# Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$? I noticed that magma can compute 2 Selmer groups for elliptic curves over $\mathbb{F}_p(t)$, so I was wondering if that would give me some info on the rank over $\mathbb{Q}(t)$.

Or more generally, how would you go about computing the rank of an elliptic curve over $\mathbb{Q}(t)$?

Note: I've read Stewart and Top's paper "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms", but am wondering about other techniques.

• Section II.2 in Silverman's "Advanced topics in the arithmetic of elliptic curves" Dec 3, 2015 at 22:45
• Oh, sorry that should be "Section III.2". Dec 3, 2015 at 23:11
• It seems III.9.3 is more relevant (caveat: Ch. III only considers characteristic 0, due to background issues). Fix a regular proper elliptic surface $\mathscr{E} \rightarrow \mathbf{P}^1_{\mathbf{Q}}$ with generic fiber $E$, so $\mathscr{E}$ is $\mathbf{Q}$-smooth, hence spreads to a $\mathbf{Z}[1/N]$-smooth proper flat elliptic fibration over $\mathbf{P}^1_{\mathbf{Z}[1/N]}$ for sufficiently divisible $N$. By III.9.3 and preservation of "degree" in flat families, the canonical height is preserved by reduction mod $p$ for all $p\nmid N$, so mod-$p$ reduction preserves linear independence. Dec 3, 2015 at 23:20
• If your curve has at least one Q(t)-rational 2-torsion point, you may try to use algorithms by I. Gusic and P. Tadic. See web.math.pmf.unizg.hr/glasnik/vol_47/no2_03.html sciencedirect.com/science/article/pii/S0022314X14003187
– duje
Dec 21, 2015 at 16:30