The Lang–Vojta Conjectures on Projective Pseudo-Hyperbolic Varieties

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

“…Conjecture 1.3 has a wide range of important consequences; we mention for example the abc conjecture of Masser-Oesterlé [33,Conjecture 3], the Bombieri-Lang conjecture [24, Conjecture F.5.2.1] and the Lang-Vojta conjecture [24,Conjecture F.5.3.6]. We refer to [1,2,4,25] for further consequences.…”

Construction of Arithmetic Teichmuller Spaces and some applications

“…§ 6.7 As an aside let me say that I have used the case of hyperbolic varieties here because of their relevance in Diophantine Geometry via Lang's Conjecture [Lang, 1986] and [Faltings, 1991]. One can also use other related hypothesis as a substitute for the hyperbolic hypothesis, for example, one can work with groupless varieties instead of hyperbolic varieties (see [Javanpeykar, 2020] for other related hypothesis which may be used here instead). § 6.8 An important property in many anabelian considerations is the following: a profinite group Π is said to be a slim profinite group (or simply Π is slim) if every open subgroup of Π has trivial center.…”

Arithmetic hyperbolicity: automorphisms and persistence

“…In our work with Junyi Xie, we build on this result and prove the stronger statement that a quasi-projective variety X over K with X (L) finite for every number field L/K has, in fact, only finitely many birational self-maps; see [30,Theorem 1.3]. For a survey of our work on self-maps of hyperbolic varieties, we refer the reader to [21,Sect. 15].…”

“…For a survey of our work on the Persistence Conjecture and its relation to Lang's conjecture, we refer the reader to [21,Sect. 17].…”