Is there any method to determine which elliptic curves over ${\mathbb Q}(t)$ have larger rank just from their equations—without knowing their exact rank—as with Mestres sums for elliptic curves?


For example, the BSD conjecture says that potentially high-rank curves admit many points in the finite fields at most primes.



Rank of elliptic surfaces
Felipe already gave a reference as a comment (not sure why he didn't write it as an answer). Anyway, here are the details. Let
$$ E:y^2 = x^3 + A(T)x + B(T) $$
be an elliptic curve over $\mathbb Q(T)$. For each prime $p$ and each $t\in\mathbb F\_p$, let
$$
a\_p(t) = p+1-\#E(\mathbb F\_p)
$$
and then let
$$
A\_p = \frac{1}{p}\sum\_{t\in\mathbb F\_p} a\_p(t)
$$
be the average trace of Frobenius over the fibers of $E$ modulo $p$. Then Nagao's conjecture is that
$$
\lim\_{T\to\infty} \frac{1}{T}\sum\_{p<T} -A\_p\cdot\log p 
= \text{rank }E\bigl(\mathbb Q(T)\bigr).\quad (\*)
$$
Rosen and I proved in [1] that Nagao's formula follows if one knows Tate's conjecture for the elliptic surface $E$. In particular, $(\*)$ is true if $\deg A(T)\le3$ and $\deg B(T)\le5$, since those conditions imply that the elliptic surface is birational to $\mathbb P^2$, and hence (with some work) Tate's conjecture holds.


EDIT: In answer to the OP's question/comment, Rania Wazir, in her thesis, gave a similar formula for elliptic 3-folds over $\mathbb Q(T\_1,T\_2)$; see [2]. Subsequently, she, Hindry, and Pacheco proved a very general version for fibrations of arbitrary dimension over bases of arbitrary dimension; see [3].


[1] MR1626465 Rosen, M. and Silverman, J.H.. On the rank of an elliptic surface, *Inv. Math* **133** (1998) 43-67.


[2] MR2041769 Wazir, Rania Arithmetic on elliptic threefolds. *Compos. Math.* **140** (2004), no. 3, 567–580. 


[3] MR2141536 Hindry, Marc; Pacheco, Amílcar; Wazir, Rania Fibrations et conjecture de Tate. (French) [Fibrations and the Tate conjecture] *J. Number Theory* **112** (2005), no. 2, 345–368. 


