Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t), $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly interested in the case when $\mathcal{E}$ is a rational surface and $\mathrm{char}(\mathbb{F}_q) > 3$. Also, let $$ P_0 = \big( x_0(t), y_0(t) \big),\qquad P_1 = \big( x_1(t), y_1(t) \big) $$ be two sections of $\mathcal{E}$ of infinite order in the Mordell-Weil group $\mathrm{MW}(\mathcal{E})$. What conditions are sufficient for $P_0$, $P_1$ to be independent in $\mathrm{MW}(\mathcal{E})$?
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$\begingroup$ If we had one element and wanted to detect wether it was torsion, we could specialize to various t, compute the orders and if for some set of specializations, the orders turned out to be coprime, then we know the original point was non torsion. For two points, we can quotient out by one point and test wether the other point is torsion. I don't know how practical this is. $\endgroup$– AsvinCommented May 5, 2020 at 14:49
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$\begingroup$ Nonzero height pairing determinant is one. Otherwise if you had to calculate a Selmer group anyway on your way to determine the rank, then you could be lucky that they are independent in there. Just like over number fields, I don't hink specialisations help. $\endgroup$– Chris WuthrichCommented May 5, 2020 at 14:51
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$\begingroup$ To add to Chris's answer, the height pairing reduces to intersection theory in the elliptic surface world. If you understand the geometry of the special fibers well, this should not be difficult at all to compute for a rational surface, and the determinant is an if-and-only-if condition so once you compute it you are done. $\endgroup$– Will SawinCommented May 5, 2020 at 14:56
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$\begingroup$ Could you please clarify your comment. Points $P_0, P_1$ are independent if and only if $\mathrm{det} = 1$? $\endgroup$– Dimitri KoshelevCommented May 5, 2020 at 15:45
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$\begingroup$ det non-zero. The intersection pairing is a naive height which has to be modified slightly to get the canonical height which yields a non-degenerated bilinear form. magma has it all implemented : magma.maths.usyd.edu.au/magma/handbook/… $\endgroup$– Chris WuthrichCommented May 5, 2020 at 18:04
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