🔢Arithmetic Geometry Unit 9 – Algebraic Varieties: Arithmetic Properties

Key Concepts and Definitions

• Algebraic varieties defined as geometric objects that can be described by polynomial equations
• Affine varieties consist of the set of solutions to a system of polynomial equations in affine space
• Projective varieties defined as the set of solutions to a system of homogeneous polynomial equations in projective space
• Rational points on a variety are points whose coordinates are rational numbers
• Integral points on a variety are points whose coordinates are integers
• Height functions measure the complexity or size of rational points on a variety
  • Logarithmic height is a common choice of height function
  • Weil height is another important height function used in arithmetic geometry
• Mordell-Weil theorem states that the group of rational points on an elliptic curve is finitely generated

Historical Context and Development

• Diophantine equations, polynomial equations with integer coefficients, have been studied since ancient times (Diophantus of Alexandria)
• Fermat's Last Theorem, stating that $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$, was a famous unsolved problem for centuries
• Hilbert's tenth problem, asking for an algorithm to determine if a Diophantine equation has integer solutions, was shown to be unsolvable (Matiyasevich, Davis, Putnam, Robinson)
• Mordell-Weil theorem, proving that the group of rational points on an elliptic curve is finitely generated, was a major breakthrough (Mordell, Weil)
• Faltings' theorem, proving the Mordell conjecture that curves of genus greater than 1 have only finitely many rational points, was another significant development
• Wiles' proof of Fermat's Last Theorem, using techniques from arithmetic geometry and modular forms, was a landmark achievement (Wiles, Taylor)

Algebraic Varieties: Basic Structure

• Algebraic varieties can be studied locally using affine charts or globally using projective embeddings
• Smooth varieties are varieties without singularities, where the tangent space at each point has the same dimension as the variety
• Singular varieties have points where the tangent space has higher dimension than the variety
• Dimension of a variety is the maximum dimension of a tangent space at a non-singular point
• Degree of a projective variety is the number of intersection points with a generic linear subspace of complementary dimension
• Genus of a smooth projective curve is a measure of its complexity, related to the degree and singularities
  • Genus 0 curves are rational curves, isomorphic to the projective line
  • Genus 1 curves are elliptic curves, with a group structure on their rational points

Arithmetic Properties of Varieties

• Rational points on varieties can be studied using height functions and other arithmetic invariants
• Density of rational points on a variety measures how many rational points of bounded height exist
• Zariski density of rational points means that the rational points are dense in the Zariski topology
• Siegel's theorem states that affine curves of genus greater than 0 have only finitely many integral points
• Manin-Mumford conjecture, now a theorem, characterizes the distribution of torsion points on abelian varieties
• Vojta's conjectures relate the distribution of rational points on varieties to value distribution theory in complex analysis
• Batyrev-Manin conjecture predicts the asymptotic behavior of rational points of bounded height on Fano varieties

Computational Techniques

• Diophantine approximation methods, such as the LLL algorithm, can be used to find rational points on varieties
• Elliptic curve method uses the group structure of elliptic curves to solve certain Diophantine equations
• Descent techniques, such as 2-descent and Selmer groups, can be used to study the rational points on elliptic curves and other abelian varieties
• Chabauty's method uses p-adic analysis to study rational points on curves of genus greater than 1
• Brauer-Manin obstruction is a cohomological obstruction to the existence of rational points on varieties
• Hasse principle, when satisfied, implies that a variety has a rational point if and only if it has a point over every completion of the base field
  • Counterexamples to the Hasse principle, such as the Selmer curve, show that local-global principles can fail

Applications in Number Theory

• Elliptic curves are used in elliptic curve cryptography, providing a secure way to exchange keys over insecure channels
• Modular curves, such as $X_0(N)$, parameterize elliptic curves with certain torsion structures and play a crucial role in the proof of Fermat's Last Theorem
• Shimura varieties are higher-dimensional analogues of modular curves, related to the Langlands program and automorphic forms
• Diophantine equations arising from Fermat-type equations, such as the Fermat-Catalan conjecture, can be studied using techniques from arithmetic geometry
• ABC conjecture, if true, would have significant implications for the distribution of rational points on curves and the behavior of Diophantine equations
• Birch and Swinnerton-Dyer conjecture relates the rank of the group of rational points on an elliptic curve to the behavior of its L-function, connecting arithmetic geometry to analytic number theory

Advanced Topics and Open Problems

• Grothendieck's section conjecture predicts that rational points on a hyperbolic curve correspond to splittings of its fundamental exact sequence
• Minhyong Kim's non-abelian Chabauty method uses ideas from the section conjecture and p-adic Hodge theory to study rational points on higher-genus curves
• Langlands program seeks to unify various areas of mathematics, including arithmetic geometry, representation theory, and automorphic forms
• Tate conjecture relates the Galois action on the étale cohomology of a variety to the existence of algebraic cycles, connecting arithmetic geometry to algebraic cycles and motives
• Hodge conjecture predicts that certain cohomology classes on complex algebraic varieties are algebraic, relating Hodge theory to algebraic cycles
• Birational anabelian geometry studies the extent to which the arithmetic and geometric properties of a variety are determined by its étale fundamental group
• Dynamical systems on varieties, such as rational maps and automorphisms, can exhibit complex behavior and connections to arithmetic geometry

Real-World Examples and Case Studies

• Cryptographic protocols based on elliptic curves (Elliptic Curve Diffie-Hellman, Elliptic Curve Digital Signature Algorithm) are widely used in secure communication and digital signatures
• Fermat's Last Theorem inspired centuries of mathematical research and led to the development of new techniques in arithmetic geometry and number theory
• Solving Diophantine equations has applications in computer science, such as in integer programming and optimization problems
• Studying rational points on varieties can help understand the behavior of solutions to systems of polynomial equations, which arise in various fields (robotics, computer vision, economics)
• Elliptic curve factorization method is a fast integer factorization algorithm that uses elliptic curves to find factors of large composite numbers
• Modular forms and elliptic curves have been used to prove significant results in number theory, such as the Taniyama-Shimura conjecture and the Sato-Tate conjecture
• Arithmetic geometry techniques have been applied to the study of Diophantine equations arising in mathematical physics, such as the Painlevé equations and the Navier-Stokes equations