9.7 Arithmetic threefolds
7 min read•august 21, 2024
Arithmetic threefolds blend and , offering insights into three-dimensional spaces with both geometric and arithmetic properties. These objects, defined over number fields or finite fields, exhibit rich structures including singularities, fibrations, and .
are powerful tools for studying arithmetic threefolds, encoding information about geometry and point counts over finite fields. Rational points on threefolds are a central focus, with density theorems and height bounds providing key insights into their distribution and properties.
Definition of arithmetic threefolds
- Arithmetic threefolds form a crucial area of study in Arithmetic Geometry combines algebraic geometry and number theory
- These mathematical objects provide insights into the interplay between geometric structures and arithmetic properties in three-dimensional spaces
Key characteristics
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- Three-dimensional algebraic varieties defined over number fields or finite fields
- Possess both geometric and arithmetic properties allowing for deep analysis
- Exhibit rich structure including singularities, fibrations, and rational points
- Zariski topology used to study local properties and global behavior
- Coherent sheaves provide tools for analyzing vector bundles and cohomology
Historical context
- Emerged from the intersection of algebraic geometry and number theory in the mid-20th century
- Developed as a natural extension of arithmetic surfaces to higher dimensions
- Influenced by works of André Weil, Alexander Grothendieck, and Pierre Deligne
- Gained prominence through connections to the Langlands program and
- Continues to evolve with advancements in cohomology theories and birational geometry
Arithmetic vs geometric threefolds
- Arithmetic threefolds focus on number-theoretic properties while geometric threefolds emphasize topological and analytic aspects
- Both types of threefolds contribute to our understanding of higher-dimensional algebraic varieties in Arithmetic Geometry
Structural differences
- Arithmetic threefolds defined over number fields or finite fields while geometric threefolds over algebraically closed fields
- Arithmetic versions consider Galois action and reduction modulo primes
- Geometric threefolds emphasize complex analytic properties (Hodge theory)
- Arithmetic threefolds involve consideration of integral models and good reduction
- Geometric counterparts focus on deformation theory and moduli spaces
Conceptual distinctions
- Arithmetic approach emphasizes number-theoretic invariants (zeta functions, )
- Geometric perspective prioritizes topological and analytic properties (fundamental group, Chern classes)
- Arithmetic threefolds study rational points and their distribution
- Geometric threefolds focus on complex geometry and Kähler structures
- Arithmetic versions consider reduction modulo different primes while geometric do not
Zeta functions of threefolds
- Zeta functions serve as powerful tools for studying arithmetic properties of threefolds
- These functions encode deep information about the geometry and arithmetic of the variety
Hasse-Weil zeta function
- Generalizes the Riemann zeta function to algebraic varieties over number fields
- Defined as an Euler product over all primes of good reduction
- Encodes information about the number of points on the threefold over finite fields
- Conjectured to have meromorphic continuation and functional equation
- Zeros and poles of the zeta function relate to cohomological information
L-functions and modularity
- L-functions associated to cohomology groups of the threefold
- Modularity conjectures predict relationships between L-functions and automorphic forms
- describes the distribution of Frobenius eigenvalues
- relates L-function behavior to rational points
- Modularity of elliptic curves over Q extends to certain threefolds (Calabi-Yau)
Rational points on threefolds
- Rational points form a central object of study in arithmetic geometry
- Understanding their distribution and properties provides insights into the arithmetic nature of threefolds
Density theorems
- Manin-Mumford conjecture for subvarieties of abelian varieties
- on rational points of varieties of general type
- Potential density for certain classes of rationally connected varieties
- Campana's conjecture relating rational points to the fundamental group
- Bombieri-Lang conjecture predicting scarcity of rational points on varieties of general type
Height bounds
- ensures finiteness of rational points of bounded height
- Vojta's height inequality relates heights to discriminants and regulators
- (Mordell conjecture) bounds heights on curves of genus > 1
- Heath-Brown's determinant method for bounding rational points
- Batyrev-Manin conjecture predicts asymptotic behavior of counting functions
Birational geometry of threefolds
- Birational geometry studies properties invariant under birational transformations
