12.7 Dynamical Manin-Mumford conjecture

The Dynamical Manin-Mumford conjecture explores the intersection of preperiodic points with subvarieties in algebraic dynamics. It extends classical results about torsion points on abelian varieties to broader dynamical contexts, bridging number theory and algebraic geometry.

This conjecture provides insights into the distribution of special points on algebraic varieties. It serves as a testing ground for new techniques in arithmetic geometry and algebraic dynamics, offering a fresh perspective on the behavior of iterated polynomial maps.

Origins and context

• Arithmetic geometry bridges number theory and algebraic geometry, providing a framework for studying arithmetic properties of geometric objects
• Dynamical Manin-Mumford conjecture emerges from this intersection, focusing on the behavior of certain points under iteration of polynomial maps
• Builds upon classical results in diophantine geometry, extending them to dynamical systems

Historical development

Top images from around the web for Historical development

• 

  Number of Points on Curves: a Conjecture of Mazur - TIB AV-Portal View original

  Is this image relevant?

• 

  On the Mumford–Tate conjecture for hyperkähler varieties | SpringerLink View original

  Is this image relevant?

• 

  Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of ... View original

  Is this image relevant?

• 

  Number of Points on Curves: a Conjecture of Mazur - TIB AV-Portal View original

  Is this image relevant?

• 

  On the Mumford–Tate conjecture for hyperkähler varieties | SpringerLink View original

  Is this image relevant?

1 of 3

Top images from around the web for Historical development

• 

  Number of Points on Curves: a Conjecture of Mazur - TIB AV-Portal View original

  Is this image relevant?

• 

  On the Mumford–Tate conjecture for hyperkähler varieties | SpringerLink View original

  Is this image relevant?

• 

  Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of ... View original

  Is this image relevant?

• 

  Number of Points on Curves: a Conjecture of Mazur - TIB AV-Portal View original

  Is this image relevant?

• 

  On the Mumford–Tate conjecture for hyperkähler varieties | SpringerLink View original

  Is this image relevant?

1 of 3

• Originates from Barry Mazur's observations in the 1990s about potential connections between torsion points on elliptic curves and periodic points in dynamical systems
• Formalized by Lucien Szpiro and Thomas Tucker in 2003, drawing parallels with the classical Manin-Mumford conjecture
• Gained significant attention after Zhang's work on equidistribution of small points in 2006

Relation to Manin-Mumford conjecture

• Classical Manin-Mumford conjecture deals with torsion points on abelian varieties embedded in their Jacobians
• Dynamical version replaces torsion points with preperiodic points of a polynomial map
• Both conjectures explore the intersection of special points (torsion or preperiodic) with subvarieties

Importance in arithmetic geometry

• Provides a bridge between classical diophantine geometry and modern dynamical systems theory
• Offers new insights into the distribution of special points on algebraic varieties
• Serves as a testing ground for developing new techniques in arithmetic geometry and algebraic dynamics

Dynamical systems fundamentals

• Dynamical systems theory studies the long-term behavior of evolving systems, crucial for understanding the Dynamical Manin-Mumford conjecture
• In arithmetic geometry, dynamical systems often involve iterating polynomial or rational maps over number fields or function fields
• Understanding these fundamentals lays the groundwork for exploring more complex phenomena in algebraic dynamics

Discrete dynamical systems

• Consist of a set X (state space) and a function f: X → X (evolution rule)
• Iteration of f generates orbits: x, f(x), f(f(x)), f(f(f(x))), ...
• In arithmetic contexts, X often represents points on an algebraic variety, and f a polynomial or rational map

Fixed points and periodic orbits

• Fixed points satisfy f(x) = x, remaining unchanged under iteration
• Periodic points of period n satisfy f^n(x) = x, where f^n denotes n-fold composition of f
• Play a crucial role in understanding the global behavior of dynamical systems
• In algebraic settings, fixed and periodic points often have special arithmetic properties

Attractors and repellers

• Attractors draw nearby points towards them under iteration (stable fixed points)
• Repellers push nearby points away under iteration (unstable fixed points)
• Basin of attraction consists of all points that converge to a given attractor
• In complex dynamics, Julia sets form the boundaries between basins of attraction

Algebraic dynamics

• Combines techniques from dynamical systems theory with algebraic geometry and number theory
• Studies iteration of polynomial or rational maps over algebraically closed fields or number fields
• Central to formulating and approaching the Dynamical Manin-Mumford conjecture

Polynomial and rational maps

• Polynomial maps f: A^n → A^n defined by polynomials in n variables
• Rational maps f: P^n → P^n given by homogeneous polynomials of the same degree
• Degree of a map crucial for understanding its dynamical behavior
• Examples include quadratic maps like f(z) = z^2 + c, central to complex dynamics

