairings. We provide examples of our implementation, including one for a prime p greater than 106, and explain how similar techniques can be used to compute p-adic integrals of differentials of the first, second and third kind on C independently of the reduction type. As an application, we also give an explicit quadratic Chabauty function vanishing on the rational points on certain genus 4 bihyperelliptic curves.