- This field provides powerful tools for classifying and understanding the structure of threefolds
Minimal model program
- Aims to find a "simplest" birational model for each threefold
- provides framework for contractions and flips
- Termination of flips proved for threefolds (not known in higher dimensions)
- Minimal models exist for varieties of non-negative Kodaira dimension
- relates minimal models to Iitaka fibrations
Fano varieties
- Fano threefolds have ample anticanonical bundle
- Classification of smooth Fano threefolds completed by Mori and Mukai
- Bounded by degree and Picard number
- Weak Fano threefolds allow for mild singularities
- Play crucial role in birational geometry and Mori fiber spaces
Cohomology theories for threefolds
- Cohomology theories provide powerful invariants for studying threefolds
- These theories connect geometric properties to algebraic and arithmetic structures
Étale cohomology
- Generalizes singular cohomology to algebraic varieties in any characteristic
- arise from groups
- Weil conjectures proved using étale cohomology (Deligne)
- -adic cohomology provides a good notion of Betti numbers in positive characteristic
- Poincaré duality and Künneth formula hold in étale cohomology
Crystalline cohomology
- Developed by Grothendieck as -adic analogue of de Rham cohomology
- Provides cohomology theory for varieties in characteristic
- Dieudonné module structure encodes information about -divisible groups
- Crystalline Frobenius action relates to zeta functions
- De Rham-Witt complex provides explicit construction of
Arithmetic intersection theory
- Arithmetic extends classical intersection theory to arithmetic settings
- This theory provides tools for studying heights and intersection numbers on arithmetic varieties
Arakelov geometry
- Combines algebraic geometry with Hermitian geometry on complex fibers
- Arithmetic Chow groups define classes of arithmetic cycles
- Arithmetic Riemann-Roch theorem generalizes classical version
- Faltings' heights defined using Arakelov theory
- Bost-Gillet-Soulé theorem on arithmetic Hilbert-Samuel formula
Height pairings
- on abelian varieties
- Generalizes to cycles of complementary dimension on arithmetic varieties
- Relates to L-functions via Beilinson-Bloch conjectures
- Arakelov intersection numbers provide arithmetic analogues of intersection numbers
- Gillet-Soulé arithmetic Hodge index theorem
Galois representations
- Galois representations encode deep arithmetic information about threefolds
- These representations connect geometric properties to Galois theory and number theory
Étale fundamental group
- Generalizes classical fundamental group to algebraic varieties
- Profinite completion of topological fundamental group in characteristic 0
- Grothendieck's anabelian geometry program studies reconstruction from fundamental groups
- Section conjecture relates rational points to sections of fundamental group exact sequence
- Tame fundamental group considers coverings with controlled ramification
Galois action on cohomology
- Galois group acts on étale cohomology groups
- -adic representations arise from this action
- Tate twists describe scaling of Galois action
- Fontaine's -adic Hodge theory relates various -adic cohomology theories
- Motivic Galois group encapsulates Galois actions on all cohomology theories
Modularity of threefolds
- Modularity connects arithmetic properties of threefolds to automorphic forms
- This area extends classical results on elliptic curves to higher-dimensional varieties
Langlands program connections
- Langlands reciprocity conjecture relates Galois representations to automorphic forms
- Modularity of elliptic curves over Q (Wiles, Taylor-Wiles) as starting point
- Potential modularity results for certain classes of threefolds (Calabi-Yau)
- Fontaine-Mazur conjecture on geometric Galois representations
- Serre's modularity conjecture and its higher-dimensional analogues
Sato-Tate conjecture
- Describes distribution of Frobenius eigenvalues for threefolds over number fields
- Generalized Sato-Tate groups capture symmetries of the Galois representations
- Proved for elliptic curves over totally real fields (L. Clozel, M. Harris, N. Shepherd-Barron, R. Taylor)
- Extends to certain families of higher-dimensional varieties
- Relates to equidistribution of Hecke eigenvalues of modular forms
Applications in cryptography
- Arithmetic threefolds find applications in modern cryptographic systems
- These geometric objects provide rich structures for designing secure encryption schemes
Threefolds in encryption schemes
- Hyperelliptic curve cryptography generalizes elliptic curve methods
- Jacobians of curves of genus 3 provide higher-dimensional analogues
- Kummer surfaces and their higher-dimensional generalizations used in cryptosystems
- Torus-based cryptography utilizes algebraic tori related to threefolds
- Pairing-based cryptography employs Weil and Tate pairings on threefolds
Security considerations
- Discrete logarithm problem in Jacobians of curves
- Index calculus attacks on higher-genus curves
