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.

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    $\begingroup$ There is a conjectural formula, first proposed by Nagao, then proved to be a consequence of Tate's conjecture by Rosen and Silverman. Inv. Math 133 (1998) 43-67. $\endgroup$ Apr 3, 2017 at 20:59

1 Answer 1


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.

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    $\begingroup$ To write it as an answer, I'd have to look up the precise statement and then TeX it :-) $\endgroup$ Apr 4, 2017 at 1:44
  • $\begingroup$ thank you , but how about $Q(t_1,...,t_n)$? $\endgroup$
    – user371596
    Apr 4, 2017 at 6:56
  • $\begingroup$ Thank you Prof. Silverman. I really appreciate for references. $\endgroup$
    – user371596
    Apr 5, 2017 at 5:57

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