9.1 Rational points

Rational points are the cornerstone of arithmetic geometry, bridging number theory and algebraic geometry. They represent solutions to equations with rational coordinates, providing insights into the arithmetic properties of algebraic varieties over number fields.

This topic explores the definition and properties of rational points, their connection to Diophantine equations, and their behavior on various geometric objects. We'll examine counting methods, height functions, and density properties, as well as applications in cryptography and coding theory.

Definition of rational points

• Rational points form a fundamental concept in arithmetic geometry connecting algebraic geometry with number theory
• Study of rational points provides insights into the arithmetic properties of algebraic varieties defined over number fields
• Exploration of rational points leads to deep connections between geometry, algebra, and number theory

Coordinates in rational numbers

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• Points with coordinates expressible as fractions of integers ($\frac{a}{b}$ where a and b are integers and b ≠ 0)
• Represent solutions to equations with coefficients in the field of rational numbers $\mathbb{Q}$
• Finite representation allows for exact computations without rounding errors
• Closed under addition, subtraction, multiplication, and division (except division by zero)

Rational points on curves

• Solutions to polynomial equations in two variables with rational coefficients
• Studied extensively in the context of elliptic curves and higher genus curves
• Mordell's conjecture (now Faltings' theorem) states that curves of genus > 1 have finitely many rational points
• Rational points on curves often form a finitely generated abelian group (Mordell-Weil theorem for elliptic curves)

Rational points on surfaces

• Solutions to polynomial equations in three variables with rational coefficients
• Classification more complex than for curves due to higher dimensionality
• K3 surfaces and abelian surfaces are particularly important in arithmetic geometry
• Density of rational points on surfaces relates to the Zariski topology and weak approximation

Diophantine equations

• Diophantine equations form the historical foundation of the study of rational points
• These equations bridge number theory and algebraic geometry, leading to many deep conjectures
• Understanding Diophantine equations has practical applications in cryptography and coding theory

Connection to rational points

• Diophantine equations seek integer or rational solutions to polynomial equations
• Rational points on varieties correspond to solutions of Diophantine equations
• Studying rational points geometrically often provides insights into Diophantine problems
• Techniques from algebraic geometry applied to solve Diophantine equations (resolution of singularities)

Types of Diophantine equations

• Linear Diophantine equations ($ax + by = c$ where a, b, c are integers)
• Quadratic equations (Pell's equation $x^2 - dy^2 = 1$)
• Higher degree equations (Fermat's Last Theorem $x^n + y^n = z^n$)
• Exponential Diophantine equations involve variables in exponents ($a^x = y^b + 1$)

Historical significance

• Ancient Greek mathematicians studied Diophantine equations (Pythagorean triples)
• Fermat's Last Theorem remained unsolved for over 350 years until Andrew Wiles' proof in 1995
• Led to development of algebraic number theory and arithmetic geometry
• Hilbert's tenth problem on algorithmic solvability of Diophantine equations proved undecidable

Rational points on elliptic curves

• Elliptic curves represent a central object of study in arithmetic geometry and number theory
• Rational points on elliptic curves have applications in cryptography and coding theory
• Understanding rational points on elliptic curves led to breakthroughs in solving Diophantine equations

Group structure

• Rational points on an elliptic curve form an abelian group under the chord-and-tangent method
• Group law defined geometrically: three collinear points sum to zero in the group
• Identity element is the point at infinity
• Group structure allows for efficient algorithms for point addition and scalar multiplication

Mordell-Weil theorem

• States that the group of rational points on an elliptic curve over a number field is finitely generated
• Decomposes the group into a direct sum of a finite torsion subgroup and a free abelian group of finite rank
• Rank of the elliptic curve defined as the rank of the free part of the Mordell-Weil group
• Determining the rank and generators of the Mordell-Weil group remains a challenging computational problem

Torsion points

• Points of finite order in the group of rational points on an elliptic curve
• Mazur's theorem bounds the possible torsion subgroups for elliptic curves over $\mathbb{Q}$
• Torsion points play a crucial role in the theory of elliptic curve cryptography
• Merel's theorem generalizes Mazur's result to elliptic curves over arbitrary number fields