- Point counting algorithms for curves and higher-dimensional varieties
- Schoof's algorithm and its generalizations for efficient point counting
- Quantum algorithms (Shor's algorithm) and post-quantum cryptography considerations
Open problems and conjectures
- Numerous open problems and conjectures drive research in arithmetic geometry of threefolds
- These unresolved questions connect various areas of mathematics and inspire new developments
Birch and Swinnerton-Dyer conjecture
- Relates rank of abelian varieties to behavior of L-functions at s=1
- Millennium Prize Problem with partial results known
- Generalizes to higher-dimensional varieties (Bloch-Kato conjecture)
- Connects arithmetic geometry to analytic number theory and algebraic K-theory
- Refined versions predict precise leading term of L-function at s=1
Bombieri-Lang conjecture
- Predicts that varieties of general type have non-dense set of rational points
- Generalizes Faltings' theorem (Mordell conjecture) to higher dimensions
- Implies finiteness of rational points on surfaces of general type
- Related to Lang's conjecture on entire curves
- Connects to hyperbolicity in complex geometry
Key Terms to Review (35)
Abundance Conjecture: The Abundance Conjecture is a hypothesis in the field of algebraic geometry that suggests a relationship between the algebraic dimension of a projective variety and its Kodaira dimension. It posits that the abundance of a variety, which refers to the number of effective divisors, should reflect its geometric properties, particularly in terms of birational geometry and the classification of algebraic varieties.
Algebraic Geometry: Algebraic geometry is a branch of mathematics that studies the solutions to polynomial equations and their geometric properties. It connects abstract algebra, especially commutative algebra, with geometry, allowing for a deeper understanding of shapes and their equations. This field provides tools to tackle questions about rational solutions, which are significant in various mathematical contexts, such as number theory and complex analysis.
Arakelov geometry: Arakelov geometry is a branch of mathematics that blends algebraic geometry with number theory, focusing on the study of arithmetic varieties, particularly in the context of higher-dimensional spaces. It introduces tools and concepts from both disciplines to analyze properties of schemes over arithmetic fields, enriching the understanding of their geometric structures and providing insights into their arithmetic aspects.
Bertini's Theorem: Bertini's Theorem is a fundamental result in algebraic geometry that asserts the general position of points in projective space, stating that a generic hyperplane section of a projective variety is smooth, provided that the variety itself is irreducible. This theorem has important implications for understanding the properties of varieties, particularly in weak approximation, arithmetic surfaces, and arithmetic threefolds, as it relates to the behavior of these structures under various conditions.
Birch and Swinnerton-Dyer Conjecture: The Birch and Swinnerton-Dyer Conjecture is a major unsolved problem in number theory that relates the number of rational points on an elliptic curve to the behavior of its L-function at a specific point. This conjecture suggests that the rank of an elliptic curve, which measures the size of its group of rational points, is linked to the vanishing order of its L-function at the point s=1.
Calabi-Yau Threefold: A Calabi-Yau threefold is a special type of geometric structure that is a compact, complex manifold of dimension three, characterized by its Ricci-flat metric and holomorphic volume form. These manifolds are significant in string theory and algebraic geometry, serving as potential candidates for the compactification of extra dimensions.
Crystalline cohomology: Crystalline cohomology is a cohomology theory for schemes over a field of positive characteristic, primarily used to study the properties of algebraic varieties in the context of p-adic numbers. It provides insights into the structure of these varieties by connecting their geometric and arithmetic aspects through a framework that incorporates both algebraic and topological methods.
David Mumford: David Mumford is a prominent mathematician known for his work in algebraic geometry, particularly in the areas of modular forms and algebraic curves. His contributions have significantly advanced the understanding of complex tori, modular curves, and other structures relevant to arithmetic geometry.
Descent Theory: Descent theory is a method in algebraic geometry that studies the properties of schemes by relating them to simpler schemes through a process known as descent. It connects local properties of varieties over various base fields to global properties, allowing for a better understanding of rational points and morphisms, which are crucial in different contexts such as the study of abelian varieties, surfaces, and higher-dimensional varieties.
