Papers › p-Adic sigma functions and heights on Jacobians of genus 2 curves
p-Adic sigma functions and heights on Jacobians of genus 2 curves
Francesca Bianchi
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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 hₚJ(ℚ)→ℚₚ by means of p-adic analogues of N\'eron functions of divisor 2Θ. We define such N\'eron 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 N\'eron 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 \`a la Colmez. We use this to relate λᵥ and hₚ to local and global extended Coleman-Gross (and hence Nekov\'a\v{r}) 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.
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