Abstract
We describe a computation of rational points on genus 3 hyperelliptic curves C defined over \(\mathbb {Q}\) whose Jacobians have Mordell–Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in SageMath to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in \(C(\mathbb {Q})\).
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- 1.
We focus on hyperelliptic curves defined over \(\mathbb {Q}\) because SageMath has an implementation of Coleman integration precisely for odd degree hyperelliptic curves given by a monic equation defined over \(\mathbb {Z}_p\) for odd primes p of good reduction.
- 2.
Note that when the derivative of p −kf M(pt) at the root is non-zero modulo p which happens quite often, then the roots that we find are in fact correct modulo \(p^{n^{\prime }-k}\).
- 3.
The hyperelliptic curve C is defined over \(\mathbb {Q}\). Thus the fact that \([Q-\infty ]\in J(\mathbb {Q}_p)_{\text{tors}}\) forces Q to have coordinates in \(\overline {\mathbb {Q}}\cap \mathbb {Q}_p\).
- 4.
Our computations show that the \(\mathbb {Q}\)-rational points of highest absolute logarithmic height (with respect to an odd degree model) on a curve among the 16, 977 hyperelliptic curves that we considered are
$$\displaystyle \begin{aligned} \left(-\frac{49}{18} , -\frac{339563}{11664}\right), \left(-\frac{49}{18},-\frac{1600445}{52488}\right) \end{aligned}$$on the hyperelliptic curve
$$\displaystyle \begin{aligned} C:y^2 + (x^4+x^2+x)y= x^7-x^6-5x^5+5x^3-3x^2-x, \end{aligned}$$which has absolute minimal discriminant 5326597 and whose Jacobian has conductor 5326597.
- 5.
We are grateful to Andrew Sutherland for kindly computing real endomorphism algebras (using the techniques of [9, 18]) for a number of curves that produced Chabauty–Coleman output similar to this example, which greatly assisted in understanding the structure of their Jacobians. (Work by Costa et al. [13] provides further data on endomorphisms, can be made rigorous, and is freely available on GitHub.) We would also like to thank Bjorn Poonen for a very helpful discussion about this phenomenon.
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Balakrishnan, J.S., Bianchi, F., Cantoral-Farfán, V., Çiperiani, M., Etropolski, A. (2019). Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves. In: Balakrishnan, J., Folsom, A., Lalín, M., Manes, M. (eds) Research Directions in Number Theory. Association for Women in Mathematics Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-19478-9_3
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