étale cohomology: Étale cohomology is a powerful tool in algebraic geometry that allows for the study of algebraic varieties over fields, especially in relation to their rational points and the geometric properties they possess. It provides a way to compute cohomological invariants of schemes that are not necessarily smooth or projective, enabling a deeper understanding of their structure through a more flexible approach compared to classical cohomology theories.
étale fundamental group: The étale fundamental group is an algebraic structure that generalizes the notion of the fundamental group in topology to the setting of algebraic geometry, particularly for schemes over a field. This group captures the symmetries and covering properties of a scheme, especially in the étale topology, allowing mathematicians to study geometric objects and their morphisms in a more profound way. It serves as a crucial tool in understanding the Galois actions on points of schemes and plays a significant role in arithmetic geometry.
Faltings' Theorem: Faltings' Theorem states that any curve of genus greater than one defined over a number field has only finitely many rational points. This theorem fundamentally connects the geometry of algebraic curves with number theory, revealing deep insights about the distribution of rational solutions on these curves and influencing various areas such as the study of Mordell-Weil groups, modular forms, and arithmetic geometry.
Fano threefold: A Fano threefold is a specific type of algebraic variety that is a three-dimensional projective manifold with positive first Chern class, which implies that it has ample anticanonical divisor. These varieties are important in algebraic geometry due to their rich structure and connections to various geometric concepts such as birational geometry, moduli spaces, and mirror symmetry.
Fano varieties: Fano varieties are a special class of algebraic varieties characterized by having ample anticanonical bundles. This property makes them particularly interesting in the study of algebraic geometry as they exhibit rich geometric structures and have connections to various concepts, including the positivity of curvature and rationality. Their importance extends into different areas of mathematics, such as birational geometry and the theory of moduli spaces.
Galois representations: Galois representations are mathematical objects that encode the action of a Galois group on a vector space, typically associated with algebraic objects like number fields or algebraic varieties. These representations allow for the study of symmetries in arithmetic, relating number theory and geometry through various structures such as modular forms and L-functions.
Hasse-Weil Zeta Function: The Hasse-Weil zeta function is a key analytic tool in number theory and algebraic geometry, which encodes important information about the number of solutions to polynomial equations over finite fields. This function is defined for a variety of algebraic varieties and plays a significant role in understanding the properties of these varieties, particularly in relation to their reduction modulo prime numbers, their connections to class groups, and their arithmetic structure in higher dimensions.
Hodge Conjecture: The Hodge Conjecture is a central statement in algebraic geometry and mathematical physics, positing that certain classes of cohomology groups can be represented by algebraic cycles. It connects the geometry of a projective algebraic variety with its topological properties, asserting that the Hodge classes correspond to the algebraic cycles, which deepens the understanding of the relationship between geometry and topology in higher dimensions.
Intersection Theory: Intersection theory studies how various geometric objects intersect within algebraic geometry, providing a framework for understanding the relationships and dimensions of these intersections. It helps in computing intersection numbers, defining cycles, and analyzing the properties of varieties, which are crucial in understanding various structures such as Jacobian varieties and arithmetic surfaces.
K3 surface: A k3 surface is a special type of complex algebraic surface that has trivial canonical bundle and vanishing first Betti number. These surfaces play a significant role in algebraic geometry due to their rich structure and the connections they have with various areas such as mirror symmetry, moduli theory, and string theory.
L-functions: L-functions are complex analytic functions that arise in number theory, particularly in the study of the distribution of prime numbers and modular forms. These functions generalize the Riemann zeta function and encapsulate deep arithmetic properties, connecting number theory with algebraic geometry and representation theory.
Lang's Conjecture: Lang's Conjecture is a hypothesis in arithmetic geometry that posits certain relationships between algebraic varieties and their rational points. Specifically, it suggests that for a given variety defined over a number field, the set of its rational points should be closely linked to the geometry of the variety and the arithmetic properties of the field. This conjecture bridges concepts from Diophantine geometry and the study of algebraic curves, particularly in contexts involving higher-dimensional varieties.
