A survey of local-global methods for Hilbert's tenth problem

DOI10.1007/978-3-031-52163-8_2MaRDI QIDQ6611633

Margherita Pagano, Laura Paladino, Zeynep Kisakürek, Vlerë Mehmeti, Valentijn Karemaker, Sylvy Anscombe

Publication date: 27 September 2024

Decidability (number-theoretic aspects) (11U05) Rational points (14G05) Elliptic curves over global fields (11G05) Decidability of theories and sets of sentences (03B25)

Cites Work

Elementary geometric local-global principles for fields
On the minimal set for counterexamples to the local-global principle
On the local-global divisibility of torsion points on elliptic curves and $\operatorname{GL}_{2}$-type varieties
A local-global principle for rational isogenies of prime degree
On the local-global principle for divisibility in the cohomology of elliptic curves
Hilbert's Tenth problem for function fields of varieties over number fields and $p$-adic fields
Hilbert's tenth problem for fields of rational functions over finite fields
On a dynamical Brauer-Manin obstruction
The decision problem for exponential diophantine equations
Galois properties of torsion points on abelian varieties
Diophantine undecidability of ${\mathbb{C}{}}(t_ 1,t_ 2)$
An analogue for elliptic curves of the Grunwald--Wang example.
Hilbert's tenth problem for algebraic function fields over infinite fields of constants of positive characteristic
Universally and existentially definable subsets of global fields
Diophantine unsolvability over $p$-adic function fields
Universally defining finitely generated subrings of global fields
Local-global principles for constant reductive groups over semi-global fields
Local-global questions for divisibility in commutative algebraic groups
Local-global principle for classical groups over function fields of $p$-adic curves
Hilbert's tenth problem over function fields of positive characteristic not containing the algebraic closure of a finite field
On a dynamical local-global principle in Mordell-Weil type groups
A universal first-order formula defining the ring of integers in a number field
Integrality at a prime for global fields and the perfect closure of global fields of characteristic $p > 2$
Hasse principle for Kummer varieties
On existential definitions of c.e. subsets of rings of functions of characteristic 0
Torsors and rational points
Local-global divisibility of rational points in some commutative algebraic groups
On local-global divisibility by $p^n$ in elliptic curves
Elliptic curve points and Diophantine models of $\mathbb Z$ in large subrings of number fields
A local‐global principle for isogenies of composite degree
On a local-global principle for the divisibility of a rational point by a positive integer
Field Arithmetic
Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers
The Arithmetic of Elliptic Curves
Characterizing integers among rational numbers with a universal-existential formula
A Hasse principle for two dimensional global fields.
Arithmetic over the rings of all algebraic integers.
Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
Undecidability and Definability for the Theory of Global Fields
Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of $\mathbb {Q}$
Irreducibility of polynomials over global fields is diophantine
Model Theory
The Brauer–Grothendieck Group
Quaternion Algebras
A p‐adic analogue of Siegel's theorem on sums of squares
Local-global principles for tori over arithmetic curves
Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields
Undecidability in Number Theory
Central Simple Algebras and Galois Cohomology
Local-global conjectures in the representation theory of finite groups
Hilbert’s Tenth Problem
On counterexamples to local-global divisibility in commutative algebraic groups
Elliptic divisibility sequences and undecidable problems about rational points
Explicit n-descent on elliptic curves, I. Algebra
Diophantine Problems Over Local Fields I
Abelian varieties over finite fields
Definability and decision problems in arithmetic
Defining $\mathbb Z$ in $\mathbb Q$
$\mathbb Q\setminus \mathbb Z$ is diophantine over $\mathbb Q$ with $32$ unknowns
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)

This page was built for publication: A survey of local-global methods for Hilbert's tenth problem

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6611633)