If \(X \to \AA_{\CC}^1\) is a family of elliptic curves having a section, show that the family is trivial.
Hints: Use the section to fix the group structure on the fibres. Show that the points of order 2 on the fibres form an étale cover of \(\AA_{\CC}^1\), which must be trivial, since \(\AA_{\CC}^1\) is simply connected. This implies that \(\lambda\) can be defined on the family, so it gives a map \(\AA_{\CC}^1 \to \AA_{\CC}^1-\ts{0,1}\). Any such map is constant, so \(\lambda\) is constant, so the family is trivial.
Solution
Put \[ S=\AA^1_{\CC}=\Spec\CC[t], \] write \[ \pi:E\longrightarrow S \] for the family, and let \[ O:S\longrightarrow E \] be the given section.
1. The section \(O\) makes \(E/S\) an elliptic scheme, and its \(2\)-torsion\[ E[2]=\ker[2] \]is a finite etale cover of \(S\) of degree \(4\).
The section chooses an origin on every fibre, so the fibrewise elliptic-curve group laws assemble to the group law on the elliptic scheme \(E/S\).
Multiplication by \(2\) on an elliptic scheme is finite locally free of degree \(4\). Since \(2\) is invertible on \(S\), its differential on the relative tangent line is multiplication by \(2\), hence an isomorphism. Thus \[ [2]:E\longrightarrow E \] is etale. Base-changing along the zero section gives \[ E[2]\longrightarrow S, \] which is therefore finite etale of degree \(4\).
2. There are four pairwise disjoint sections\[ O,T_0,T_1,T_\lambda:S\longrightarrow E \]whose images are exactly \(E[2]\).
The affine line over \(\CC\) has no nontrivial connected finite etale covers. Hence every finite etale cover of \(S\) is a disjoint union of copies of \(S\). Applying this to the degree-\(4\) cover in step 1 gives four sections. One is the zero section \(O\); label the other three \[ T_0,T_1,T_\lambda. \] Because \(E[2]\to S\) is etale, these components are disjoint.
3. The complete linear system\[ \abs{2O} \]defines a finite flat double cover\[ q:E\longrightarrow\PP^1_S \]whose branch sections are exactly the images of\[ O,T_0,T_1,T_\lambda. \]After an automorphism of \(\PP^1_S\), the first three branch sections are\[ \infty,\qquad 0,\qquad 1. \]
On every elliptic fibre \(E_s\), the line bundle \[ \OO_{E_s}(2O_s) \] has degree \(2\) and two independent sections. Its complete linear system is the quotient map by the involution \[ [-1]:E_s\longrightarrow E_s. \] By cohomology and base change, the rank-\(2\) bundle \[ \pi_*\OO_E(2O) \] and its fibrewise linear systems give the relative quotient \[ q:E\longrightarrow\PP\bigl(\pi_*\OO_E(2O)\bigr), \] whose restriction to every fibre has degree \(2\). This morphism is proper and fibrewise finite, hence finite. The fibrewise flatness criterion then shows that it is flat. Thus it is finite flat of degree \(2\).
Every vector bundle on \[ S=\Spec\CC[t] \] is free, so the target is isomorphic to \(\PP^1_S\). The fixed points of \([-1]\) are precisely the points killed by \(2\); hence the ramification sections of \(q\) are the four sections in step 2, and their images are the four branch sections.
It remains to normalize three branch sections globally. Since \(\Pic(S)=0\), each section of \(\PP^1_S\) can be represented by a unimodular column vector in \(\CC[t]^2\). Choose vectors \(v_\infty,v_0\) representing the images of \(O,T_0\). Their sections are disjoint, so \[ \det(v_\infty,v_0) \] vanishes nowhere on \(S\) and hence is a unit of \(\CC[t]\). Thus the matrix with columns \(v_\infty,v_0\) lies in \(\GL_2(\CC[t])\). Its inverse gives an automorphism of \(\PP^1_S\) carrying these two sections to \[ \infty=[1:0], \qquad 0=[0:1]. \]
Write the image of \(T_1\) in these coordinates as \([a:b]\). Disjointness from \(\infty\) and \(0\) says respectively that \(b\) and \(a\) vanish nowhere, so \[ a,b\in\CC[t]^*=\CC^*. \] A diagonal automorphism preserving \(\infty\) and \(0\) now carries \([a:b]\) to \[ 1=[1:1]. \] This gives the required global normalization.