Langlands Program Connections: The Langlands Program is a set of conjectures and theories that establish deep relationships between number theory and representation theory, particularly through the study of Galois groups and automorphic forms. This program seeks to bridge the gap between various areas of mathematics, connecting arithmetic objects like varieties and their symmetries with analytic objects like modular forms and L-functions.
Minimal Model Program: The Minimal Model Program (MMP) is a framework in algebraic geometry that aims to classify algebraic varieties by finding simpler models of them. It focuses on understanding the birational geometry of these varieties, particularly by constructing minimal models and canonical models. This program plays a crucial role in the study of threefolds, especially in the context of their singularities, canonical rings, and their behavior under various morphisms.
Modularity Conjectures: The Modularity Conjectures propose a deep connection between number theory and algebraic geometry, suggesting that every rational elliptic curve is modular. This means that it can be associated with a modular form, which is a special kind of complex function that has certain symmetries. These conjectures have profound implications for the Langlands program and the study of Diophantine equations.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over a number field is finitely generated. This fundamental result connects the theory of elliptic curves with algebraic number theory, revealing the structure of rational solutions and their relationship to torsion points and complex multiplication.
Mori Theory: Mori theory is a branch of algebraic geometry that focuses on the study of higher-dimensional algebraic varieties, particularly in relation to their structure and classification. It introduces important concepts such as the minimal model program, which aims to systematically transform varieties into simpler forms while preserving their essential features. This approach is particularly useful in understanding the geometry of arithmetic threefolds, where one can analyze singularities and their resolutions.
Néron-Tate Height Pairing: The Néron-Tate height pairing is a bilinear form that measures the arithmetic intersection of two points on an abelian variety, playing a crucial role in understanding the heights of rational points. This pairing provides a way to relate the algebraic properties of these points to their geometric representation on the variety, often utilized in the study of rational points over number fields and in the context of arithmetic geometry. It encapsulates important information about the geometry of the abelian variety, allowing for deeper insights into the structure of rational points.
Northcott Property: The Northcott Property refers to a condition in number theory related to the boundedness of rational points on algebraic varieties. Specifically, it states that for a given projective variety defined over a number field, the set of rational points of bounded height is finite. This concept is particularly relevant in understanding the behavior of points on arithmetic threefolds, as well as in the dynamics of height functions and their periodic points.
Number Theory: Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. It explores concepts like divisibility, prime numbers, and the solutions to equations in whole numbers. This field is fundamental in understanding the underlying structures in mathematics, influencing various areas such as cryptography, algebra, and geometry.
Rational Points: Rational points are solutions to equations that can be expressed as fractions of integers, typically where the coordinates are in the form of rational numbers. These points are crucial in the study of algebraic varieties, especially in understanding the solutions over different fields, including the rational numbers, which can reveal deeper properties of the geometric objects involved.
Sato-Tate Conjecture: The Sato-Tate Conjecture is a conjecture in number theory that predicts the distribution of normalized Frobenius angles associated with elliptic curves over finite fields. It states that if you take an elliptic curve defined over a rational field, the angles formed by the Frobenius endomorphism are equidistributed according to the Sato-Tate measure, which is a specific probability measure on the unit circle. This conjecture connects deeply with several areas of arithmetic geometry and number theory.
Shigefumi Mori: Shigefumi Mori is a prominent mathematician known for his work in algebraic geometry, particularly regarding the geometry of algebraic varieties and the theory of moduli spaces. His contributions include deep insights into the structure of polarizations and arithmetic geometry, which connect algebraic geometry with number theory and arithmetic applications.
Singularity: In the context of arithmetic geometry, a singularity refers to a point on an algebraic variety where the mathematical object fails to be well-behaved, often characterized by a loss of differentiability or where certain geometric properties break down. Singularities can provide crucial insights into the structure and behavior of varieties, particularly in understanding their local and global properties.
Theory of motives: The theory of motives is a framework in mathematics that seeks to understand the relationships between algebraic varieties through the use of abstract objects called motives. This theory aims to unify various areas in algebraic geometry and number theory, bridging the gap between geometric, topological, and arithmetic properties of these varieties.
Zeta Functions: Zeta functions are complex functions that encode important number-theoretic properties, often used to study the distribution of prime numbers and other arithmetic properties. They provide a bridge between algebraic geometry and number theory, enabling deeper insights into the structure of varieties and schemes over number fields.