Chapter 5. Arithmetic Geometry and Modern Directions

Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations....

41 items

Section	Title
1	Chapter 5. Arithmetic Geometry and Modern Directions
2	Schemes
3	Morphisms and Fibers
4	Curves over Fields
5	Arithmetic Surfaces
6	Étale Cohomology
7	Weil Conjectures
8	Representation Theory Background
9	Automorphic Representations
10	Adelic Methods
11	Langlands Program
12	Functoriality
13	Trace Formula
14	Fast Integer Arithmetic
15	Primality Testing
16	Integer Factorization
17	Lattice Reduction
18	Algorithms for Modular Forms
19	Algorithms for Elliptic Curves
20	Symbolic and Numeric Computation
21	RSA Cryptosystem
22	Diffie-Hellman Key Exchange
23	Elliptic Curve Cryptography
24	Pairing-Based Cryptography
25	Lattice Cryptography
26	Post-Quantum Cryptography
27	Zero-Knowledge Proofs
28	Random Integers
29	Smooth Numbers
30	Probabilistic Primality
31	Probabilistic Algorithms
32	Random Matrices and Zeta Zeros
33	Probabilistic Models for Primes
34	Arithmetic Statistics
35	Open Problems in Number Theory
36	Fermat's Last Theorem
37	The Riemann Hypothesis
38	The Birch and Swinnerton-Dyer Conjecture
39	The Langlands Program
40	Future Directions in Number Theory
41	Appendix A.1 Sets and Functions

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Chapter 5. Arithmetic Geometry and Modern Directions

Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations....

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Schemes

Classical algebraic geometry studies varieties defined by polynomial equations. This theory works well over algebraically closed fields, especially over $\mathbb{C}$. However,...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Morphisms and Fibers

Geometry is not only concerned with spaces themselves, but also with maps between spaces. In algebraic geometry and arithmetic geometry, these maps are called morphisms.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Curves over Fields

An algebraic curve is a geometric object whose dimension is one. Curves are among the oldest and most important objects in number theory and algebraic geometry.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Arithmetic Surfaces

Arithmetic geometry often studies families of algebraic curves varying over arithmetic bases. The most important base is

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Étale Cohomology

Classical topology studies geometric spaces using invariants such as homology and cohomology. Over the complex numbers, algebraic varieties can often be viewed as topological...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Weil Conjectures

One of the central problems in arithmetic geometry is understanding the number of solutions of polynomial equations over finite fields.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Representation Theory Background

Representation theory studies abstract algebraic objects by expressing them as linear transformations of vector spaces.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Automorphic Representations

Classically, number theory studied special analytic functions such as modular forms. These functions satisfy strong symmetry conditions under actions of arithmetic groups.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Adelic Methods

Number theory studies arithmetic simultaneously at two levels:

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Langlands Program

The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Functoriality

Functoriality is the unifying mechanism of the Langlands program. It predicts systematic relationships between automorphic representations attached to different algebraic groups.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Trace Formula

One of the central ideas of modern analysis is that functions may be decomposed spectrally into elementary pieces.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Fast Integer Arithmetic

Modern computational number theory depends fundamentally on efficient arithmetic with large integers.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Primality Testing

A prime number is an integer greater than $1$ whose only positive divisors are

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Integer Factorization

Integer factorization asks for the prime decomposition of a positive integer. Given

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Lattice Reduction

A lattice is a discrete additive subgroup of Euclidean space. More concretely, let

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Algorithms for Modular Forms

Modular forms are highly structured analytic functions with deep arithmetic properties. Although their definitions involve complex analysis and group actions, modular forms...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Algorithms for Elliptic Curves

Elliptic curves occupy a central position in modern number theory, arithmetic geometry, and cryptography.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Symbolic and Numeric Computation

Modern number theory relies heavily on computation. Two broad computational paradigms dominate the subject:

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

RSA Cryptosystem

Classical cryptography uses a shared secret key. Both sender and receiver must know the same secret information in advance.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Diffie-Hellman Key Exchange

Secure communication requires two parties to share secret information. In classical symmetric cryptography, both parties must already possess the same secret key before...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Elliptic Curve Cryptography

Elliptic curve cryptography is a public-key cryptographic framework based on the arithmetic of elliptic curves over finite fields.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Pairing-Based Cryptography

Pairing-based cryptography uses special maps defined on elliptic curve groups. A pairing is a function

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Lattice Cryptography

Lattice cryptography is a family of cryptographic systems based on the presumed hardness of computational problems on high-dimensional lattices.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Post-Quantum Cryptography

Modern public-key cryptography relies heavily on two computational assumptions:

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Zero-Knowledge Proofs

A zero-knowledge proof allows one party to convince another that a statement is true without revealing why it is true.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Random Integers

Number theory often studies exact statements about individual integers. For example, one may ask whether a given integer is prime, squarefree, smooth, or representable as a...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Smooth Numbers

A positive integer is called $y$-smooth if all of its prime factors are at most $y$.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Probabilistic Primality

A primality test determines whether an integer is prime.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Probabilistic Algorithms

A probabilistic algorithm uses random choices during its execution. In number theory, this is often a practical advantage rather than a weakness.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Random Matrices and Zeta Zeros

The Riemann zeta function is defined for $\operatorname{Re}s>1$ by

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Probabilistic Models for Primes

Prime numbers are deterministic objects, but many aspects of their distribution resemble random behavior.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Arithmetic Statistics

Arithmetic statistics studies the distribution of arithmetic objects inside large families.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Open Problems in Number Theory

Number theory contains some of the oldest and deepest unsolved problems in mathematics.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Fermat's Last Theorem

Fermat's Last Theorem states that there are no positive integers

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Riemann Hypothesis

The Riemann zeta function is one of the central objects in mathematics.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Birch and Swinnerton-Dyer Conjecture

An elliptic curve over $\mathbb{Q}$ may be written in Weierstrass form

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Langlands Program

The Langlands program is one of the largest and most influential research programs in modern mathematics.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Future Directions in Number Theory

Modern number theory continues to evolve rapidly.

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Appendix A.1 Sets and Functions

A set is a collection of objects called elements.