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The variation of the rank of elliptic curves over $\mathbb{Q}$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over $\mathbb{Q}$ admits infinitely many quadratic twists of rank $\ge 1$. Most elliptic curves have even infinitely many twists of rank $\ge 2$ and examples of elliptic curves with infinitely many twists of rank $\ge 4$ are known. There are also certain density results. This paper studies the variation of the...

The paper contains an expanded version of the talk delivered by the first author during the conference ALANT3 in Będlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to ℓ-adic cohomology of schemes. Some other topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted a variety of open questions which can lead to interesting...

A variety $X$ over a field $K$ is of Hilbert type if $X\left(K\right)$ is not thin. We prove that if $f:X\to S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^{-1}\left(s\right)$, $s\in S\left(K\right)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.