🔢Arithmetic Geometry Unit 6 – L-functions and Zeta Functions in Arithmetic

Definition and Basic Properties

• L-functions generalize the Riemann zeta function and Dirichlet L-functions to more complex mathematical objects
• Defined as Dirichlet series $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ where $a_n$ are complex coefficients and $s$ is a complex variable
• Converge in a half-plane $\Re(s) > \sigma$ for some real number $\sigma$, called the abscissa of convergence
• Satisfy a functional equation relating $L(s)$ to $L(1-s)$, often involving a gamma factor and a conductor term
• Have an Euler product representation $L(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}$ over primes $p$, reflecting the multiplicative structure of the coefficients
  • Euler product converges absolutely for $\Re(s) > 1$, providing a link between analytic and arithmetic properties
• Expected to satisfy the Riemann Hypothesis, stating that all non-trivial zeros lie on the critical line $\Re(s) = \frac{1}{2}$
• Zeta functions are a special case of L-functions, associated with algebraic varieties over finite fields or number fields

Historical Context and Development

• Riemann introduced the zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ in 1859 to study the distribution of prime numbers
• Dirichlet L-functions, defined as $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$ for Dirichlet characters $\chi$, used to prove Dirichlet's theorem on primes in arithmetic progressions
• Hecke L-functions, associated with modular forms, introduced by Erich Hecke in the early 20th century
• Artin L-functions, attached to representations of Galois groups, developed by Emil Artin in the 1920s
• Hasse-Weil L-functions, associated with algebraic varieties over number fields, introduced by Helmut Hasse and André Weil in the 1930s
• Langlands program, formulated by Robert Langlands in the 1960s, seeks to unify various types of L-functions and establish reciprocity laws between them
• Birch and Swinnerton-Dyer conjecture, relating the rank of elliptic curves to the order of vanishing of their L-functions at $s=1$, posed in the 1960s
• Modularity theorem, proving that all elliptic curves over $\mathbb{Q}$ are modular (Taniyama-Shimura conjecture), a key step in Andrew Wiles' proof of Fermat's Last Theorem in 1995

Types of L-functions and Zeta Functions

• Riemann zeta function $\zeta(s)$, the prototypical example of an L-function
• Dirichlet L-functions $L(s, \chi)$, associated with Dirichlet characters $\chi$
• Dedekind zeta functions $\zeta_K(s)$, associated with number fields $K$, generalizing the Riemann zeta function
• Hecke L-functions $L(s, f)$, associated with modular forms $f$
  • Eisenstein series and cusp forms give rise to distinct types of Hecke L-functions
• Artin L-functions $L(s, \rho)$, associated with representations $\rho$ of Galois groups
• Hasse-Weil L-functions $L(s, X)$, associated with algebraic varieties $X$ over number fields
  • Elliptic curve L-functions are a special case, with deep connections to the Birch and Swinnerton-Dyer conjecture
• Automorphic L-functions, associated with automorphic representations of reductive groups, central to the Langlands program
• Motivic L-functions, associated with motives in algebraic geometry, unifying various types of L-functions

Analytic Continuation and Functional Equations

• L-functions initially defined as Dirichlet series in a half-plane of convergence
• Analytic continuation extends L-functions to meromorphic functions on the entire complex plane
  • Riemann zeta function has a simple pole at $s=1$, while other L-functions are often entire (no poles)
• Functional equations relate values of L-functions at $s$ and $1-s$, often taking the form $\Lambda(s) = \varepsilon \Lambda(1-s)$, where $\Lambda(s)$ is a completed L-function involving gamma factors and $\varepsilon$ is a complex number of absolute value 1
• Gamma factors in functional equations reflect the archimedean part of the L-function, related to the infinite places of a number field
• Conductor term in functional equations measures the complexity of the arithmetic object associated with the L-function
• Explicit formulas relate sums over zeros of L-functions to sums over prime powers, providing a link between analytic and arithmetic information
• Functional equations and explicit formulas are key tools in studying the distribution of zeros and the behavior of L-functions on the critical line

