Generalized Nagao conjecture for hyperelliptic Jacobians

Let $\mathcal{X}$ be a hyperelliptic curve defined over $\mathbb{Q}(T)$ whose Jacobian $J_{\mathcal{X}}$ has no subvariety defined over $\mathbb{Q}$. For an integer $t$, let $a_{\mathcal{X}_t}(p)=p+1-\#\mathcal{X}_t(\mathbb{F}_p)$ and define

$$A_{\mathcal{X},1}(p)=\frac{1}{p}\sum_{t=0}^{p-1}a_{\mathcal{X}_t}(p).$$

Generalized Nagao conjecture.

$$\lim_{P \to \infty} \frac{1}{P} \sum_{p \leq P} -A_{\mathcal{X},1}(p) \log p=\operatorname{rank} J_{\mathcal{X}}(\mathbb{Q}(T)).$$

The conjecture is proposed as the rank-detecting generalization of Nagao's conjecture for hyperelliptic Jacobians. The paper uses it conditionally to construct high-rank examples, and no resolution status is supplied here.