Height functions

• Measure arithmetic complexity of points in projective space over number fields
• Logarithmic height h(P) for P = [x_0 : ... : x_n] in P^n(Q) defined as: $h(P) = \log(\max\{|x_0|, ..., |x_n|\})$
• Extend to number fields using local absolute values
• Play key role in formulating finiteness statements in arithmetic geometry

Canonical heights

• Refinement of standard height functions adapted to specific dynamical systems
• For a map f of degree d, canonical height ĥ_f satisfies: $ĥ_f(f(P)) = d · ĥ_f(P)$
• Capture long-term arithmetic behavior of points under iteration
• Vanish precisely on preperiodic points, crucial for Dynamical Manin-Mumford conjecture

Formulation of the conjecture

• Dynamical Manin-Mumford conjecture synthesizes ideas from algebraic dynamics and diophantine geometry
• Addresses the intersection of preperiodic points with subvarieties in a dynamical setting
• Generalizes classical results about torsion points on abelian varieties to broader dynamical contexts

Statement of the conjecture

• Let X be an algebraic variety over a number field K
• Consider a dominant rational map f: X → X defined over K
• Let V be a subvariety of X not contained in the indeterminacy locus of f
• Conjecture: The Zariski closure of the set of preperiodic points of f contained in V is a finite union of preperiodic subvarieties of V

Key components and terms

• Preperiodic points eventually enter a periodic orbit under iteration of f
• Zariski closure takes the smallest algebraic set containing a given set of points
• Preperiodic subvarieties mapped to themselves by some iterate of f
• Dominant map ensures the image is dense in the target space

Variations and generalizations

• Polarized version considers pairs (X, L) where L is an ample line bundle
• Function field analogue replaces number fields with function fields of curves
• Adelic formulation uses adelic metrics on line bundles instead of height functions
• p-adic versions study dynamics over p-adic fields or their algebraic closures

Preperiodic points

• Central objects in the study of algebraic dynamics and the Dynamical Manin-Mumford conjecture
• Generalize the notion of torsion points on abelian varieties to arbitrary dynamical systems
• Exhibit special arithmetic properties, often with bounded height in suitable coordinates

Definition and properties

• Point P is preperiodic for f if there exist integers m ≥ 0, n > 0 such that f^m(P) = f^(m+n)(P)
• Includes both periodic points (m = 0) and points that eventually become periodic (m > 0)
• Form a set PrePer(f) that is invariant under f and its iterates
• In many cases, PrePer(f) is a set of algebraic numbers when f is defined over Q

Finiteness conditions

• Northcott property: for maps f of degree > 1, PrePer(f) ∩ X(K) is finite for any number field K
• Height bounds: preperiodic points characterized by bounded canonical height
• Local-global principles: local preperiodicity at finitely many places often implies global preperiodicity

Examples in various systems

• Multiplication-by-2 map on P^1: preperiodic points are 0, ∞, and roots of unity
• Chebyshev polynomials: preperiodic points related to algebraic integers in cyclotomic fields
• Lattès maps: arise from elliptic curves, all preperiodic points are algebraic

Algebraic geometry connections

• Dynamical Manin-Mumford conjecture draws inspiration from classical results in algebraic geometry
• Explores analogies between dynamics on general varieties and the theory of abelian varieties
• Provides a framework for understanding special points in both static and dynamic settings

Abelian varieties

• Projective algebraic groups, generalizing elliptic curves to higher dimensions
• Possess a rich theory of torsion points and their distribution
• Jacobians of curves serve as prototypical examples of abelian varieties
• Mordell-Weil theorem establishes finiteness of rational points of bounded height

Torsion points

• Elements of finite order in the group law of an abelian variety
• Analogous to roots of unity in the multiplicative group
• Dense in the complex topology but form a discrete set in the Zariski topology
• Raynaud's theorem (proving Manin-Mumford) shows torsion points lying on a subvariety form a finite union of translates of abelian subvarieties

Mordell-Lang conjecture vs Dynamical MM

• Mordell-Lang deals with intersections of subgroups of abelian varieties with subvarieties
• Dynamical MM replaces group structure with action of a rational map
• Both conjectures predict "unlikely intersections" are governed by underlying algebraic structure
• Techniques developed for Mordell-Lang (e.g., equidistribution) inspire approaches to Dynamical MM

Proof techniques and approaches

• Dynamical Manin-Mumford conjecture requires a diverse set of tools from various areas of mathematics
• Combines methods from ergodic theory, p-adic analysis, and arithmetic geometry
• Successful approaches often involve translating dynamical problems into questions about distribution of points

