This SageMath code computes the local cyclotomic p-adic Coleman--Gross height pairing $h_p$ on a hyperelliptic curve $C:y^2=f(x)$ over $\mathbb{Q}_p$ with good reduction. More precisely, it computes $h_p(P-Q,R-S)$ for $P,Q,R,S\in C(\mathbb{Q}_p)$ satisfying the following conditions:

• $f$ is monic (one can use move_to_monic1 or move_to_monic2 if this is not satisfied);
• the residue discs $D(P)$ and $D(Q)$ of $P$ and $Q$ are distinct from $D(R)$, $D(\iota(R))$, $D(S)$ and $D(\iota(S))$, where $\iota$ is the hyperelliptic involution.

Dependencies

Jennifer Balakrishnan's code for even degree Coleman integrals is required. Download it from https://github.com/jbalakrishnan/AWS and follow the instructions given there.

Main functions

• height_infinities(P, Q) computes $h_p(\infty_- - \infty_+, P-Q)$ for even degree models.
• height_four_affine_points(P, Q, R, S) computes $h_p(P-Q, R-S)$ for affine $P,Q,R,S$.
• height_divisors(D1, D2) computes $h_p(D_1, D_2)$ for two degree 0 divisors $D_1,D_2$ on $C$ with disjoint an pointwise $\mathbb{Q}_p$-rational
• height_infinities_residuedisc_to_z(P) computes $h_p(\infty_- - \infty_+, P(z)-P)$, where $P(z)$ is a parametric point in the residue disc of $P$ (used for quadratic Chabauty).

Examples

• $X_0^+(107)$: To compute the rational points on X0+(107), load qc_X0107plus_p7.m into magma. This requires the QCMod package, available from https://github.com/steffenmueller/QCMod. To verify that the coefficients of the global height as a bilinear pairing are as claimed in qc_X0107plus_p7.m, run solve_for_height_X0107plus.sage.
• $X_0^+(67)$: To verify the p-adic BSD conjecture for X0+(67) and p = 11, 29, 31, 71, 89, run bsd_X067plus.sage.
• Tests: The file examples.sage contains many tests and sanity checks, including comparisons with Balakrishnan's implementation of the algorithm of Balakrishnan--Besser for odd degree.

Authors Stevan Gajović and Steffen Müller