4. The fourth branch section defines a morphism\[ \boxed{ \lambda:S\longrightarrow\AA^1_{\CC}\setminus\{0,1\}. } \]
In the coordinate fixed in step 3, write the fourth branch section as \[ x=\lambda. \] It is disjoint from the section at infinity, so it takes values in the affine chart \(\AA^1_S\). It is also disjoint from the branch sections \(x=0\) and \(x=1\). Therefore its coordinate is a regular function \[ \lambda\in\Gamma(S,\OO_S)=\CC[t] \] such that neither \(\lambda\) nor \(\lambda-1\) vanishes anywhere on \(S\). Equivalently, it is the displayed morphism.
5. The function \(\lambda\) is constant.
Since the map in step 4 avoids \(0\) and \(1\), both \[ \lambda \qquad\text{and}\qquad \lambda-1 \] are units of \(\CC[t]\). But \[ \CC[t]^*=\CC^*. \] Hence \(\lambda\in\CC\setminus\{0,1\}\) is constant.
6. The double cover \(q\) is the product over \(S\) of the Legendre doublecover\[ E_\lambda: y^2=x(x-1)(x-\lambda) \longrightarrow \PP^1_{\CC}. \]
By step 5 the branch divisor on \(\PP^1_S\) is the constant divisor \[ B = \{0\}+\{1\}+\{\lambda\}+\{\infty\}. \] Because \(2\) is invertible, the finite flat double cover has the trace decomposition \[ q_*\OO_E \cong \OO_{\PP^1_S}\oplus\mcl^{-1}, \] where multiplication on the trace-zero summand is specified by a section \[ s\in H^0(\PP^1_S,\mcl^{\tensor2}) \] whose zero divisor is \(B\).
The branch divisor has relative degree \(4\), so \[ \mcl^{\tensor2}\cong\OO_{\PP^1_S}(4). \] Moreover \[ \Pic(S)=0 \] and \[ \Pic(\PP^1_S)\cong\ZZ, \] generated by \(\OO(1)\). Hence \[ \mcl\cong\OO_{\PP^1_S}(2). \]
In homogeneous coordinates \([X:Z]\), the standard section \[ s_0=XZ(X-Z)(X-\lambda Z) \] of \(\OO(4)\) has zero divisor \(B\). The sections \(s\) and \(s_0\) have the same zero divisor, so their quotient is a global unit on \(\PP^1_S\). Such a unit comes from \[ \CC[t]^*=\CC^*. \] Thus \[ s=c\,s_0 \] for some \(c\in\CC^*\). Since \(\CC\) is algebraically closed, write \[ c=u^2. \] Rescaling the trace-zero generator by \(u\) identifies the two double-cover algebras. Therefore \[ E\cong E_\lambda\times_{\Spec\CC}S \] over \(S\), with the given zero section corresponding to the point at infinity.
7. The family \(E\to\AA^1_{\CC}\) is trivial.
Step 6 gives an isomorphism of elliptic schemes \[ \boxed{ E\cong E_\lambda\times\AA^1_{\CC} } \] over the base. This is precisely the required trivialization.
8. Q.E.D.
Steps <1>1–<1>2 trivialize the relative \(2\)-torsion, steps <1>3–<1>5 produce Hartshorne’s Legendre parameter and show it is constant, and steps <1>6–<1>7 show that no residual quadratic twist remains.