Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$? 



modify a fibration with a fiber of higher multiple
The answer is yes if $n$ is a prime power by a theorem of Liu-Lorenzini-Raynaud; see p. 497, Corollary 6.7 in <http://www.math.u-bordeaux1.fr/~qliu/articles/LLR.pdf> (This answered a question of Neron.) It's true in general if your fiber $F$ is multiplicative.


By the way: the answer is no if you are considering honest elliptic fibrations (with a section) when you say "fibration". In fact, assuming you are working with $X$ non-singular, the existence of a section forces every fiber to have at least one reduced irreducible component. Thus, it can't be non-reduced everywhere. (This was already mentioned in the comments.)


Also useful might be (from the above paper): "Examples of multiple fibers can be found in [18], Theorem 0.1, in [24], Sect.8,in[23],Sect.3, and in[48],9.4.1"


