Related papers: Effective Methods for Diophantine Finiteness

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in…

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^1_K\to\mathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an…

Let $G$ be a semisimple algebraic group. We develop a machinery for manipulation and manufacture of well-rounded families $\left\{ \mathcal{B}_{T}\right\} _{T>0}\subset G$ as they were defined in a work by A. Gorodnik and A. Nevo. The…

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the…

The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…

We prove the Shafarevich conjecture for very irregular varieties of dimension less than half the dimension of their Albanese variety, subject to some mild numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in the…

In this paper, we prove the Shafarevich conjecture for certain complete intersections of hypersurfaces in abelian varieties defined over a number field $K$ using the Lawrence-Venkatesh method. The main new inputs we need are computation of…

We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a…

Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively…

In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both…

Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…

The Shafarevich conjecture for a class of varieties over a number field posits the finitude of those with good reduction outside a finite set of primes. In the case of hypersurfaces in the torus $\mathbb{G}_m^n$, a natural class to consider…

We prove a generalization of Shafarevich's Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the…

Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate-Shafarevich group of a $K$-torus $T$ is $Sha(T , V) = \ker\left(H^1(K , T) \to \prod_{v \in V} H^1(K_v , T)\right)$. We prove that if $K = k(X)$ is…

In a previous paper {GN2} an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the…

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field $K$. More precisely, we give effective bounds for the number of specializations $t\in \mathcal{O}_K$ that do not preserve the irreducibility…

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for…

This article is devoted to examples of (orbifold) K\"ahler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an…

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of…