In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$. We prove that if the Galois group has order at most $3$, it always extends to a subgroup of the Jonquières group associated with the point $P$. Conversely, with a degree of at least $4$, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$. In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

In a recent paper Moshe Jarden (Diamonds in torsion of Abelian varieties, J. Inst. Math. Jussieu9(3) (2010), 477–480) proposed a conjecture, later named the Kuykian conjecture, which states that if $A$ is an Abelian variety defined over a Hilbertian field $K$, then every intermediate field of $K({A}_{\mathrm{tor} } )/ K$ is Hilbertian. We prove that the conjecture holds for Galois extensions of $K$ in $K({A}_{\mathrm{tor} } )$.

A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

We study the Galois tower generated by iterates of a quadratic polynomial $f$ defined over an arbitrary field. One question of interest is to find the proportion $a_n$ of elements at level $n$ that fix at least one root; in the global field case these correspond to unramified primes in the base field that have a divisor at level $n$ of residue class degree one. We thus define a stochastic process associated to the tower that encodes root-fixing information at each level. We develop a uniqueness result for certain permutation groups, and use this to show that for many $f$ each level of the tower contains a certain central involution. It follows that the associated stochastic process is a martingale, and convergence theorems then allow us to establish a criterion for showing that $a_n$ tends to 0. As an application, we study the dynamics of the family $x^2 + c \in\overline{\mathbb{F}}_p[x]$, and this in turn is used to establish a basic property of the $p$-adic Mandelbrot set.

We give an explicit recipe for the determination of the images associated to the Galois action on $p$-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over $\mathbb{Q}$ without complex multiplication with conductor less than 200 and for each prime number $p$.