Related papers: Codimension two integral points on some rationally…

We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral…

In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then…

Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose…

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical…

We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a…

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical…

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a…

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists…

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin's Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree $1$ whose geometric…

We prove the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1,2). The proof uses a mixture of the circle method and techniques from the…

We study finite orbits for non-elementary groups of automorphisms of compact projective surfaces. In particular we prove that if the surface and the group are defined over a number field k and the group contains parabolic elements, then the…

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings.…

We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube translation $3$-manifold. We also extend this original result to…

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\phi:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside…

In this article, we study the Zariski closure of modular points in the two-dimensional universal deformation space when the residual Galois representation is reducible. Unlike the previous approaches in the residually irreducible case from…

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

A field $K$ is called ample if for every geometrically integral $K$-variety $V$ with a smooth $K$-point, $V(K)$ is Zariski-dense in $V$. A field $K$ is virtually ample if some finite extension of $K$ is ample. We prove that there exists a…

In this paper we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface \begin{align*} x_1y_1^2+...+x_sy_s^2 = 0…