Nagao's conjecture for the rank of an elliptic curve over $\mathbb{Q}(T)$

Let $\mathcal{E}$ be an elliptic curve over $\mathbb{Q}(T)$ as in the source. For each prime $p$, let $\mathcal{E}_t$ denote the elliptic curve over $\mathbb{Q}$ obtained by specializing $T$ at $t\in\mathbb{Q}$, and define $a_t(p)$ by

$$p+1-a_t(p)=\#\mathcal{E}_t(\mathbb{F}_p)$$

unless $p$ divides the discriminant of $\mathcal{E}_t$, in which case $a_t(p)=0$. Set

$$A_{\mathcal{E}}(p)=\frac{1}{p}\sum_{t=0}^{p-1}a_t(p).$$

Nagao's conjecture. The rank $r$ of $\mathcal{E}(\mathbb{Q}(T))$ satisfies

$$r=\lim_{Q\rightarrow\infty}\frac{\log Q}{Q}\sum_{p\leq Q}-A_{\mathcal{E}}(p),$$

where the sum runs over all primes $p\leq Q$. The conjecture relates the Mordell–Weil rank over the function field to averaged point-counting data of the specialized elliptic curves. It was proven by Rosen and Silverman for rational elliptic surfaces, while the general formulation remains beyond that result.