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...

Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence...

We give upper and lower bounds for the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.

Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}:y²=xⁿ+a$ and $C_{n,a}:y²=x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}\left(\mathbb{Q}\right)$ and $J_{n,a}\left(\mathbb{Q}\right)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}\left(\mathbb{Q}\right)$. Namely, we show that $J_{8,a}{\left(\mathbb{Q}\right)}_{tors}=J_{8,a}\left(\mathbb{Q}\right)\left[2\right]$. In addition, we characterize the torsion parts of $J_{p,a}\left(\mathbb{Q}\right)$, where p is an odd prime, and...