Zeta and L-functions are powerful tools in arithmetic geometry, encoding deep information about algebraic varieties. They connect point-counting over finite fields to complex analysis, revealing hidden structures in number theory.

These functions play a crucial role in major conjectures like Birch and Swinnerton-Dyer, linking arithmetic properties of elliptic curves to analytic behavior. They're essential for understanding the interplay between geometry and number theory.

Zeta and L-functions for Varieties

Zeta Functions

• Zeta functions are generating functions that encode arithmetic information about algebraic varieties over finite fields
• For an algebraic variety X defined over a finite field Fq, the zeta function Z(X, t) is defined as a formal power series $\sum_{n=1}^{\infty} \frac{|X(Fqn)|}{n} t^n$, where |X(Fqn)| denotes the number of points on X over the extension field Fqn
• The coefficients of the zeta function contain important arithmetic information about the underlying variety
• Example: The zeta function of the projective line P1 over Fq is $Z(P^1, t) = \frac{1}{(1-t)(1-qt)}$

L-functions

• L-functions are generalizations of zeta functions that are associated with more general arithmetic objects, such as Galois representations or automorphic forms
• L-functions can be defined for algebraic varieties equipped with additional structures, such as a Galois action or a sheaf
• The coefficients of the L-function contain important arithmetic information about the underlying arithmetic object
• Example: The Hasse-Weil L-function of an elliptic curve E over Q is defined as an Euler product $L(E, s) = \prod_p \frac{1}{1 - a_p p^{-s} + p^{1-2s}}$, where p runs over the primes and ap is related to the number of points on E modulo p
• L-functions play a central role in the Langlands program, which seeks to unify various areas of mathematics, including number theory, representation theory, and harmonic analysis

Zeta Functions and Point Counts

Zeta Functions and Point Counting

• The zeta function Z(X, t) encodes the number of points on the variety X over various finite field extensions Fqn
• The coefficient of tn in the power series expansion of Z(X, t) is equal to the number of points on X over the field Fqn, divided by n
• Example: For an elliptic curve E over Fq, the number of points on E over Fqn is related to the trace of Frobenius endomorphism, which can be computed using the zeta function
• Point counting on varieties over finite fields has important applications in cryptography, such as in the design of elliptic curve cryptosystems

Properties of Zeta Functions

• The zeta function satisfies a functional equation that relates its values at t and 1/qt, where q is the cardinality of the base field
• The zeta function can be expressed as a rational function $Z(X, t) = \frac{P(t)}{(1-t)(1-qt)}$, where P(t) is a polynomial with integer coefficients
• The degree of the polynomial P(t) is related to the dimension and geometric properties of the variety X
• Example: For a smooth projective curve C of genus g over Fq, the zeta function has the form $Z(C, t) = \frac{P(t)}{(1-t)(1-qt)}$, where P(t) is a polynomial of degree 2g
• The Riemann hypothesis for varieties over finite fields, proved by Deligne, states that the zeros of P(t) have absolute value q-1/2

Analytic Properties of Zeta Functions

Analytic Continuation and Functional Equations

• Zeta functions and L-functions can be regarded as complex analytic functions by considering their Euler product expansions and analytic continuation
• The analytic properties of L-functions, such as their meromorphic continuation and functional equation, are closely related to the arithmetic properties of the underlying varieties or arithmetic objects
• The Hasse-Weil L-function, associated with an elliptic curve, is conjectured to have an analytic continuation and satisfy a functional equation
• Example: The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ has an analytic continuation to the entire complex plane, except for a simple pole at s=1, and satisfies the functional equation $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$

Special Values and Arithmetic Significance

• The special values of L-functions at certain points, such as s=1, are conjectured to have arithmetic significance and are related to important invariants of the underlying variety or arithmetic object
• Example: The Birch and Swinnerton-Dyer conjecture relates the value of the L-function of an elliptic curve at s=1 to the rank of the elliptic curve and other arithmetic invariants
• Special values of L-functions are also related to the Bloch-Kato conjecture, which generalizes the Birch and Swinnerton-Dyer conjecture to motives
• The analytic continuation and functional equation of L-functions are crucial in the study of the Birch and Swinnerton-Dyer conjecture and other conjectures in arithmetic geometry

L-functions and the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture

• The Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve (the number of independent rational points of infinite order) to the behavior of its L-function at s=1
• The conjecture states that the rank of an elliptic curve is equal to the order of vanishing of its L-function at s=1
• The conjecture also predicts a precise formula for the leading coefficient of the Taylor expansion of the L-function at s=1 in terms of arithmetic invariants of the elliptic curve, such as the Tate-Shafarevich group and the regulator
• Example: For the elliptic curve E : y2 = x3 - x, the Birch and Swinnerton-Dyer conjecture predicts that the rank of E is 0 and that the value of its L-function at s=1 is related to the order of the Tate-Shafarevich group of E

Generalizations and Progress

• The Birch and Swinnerton-Dyer conjecture is one of the most important open problems in number theory and has far-reaching consequences in the study of elliptic curves and their arithmetic
• Progress towards the conjecture has been made in specific cases, such as for elliptic curves with complex multiplication or for certain families of elliptic curves, but the general case remains unresolved
• The conjecture has been generalized to abelian varieties and motives, leading to deep connections between L-functions and arithmetic geometry
• Example: The Gross-Zagier formula relates the heights of Heegner points on modular elliptic curves to the derivatives of L-functions, providing evidence for the Birch and Swinnerton-Dyer conjecture in the case of elliptic curves with complex multiplication
• The Birch and Swinnerton-Dyer conjecture is part of the Millennium Prize Problems, reflecting its importance and difficulty in the field of mathematics