Related papers: Maximum scattered linear sets and complete caps in…

Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois…

In this work, we introduce a natural notion concerning finite vector spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional subspace…

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension…

Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered…

We show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the…

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over…

In this article we study the bicanonical map $\phi_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $\phi_2$ has diverse behavior and exhibit most of the complexities that are possible for a bicanonical…

Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute…

We study compatible families of four-dimensional Galois representations constructed in the \'{e}tale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are…

In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of…

Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\leq e$. We show that there are infinitely many irreducible Galois representations $\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that…

Let $E\supseteq F$ be a field extension and $M$ a graded Lie algebra of maximal class over $E$. We investigate the $F$-subalgebras $L$ of $M$, generated by elements of degree $1$. We provide conditions for $L$ being either ideally…

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were…

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…

Let $R=K[t;\sigma]$ be a skew polynomial ring, where $K$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $\sigma$. We show that two irreducible similar skew polynomials $f,g\in R$ are similar if and only if…

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our…

Let m_2(n, q) be the maximum size of k for which there exists a k-cap in PG(n, q), and let m'_2(n, q) be the second largest value of k for which there exists a complete k-cap in PG(n, q). In this paper Chao's upper bound q^2 - q + 5 for…

Let $m_2(n, q), n \geq 3$, be the maximum size of k for which there exists a complete k-cap in PG(n, q). In this paper the known bounds for $m_2(n, q), n \geq 4$, q even and $q \geq 2048$, will be considerably improved.

Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ --…