Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

Question

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces.

I got quite stuck in Corollary 3.27 and Proposition 3.24. I do not understand when they say the following(line 3 Corollary 3.27):

"Dividing (the cocycle defining) S' by different primes $p>p_0$ we obtain elliptic K3 surfaces(of different degrees)".

1) What is the cocycle defining a surface?Do they mean the cocycle whose square is 0 that gives the elliptic fibration? Moreover:what is the relation between the new surface and the old, is there some natural map?

2) Then at line 5 of Corollary 3.27 they say: "${\epsilon}_p$(the one obtained dividing the cocycle by p)by deformation theory contain rational multisection of degree divisible by $d({\epsilon}_p)$". I don't understand the content of "by deformation theory". I believe they are referring to Prop 2.2 and 2.12, but there is proved (as far as I can read) that some deformed surface contains a rational curve, how can I get one back for the original surface? So I don't understand where they have really proved this statement. But since they just say "by deformation theory" I believe at some point of the paper was clear and i missed that part.

3) Last: at prop 3.24 at the very beginning of the prove they say "Let $\epsilon''={\epsilon'}^{d_{\epsilon}}$ Is this the standard notation for something? I don't understand what they are referring to.

I believe there will be someone here that has read the paper and can enlighten me on these question, or probably the question are so basic that even someone that didn't read the paper but is comfortable in algebraic geometry can answer (I'm not yet very much, but I really want to understand this paper).

So I apologize if it is not an hardly advanced research question; but I didn't have any feed back on Stack Exchange on this question and in my department there is no one comfortable with the tools they are using, so I'm self learning it.

Thanks in advance!

Edit: I apologize, reading once again I see where they prove the statement about rational curves. So question 2 is not a problem, I remain doubtful about the other 2 questions.

Answer 1

Question 1: $S'$ is an elliptic surface without a section such that it is Jacobian is $S$. The surfaces with this property are parametrized by a certain cohomology group and the cocycle the authors refer to, lives in this group. More details about non-Jacobian elliptic fibrations can be found in e.g., the final chapter of Cossec-Dolgachev Enriques surfaces. This chapter will probably also answer your third question.