I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?



(nontrivial) isotrivial family of elliptic curves
Hint: use quadratic twists.


**Edit**: So as not to drag things out, I hope it's okay if I just give you a standard example. Let


$E\_0: y^2 = x^3 + Ax + B$


be your favorite elliptic curve over $\mathbb{Q}$ (i.e., any will do). Consider the 
elliptic curve
$E: t y^2 = x^3 + Ax + B$  

over the rational function field $\mathbb{Q}(t) = \mathbb{Q}(\mathbb{P}^1)$. Spreading this out as a scheme over $\mathbb{P}^1\_{/\mathbb{Q}}$, we see that there are two singular fibers, at $t = 0$ and $t = \infty$. Discarding these we get an elliptic curve over $\mathbb{A}^1 \setminus \{0\}$ which is isotrivial -- the $j$-invariant over every fiber is $j(E\_0)$ -- but nontrivial: the isomorphism classes of the fibers are in bijection with $H^1(\mathbb{Q},\mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.


It's a good bet that you'll find this example somewhere in the chapter on elliptic surfaces in Silverman's *Advanced Topics in the Arithmetic of Elliptic Curves*.


Can it be done over any base scheme? Unless I misunderstand, of course not, e.g. not over the spectrum of a field.


