Rank of elliptic surfaces 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.
 A: 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. 
