Groups with no Parametric Galois Extension

Abstract: We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups $G$ which do not have a realization $F/\Qq(T)$ that induces all Galois extensions $L/\Qq(U)$ of group $G$ by specializing $T$ to $f(U) \in \Qq(U)$. For these groups, we produce two extensions $L/\Qq(U)$ that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups $G$ include symmetric groups $S_n$, $n\geq 7$, infinitely many $PSL_2(\Ff_p)$, the Monster. Two variants of the question with $\Qq(U)$ replaced by $\Cc(U)$ and $\Qq$ are answered similarly, the second one under a diophantine "working hypothesis" going back to a problem of Schinzel. We introduce two new tools: a comparizon theorem between the invariants of an extension $F/\Cc(T)$ and those obtained by specializing $T$ to $f(U) \in \Cc(U)$, and, given two regular Galois extensions of $k(T)$, a finite set of polynomials $P(U,T,Y)$ that say whether these extensions have a common specialization $E/k$.