Counting rational points

• Counting rational points connects arithmetic geometry to analytic number theory
• Techniques for counting rational points often involve zeta functions and L-functions
• Understanding the distribution of rational points leads to insights about the arithmetic of varieties

Local-global principle

• Relates the existence of global solutions (over $\mathbb{Q}$) to local solutions (over $\mathbb{Q}_p$ and $\mathbb{R}$)
• Originated with Hasse's work on quadratic forms and elliptic curves
• Successful for many classes of equations (quadratic forms, genus 0 curves)
• Fails for some higher genus curves and higher-dimensional varieties

Hasse principle

• States that a polynomial equation has a rational solution if and only if it has a real solution and a p-adic solution for all primes p
• Holds for quadratic forms in at least three variables (Hasse-Minkowski theorem)
• Applies to genus 0 curves and some elliptic curves
• Failure of the Hasse principle led to the development of the Brauer-Manin obstruction

Counterexamples to Hasse principle

• Selmer's example: the curve $3x^3 + 4y^3 + 5z^3 = 0$ has solutions in all local fields but no rational solutions
• Genus 1 curves (elliptic curves) can violate the Hasse principle due to non-trivial Tate-Shafarevich groups
• Higher genus curves and higher-dimensional varieties provide more complex counterexamples
• Understanding and classifying counterexamples remains an active area of research in arithmetic geometry

Height of rational points

• Height functions provide a way to measure the complexity of rational points
• Crucial tool in the study of Diophantine equations and arithmetic geometry
• Allows for finiteness results and effective bounds on rational points

Naive height

• For a rational number $\frac{a}{b}$ in lowest terms, defined as $H(\frac{a}{b}) = \max(|a|, |b|)$
• Extends to points in projective space by taking the maximum of the heights of the coordinates
• Provides a simple measure of the size or complexity of a rational point
• Useful for computational purposes but lacks some desirable theoretical properties

Logarithmic height

• Defined as $h(\frac{a}{b}) = \log \max(|a|, |b|)$ for a rational number in lowest terms
• Extends to points in projective space and abelian varieties
• Satisfies better functorial properties than the naive height
• Plays a crucial role in the formulation of many conjectures in arithmetic geometry (Lang's height conjecture)

Northcott's theorem

• States that for any positive real number B, there are only finitely many rational points with height less than B
• Implies that rational points can be enumerated in order of increasing height
• Fundamental result in the theory of heights and Diophantine geometry
• Generalizes to number fields and provides a basis for effective results in Diophantine approximation

Density of rational points

• Density of rational points provides information about the arithmetic nature of varieties
• Connects to questions of approximation and distribution of rational points
• Important in understanding the arithmetic of higher-dimensional varieties

Zariski density

• Rational points are Zariski dense if their closure in the Zariski topology is the entire variety
• Conjectured to hold for many classes of varieties (uniruled varieties, K3 surfaces with infinite automorphism group)
• Implies the existence of many rational points but does not guarantee their uniform distribution
• Related to the potential density problem for varieties over number fields

Weak approximation

• Property that rational points are dense in the product of all completions of the base field
• Holds for projective space and many rational varieties
• Fails for some elliptic curves and higher genus curves
• Obstruction to weak approximation often described by the Brauer-Manin obstruction

Strong approximation

• Stronger property than weak approximation, involving adelic points
• Holds for simply connected linear algebraic groups (Kneser-Platonov theorem)
• Fails for abelian varieties and many other classes of varieties
• Has applications in the study of integral points on varieties

Rational points in arithmetic geometry

• Rational points form a central object of study in modern arithmetic geometry
• Connects various areas of mathematics including number theory, algebraic geometry, and complex analysis
• Many deep conjectures in arithmetic geometry revolve around the behavior of rational points