Equidistribution methods

• Study limiting distribution of orbits or special points as complexity increases
• Utilize Berkovich spaces to work with non-Archimedean fields uniformly
• Apply potential theory and capacity theory to analyze canonical measures
• Key tool: Yuan's equidistribution theorem for points of small height

p-adic analysis

• Exploit ultrametric properties of p-adic absolute values
• Study dynamics over p-adic fields and their algebraic closures
• Use p-adic Fatou and Julia sets to understand local behavior of iterates
• Apply Skolem-Mahler-Lech theorem to study p-adic linear recurrence sequences

Arithmetic equidistribution theory

• Combines ideas from dynamics, ergodic theory, and arithmetic geometry
• Studies distribution of Galois orbits of special points (e.g., preperiodic points)
• Utilizes adelic metrics and height functions to quantify arithmetic complexity
• Key result: Szpiro-Ullmo-Zhang equidistribution theorem for small points

Known results and partial proofs

• While the full Dynamical Manin-Mumford conjecture remains open, significant progress has been made in various cases
• Results often combine techniques from arithmetic geometry, complex dynamics, and p-adic analysis
• Partial proofs provide insights into potential strategies for tackling the general case

Special cases and examples

• Polynomial maps on P^1: conjecture proven for unicritical polynomials (Baker-DeMarco)
• Split polynomial maps: conjecture established for products of one-dimensional maps (Ghioca-Tucker-Zieve)
• Toric varieties: results known for monomial maps (Ghioca-Tucker-Zhang)
• Abelian varieties: conjecture true for endomorphisms preserving a polarization (Ghioca-Tucker-Zhang)

Progress in dimension one

• Morton-Silverman conjecture: uniform boundedness of preperiodic points for maps of degree d > 1
• Masser-Zannier theorem: finiteness of simultaneous preperiodic points in certain families
• Pink-Roessler approach: using unlikely intersections in products of modular curves
• Heigher-Silverman result: Dynamical MM for quadratic polynomials over function fields

Higher-dimensional challenges

• Lack of a good theory of canonical heights for general rational maps
• Difficulty in understanding dynamics on varieties without additional structure (group law)
• Interplay between arithmetic and geometric degree in higher dimensions
• Need for new equidistribution results for higher-dimensional varieties

Applications and implications

• Dynamical Manin-Mumford conjecture connects various areas of mathematics, leading to unexpected applications
• Provides insights into the distribution of special points in both arithmetic and geometric settings
• Influences development of new techniques in algebraic geometry and number theory

Number theory connections

• Provides new approaches to studying rational points on varieties
• Offers insights into the distribution of algebraic numbers of small height
• Connects to the theory of unlikely intersections in arithmetic geometry
• Influences study of Galois representations associated to dynamical systems

Cryptography applications

• Preperiodic points in certain dynamical systems used in public-key cryptography
• Understanding distribution of these points crucial for assessing security of cryptosystems
• Dynamical analogues of elliptic curve cryptography based on rational maps
• Post-quantum cryptography explores higher-dimensional dynamical systems

Broader impact in mathematics

• Stimulates development of new tools in arithmetic geometry and dynamics
• Influences study of arithmetic properties of moduli spaces of dynamical systems
• Provides new perspectives on classical problems in diophantine geometry
• Encourages cross-pollination of ideas between complex dynamics and arithmetic

Open problems and future directions

• Dynamical Manin-Mumford conjecture remains a central open problem in arithmetic dynamics
• Ongoing research explores various generalizations and related conjectures
• Future work likely to involve developing new tools at the intersection of dynamics and arithmetic geometry

Unsolved cases

• General case of Dynamical MM for rational maps on projective spaces
• Uniform bounds on number of preperiodic points (dynamical analogue of Uniform Boundedness Conjecture)
• Dynamical André-Oort conjecture for more general spaces than P^1 x P^1
• Effective versions of known results, providing explicit bounds and algorithms

Related conjectures

• Dynamical Bogomolov conjecture: characterization of subvarieties with dense small points
• Pink-Zilber conjecture: unlikely intersections in mixed Shimura varieties
• Dynamical Mordell-Lang conjecture: describes orbit intersections with subvarieties
• Dynamical Lehmer conjecture: lower bounds on heights of non-preperiodic points

Potential generalizations

• Non-algebraic dynamical systems, such as transcendental entire functions
• Dynamics over more general fields (function fields, p-adic fields)
• Analogues for group actions beyond iteration of a single map
• Connections to ergodic theory and measure-preserving dynamical systems