Related papers: Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$

This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let $F$ be a field of characteristic $\neq 2$, $f\in F[x]$ be monic and quadratic…

We study the \'{e}tale fundamental groups of singular reduced connected curves defined over an algebraically closed field of arbitrary prime characteristic. It is shown that when the curve is projective, the \'{e}tale fundamental group is a…

Let $p >= 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\infty=Q(E[p^\infty])$ be a pro-$p$ Galois extension over a number field $K$. We consider $X(E/K_\infty)$, the Pontryagin dual of the $p$-Selmer…

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ with isometry group of dimension 4, in terms of their…

Let $\rho\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $\pi$. We prove that $\operatorname{tr}\rho$ is…

We work in the context of a complete totally transcendental theory $T = T^{eq}$. We consider the prime model $M_{A}$ over a set $A$. For intermediate sets $B$ with $A\subseteq B \subseteq M_{A}$ which are normal ($Aut(M_{A}/A)$-invariant)…

Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P of C, let R be the ring of functions that are regular outside P, and let K be the completion at P of the function field of…

Let $S=S_{g,p}$ be a compact, orientable surface of genus $g$ with $p$ punctures and such that $d(S):=3g-3+p>0$. The mapping class group $\textup{Mod}_S$ acts properly discontinuously on the Teichm\"uller space $\mathcal T(S)$ of marked…

We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or $\mathcal{D}$-groupoids) that are algebraic integrable, that is, isogenous to the action groupoid of an algebraic group action. Our…

It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (respectively, any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that…

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group $SUT(n,\mathbb{F})$ is minimal for every local field $\mathbb{F}$ of characteristic $\neq 2$. This result is new even…

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture.…

It is now known that for any prime p and any finite semiabelian p-group G, there exists a (tame) realization of G as a Galois group over the rationals Q with exactly d = d(G) ramified primes, where d(G) is the minimal number of generators…

We show how to count and randomly generate finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for…

We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any…

We discuss $SL(2,Z)$ subgroups appropriate for the study of $N=2$ Super Yang-Mills with $N_f=2n$ flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular,…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…

A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even…

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of…

Ideas and techniques from Khare's and Wintenberger's article on the proof of Serre's conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as Galois…