# Coefficients of elliptic curves over function fields

Consider the projective plane $$\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$$ over the algebraic closure of the function field $$\mathbb{C}(t)$$.

Take the point $$p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\mathbb{C}(t)}}$$ and eight more general points $$p_1,\dots,p_8\in \mathbb{P}^2_{\overline{\mathbb{C}(t)}}$$ which are not defined over $$\mathbb{C}(t)$$. Assume that there is a cubic curve $$C \subset \mathbb{P}^2_{\overline{\mathbb{C}(t)}}$$, defined over $$\mathbb{C}(t)$$, passing through $$p_0,p_1,\dots,p_8$$. We can write $$C$$ as $$C = \{ \alpha_0(t) x^3 + \alpha_1(t)x^2y + \alpha_2(t)x^2z + \alpha_3(t)xy^2 + \alpha_4(t)xyz + \alpha_5(t)xz^2 + \alpha_6(t)y^3 + \alpha_7(t)y^2z + \alpha_8(t)yz^2 + \alpha_9(t)z^3 = 0 \}$$ where $$\alpha_0,\dots,\alpha_9$$ are polynomials in $$t$$, and extend $$C$$ to a fibration on $$\mathbb{P}^1$$ as $$C' = \{A_0(s,t) x^3 + A_1(s,t)x^2y + A_2(s,t)x^2z + A_3(s,t)xy^2 + A_4(s,t)xyz + A_5(s,t)xz^2 + A_6(s,t)y^3 + A_7(s,t)y^2z + A_8(s,t)yz^2 + A_9(s,t)z^3 = 0 \}$$ where now $$A_0,\dots,A_9$$ are homogeneous polynomials on $$\mathbb{P}^1$$.

Can we find such a curve $$C'$$ such that the polynomials $$A_0,\dots,A_9$$ appearing in the expression of $$C'$$ are of degree one or at least $$A_9$$ is of degree one?

• Bound in terms of what? Take $C'$ with polynomials of arbitrary high degree, then choose $8$ points on it. Aug 6, 2021 at 15:55
• You are right, my question was badly written. I modified it.
– Arty
Aug 6, 2021 at 16:25
• I repeat my objection: Weierstrass form is unique! So if you start with Weierstrass form with polynomials of high degrees, it cannot be reduced to a Weierstrass form with polynomials of low degrees. Aug 6, 2021 at 17:46
• Sorry, I modified the question again. Now $C'$ has a completely general form.
– Arty
Aug 6, 2021 at 18:01

Are you asking if, for all tuples $$p_1,\dots, p_8$$, there exists such a $$C'$$ with $$A_9$$ of degree one? This is false, assuming $$A_9=0$$ does not count as degree one.
We can for example choose one of the $$p_i$$ to equal $$[0: 0 : 1]$$ for $$t=1$$ and and one to equal $$[0,0,1]$$ for $$t=2$$. Then regardless of which $$C'$$ we take, $$A_9$$ will equal $$0$$ for $$t=1$$ and $$t=2$$ and thus cannot have degree one.
• I was asking if there exists such a $C'$ for $p_1,\dots,p_8$ general not for all tuples $p_1,\dots,p_8$. Perhaps for a point $p_i$ to take the same value for two different values of $t$ is a closed condition.