Faltings' theorem

• Formerly known as Mordell's conjecture, proved by Gerd Faltings in 1983
• States that a curve of genus greater than 1 over a number field has only finitely many rational points
• Revolutionized the study of rational points on higher genus curves
• Proof uses sophisticated techniques from arithmetic geometry and Arakelov theory

abc conjecture

• Fundamental conjecture in number theory with far-reaching consequences
• States a relationship between the product of distinct prime factors of three coprime integers a, b, c satisfying a + b = c
• Implies many other important results (Fermat's Last Theorem for sufficiently large exponents)
• Proposed proof by Shinichi Mochizuki using Inter-universal Teichmüller theory remains controversial

Birch and Swinnerton-Dyer conjecture

• Relates the rank of the Mordell-Weil group of an elliptic curve to the order of vanishing of its L-function at s = 1
• One of the seven Millennium Prize Problems
• Partial results known (agreement of ranks in some cases, p-adic analogues)
• Has implications for the study of rational points on elliptic curves and their distribution

Computational aspects

• Computational methods play a crucial role in the study of rational points
• Algorithms for finding and counting rational points inform theoretical developments
• Computational experiments often lead to new conjectures and insights in arithmetic geometry

Point searching algorithms

• Methods for finding rational points on varieties (elliptic curves, higher genus curves, surfaces)
• Include naive search methods, sieving techniques, and more sophisticated approaches
• Lattice-based methods (LLL algorithm) used for finding small height solutions
• Descent methods exploit the group structure of elliptic curves to find rational points

Rational point databases

• Collections of known rational points on various types of varieties
• Include databases of elliptic curves (Cremona's tables) and higher genus curves
• Serve as testbeds for conjectures and computational methods
• Collaborative efforts in the mathematical community to expand and maintain these databases

Complexity of rational point problems

• Determining the existence of rational points on varieties often NP-hard or undecidable
• Counting rational points of bounded height has varying complexity depending on the type of variety
• Algorithms for elliptic curves over finite fields have polynomial time complexity (Schoof's algorithm)
• Complexity considerations inform the choice of curves for use in cryptography and coding theory

Applications of rational points

• Study of rational points has numerous practical applications beyond pure mathematics
• Insights from arithmetic geometry inform the development of secure cryptographic systems
• Understanding rational points crucial for advances in coding theory and communication

Cryptography

• Elliptic curve cryptography (ECC) uses the difficulty of the discrete logarithm problem on elliptic curves
• Rational points on elliptic curves over finite fields form the basis for ECC key generation and encryption
• Post-quantum cryptography explores the use of higher genus curves and higher-dimensional varieties
• Counting points on elliptic curves over finite fields crucial for assessing cryptographic security

Coding theory

• Algebraic geometry codes use rational points on curves over finite fields to construct error-correcting codes
• Goppa codes based on rational points on algebraic curves achieve good parameters
• Rational points on higher-dimensional varieties used in constructing locally decodable codes
• Understanding the distribution of rational points informs the design of efficient coding schemes

Diophantine approximation

• Study of how well real numbers can be approximated by rational numbers
• Rational points on varieties used to construct good Diophantine approximations
• Simultaneous approximation problems related to rational points on higher-dimensional varieties
• Applications in dynamical systems and ergodic theory (Littlewood conjecture)

Open problems

• Many fundamental questions about rational points remain open
• These problems drive research in arithmetic geometry and related fields
• Solutions to these problems would have far-reaching consequences in mathematics

Uniformity conjectures

• Propose uniform bounds on the number of rational points on varieties of a given type
• Uniform boundedness conjecture for torsion points on elliptic curves (proved by Merel)
• Uniform bound conjecture for the number of rational points on curves of genus > 1 (open)
• Relate to questions of effectivity in Faltings' theorem and other finiteness results

Lang's conjectures

• Series of conjectures by Serge Lang on the distribution of rational points
• Lang's height conjecture relates the canonical height of rational points to the degree of the variety
• Vojta's conjecture generalizes Lang's conjectures to arithmetic varieties
• Implications for the distribution of rational and integral points on varieties

Bombieri-Lang conjecture

• Generalizes Faltings' theorem to higher-dimensional varieties
• States that for a variety of general type, rational points are not Zariski dense
• Would imply many other important conjectures in arithmetic geometry
• Partial results known for some classes of varieties, but general case remains wide open