Connections to Number Theory

• L-functions encode arithmetic information about various mathematical objects, such as number fields, algebraic varieties, and modular forms
• Riemann zeta function encodes information about the distribution of prime numbers
  • Prime Number Theorem equivalent to the non-vanishing of $\zeta(s)$ on the line $\Re(s) = 1$
• Dirichlet L-functions used to prove Dirichlet's theorem on primes in arithmetic progressions and the non-vanishing of $L(1, \chi)$ for non-principal characters $\chi$
• Dedekind zeta functions encode information about the class number and unit group of a number field
• Hecke L-functions and modular forms are connected through the Mellin transform, with coefficients of modular forms appearing as coefficients of L-functions
• Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve to the order of vanishing of its L-function at $s=1$
  • Modularity theorem establishes a correspondence between elliptic curves and modular forms, with matching L-functions
• Langlands program seeks to establish reciprocity laws between L-functions of different types, unifying various areas of number theory

Applications in Arithmetic Geometry

• L-functions are central tools in studying arithmetic properties of algebraic varieties
• Hasse-Weil L-functions encode information about the number of points on an algebraic variety over finite fields
  • Weil conjectures, proved by Deligne, relate the zeros of Hasse-Weil L-functions to the Betti numbers of the variety
• Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve (a geometric property) to the order of vanishing of its L-function at $s=1$ (an analytic property)
  • Kolyvagin and Gross-Zagier proved special cases of the conjecture, using L-functions and Heegner points
• Modularity theorem, proving that all elliptic curves over $\mathbb{Q}$ are modular, relies on comparing L-functions of elliptic curves and modular forms
• Serre's conjecture, proved by Khare and Wintenberger, establishes a correspondence between mod $p$ Galois representations and modular forms, using L-functions
• Langlands program aims to establish deep connections between L-functions, automorphic forms, and Galois representations, with far-reaching consequences in arithmetic geometry

Key Theorems and Conjectures

• Prime Number Theorem: equivalent to the non-vanishing of the Riemann zeta function on the line $\Re(s) = 1$
• Dirichlet's Theorem on Primes in Arithmetic Progressions: proved using non-vanishing of Dirichlet L-functions at $s=1$
• Riemann Hypothesis: states that all non-trivial zeros of the Riemann zeta function lie on the critical line $\Re(s) = \frac{1}{2}$
  • Generalized Riemann Hypothesis extends this to all Dirichlet L-functions and other L-functions
• Birch and Swinnerton-Dyer Conjecture: relates the rank of an elliptic curve to the order of vanishing of its L-function at $s=1$
  • Modularity Theorem, proving the Taniyama-Shimura Conjecture, shows that all elliptic curves over $\mathbb{Q}$ are modular
• Langlands Functoriality Conjecture: predicts the existence of reciprocity laws between automorphic L-functions of different groups
  • Artin's Conjecture, a special case, states that Artin L-functions are automorphic
• Deligne's Proof of the Weil Conjectures: relates the zeros of Hasse-Weil L-functions to the Betti numbers of algebraic varieties
• Sato-Tate Conjecture: describes the distribution of angles of Frobenius elements acting on elliptic curves, proved using L-functions and automorphic forms

Computational Techniques and Examples

• Euler-Maclaurin formula provides a method to approximate values of L-functions using Bernoulli numbers and derivatives of the associated arithmetic function
• Explicit formulas express sums over zeros of L-functions in terms of sums over prime powers, allowing for numerical computations
• Algorithms for computing L-functions typically involve approximating the Dirichlet series or using the Euler product
  • Dirichlet series computation requires efficient methods for evaluating the coefficients $a_n$ and controlling the truncation error
  • Euler product computation requires efficient prime enumeration and techniques for handling the infinite product
• Example: Computing values of the Riemann zeta function
  • Euler-Maclaurin formula: $\zeta(s) \approx \sum_{n=1}^{N} \frac{1}{n^s} + \frac{N^{1-s}}{s-1} + \frac{N^{-s}}{2} + \sum_{k=1}^{m} \frac{B_{2k}}{(2k)!} s(s+1)\cdots(s+2k-2) N^{-s-2k+1}$
  • Euler product: $\zeta(s) \approx \prod_{p \leq N} \left(1 - \frac{1}{p^s}\right)^{-1}$, where $N$ is a suitable truncation point
• Example: Computing values of Dirichlet L-functions
  • Dirichlet series: $L(s, \chi) \approx \sum_{n=1}^{N} \frac{\chi(n)}{n^s}$, where $\chi$ is a Dirichlet character and $N$ is a suitable truncation point
  • Euler product: $L(s, \chi) \approx \prod_{p \leq N} \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}$, where $N$ is a suitable truncation point
• Computational methods for L-functions are essential for numerical investigations and testing conjectures, such as the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture