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Height uniformity for rational points on odd hyperelliptic curves

Determine constants c_g, d_g > 0, depending only on the genus g, such that for every polynomial f(x) = x^{2g+1} + c_2 x^{2g-1} + ⋯ + c_{2g+1} with nonzero discriminant and for every rational point P = (α, β) ∈ C_f^0(Q) on the affine curve y^2 = f(x), the logarithmic Weil height of the x-coordinate satisfies h(α) ≤ c_g · log Ht(f) + d_g.

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Background

The authors’ main results give lower bounds for heights of rational points on Jacobians in a density 1 family, and they highlight a tension with a conjectured upper bound for the x-coordinate height of rational points on the affine curve y2 = f(x).

They note that this conjectured upper bound would follow from Vojta’s conjecture, underscoring its deep arithmetic nature and relevance to uniformity questions in higher genus.

References

There is an interesting tension between this result and the conjecture that there are constants $c_g, d_g > 0$ such that for any $f \in \mathcal{F}(X)$ and $P = (\alpha, \beta) \in C_f0(Q)$, we have h(\alpha) \leq c_g \log Ht(f) + d_g (as shown in , this would follow from Vojta's conjecture), especially in light of the expectation (see Remark 10.11) that $100 \%$ of curves in the family (\ref{eqn_intro_family_of_curves}) have no rational points other than $P_\infty$.

— 100% of odd hyperelliptic Jacobians have no rational points of small height (2405.10224 - Laga et al., 16 May 2024) in Introduction, following Theorem 1.2