12.4 Weil conjectures and étale cohomology

The Weil conjectures are a set of mathematical predictions about the properties of algebraic varieties over finite fields. They link number theory, geometry, and topology, sparking the development of new tools like étale cohomology.

Étale cohomology is a powerful technique for studying algebraic varieties, especially in arithmetic geometry. It helped prove the Weil conjectures and continues to be crucial in understanding the deep connections between algebra, geometry, and number theory.

Weil Conjectures and Their Significance

The Weil Conjectures and Their Implications

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• The Weil conjectures, proposed by André Weil in 1949, are a set of four conjectures about the zeta function of a smooth projective variety over a finite field
• The first conjecture states that the zeta function of a variety over a finite field is a rational function
• The second conjecture, known as the functional equation, relates the values of the zeta function at $s$ and $1-s$
• The third conjecture, called the Riemann hypothesis for varieties over finite fields, states that the zeros of the zeta function lie on a specific vertical line in the complex plane (the line $\text{Re}(s) = \frac{1}{2}$)
• The fourth conjecture, known as the Betti number conjecture, relates the degree of the zeta function to the Betti numbers of the variety (the dimensions of the étale cohomology groups)

Significance in Arithmetic Geometry

• The Weil conjectures have significant implications for understanding the arithmetic properties of varieties over finite fields and their connections to topology and complex geometry
  • They provide a deep link between the arithmetic of varieties over finite fields and their geometric and topological properties
  • The conjectures suggest that the zeta function encodes important information about the variety, such as its number of rational points and its cohomology
• Proving the Weil conjectures was a major goal in arithmetic geometry and led to the development of powerful new tools, such as étale cohomology
• The Weil conjectures have inspired further research in areas such as the Langlands program, which aims to unify various branches of mathematics through the study of L-functions and representations of Galois groups

Étale Cohomology for Arithmetic Geometry

Definition and Properties of Étale Cohomology

• Étale cohomology is a cohomology theory for algebraic varieties that is analogous to singular cohomology in topology
• It is defined using the étale topology, which is a Grothendieck topology on the category of schemes
  • The étale topology is finer than the Zariski topology and allows for the study of arithmetic properties
• Étale cohomology groups are equipped with a Galois action, which allows for the study of arithmetic properties of varieties
  • The Galois action on étale cohomology encodes information about the action of the absolute Galois group on the geometric points of the variety
• Étale cohomology satisfies various functorial properties, such as excision and the Mayer-Vietoris sequence, which make it a powerful tool for studying varieties

Role in Proving the Weil Conjectures

• The development of étale cohomology by Grothendieck and others provided the necessary tools to prove the Weil conjectures
• The proof of the Weil conjectures using étale cohomology was completed by Deligne in the 1970s
  • Deligne's proof relied on the construction of a suitable weight filtration on the étale cohomology groups and the use of the Lefschetz fixed-point formula
• Étale cohomology allows for the definition of L-functions associated to algebraic varieties, which play a crucial role in the proof of the Weil conjectures
  • The L-function of a variety over a finite field can be expressed as an alternating product of the characteristic polynomials of the Frobenius action on the étale cohomology groups

Applying Étale Cohomology to Algebraic Varieties

Arithmetic Invariants from Étale Cohomology

• Étale cohomology can be used to define important arithmetic invariants of algebraic varieties, such as the étale fundamental group and the Brauer group
• The étale fundamental group captures information about the Galois action on the étale covers of a variety and is related to the problem of solving equations over finite fields
  • For example, the étale fundamental group of the projective line minus three points is isomorphic to the profinite completion of the free group on two generators
• The Brauer group, defined using étale cohomology, measures the obstruction to the Hasse principle for varieties over global fields
  • The Brauer-Manin obstruction, which is defined using the Brauer group, can explain the failure of the Hasse principle for certain varieties (such as the Skorobogatov surface)

Studying Rational Points and Zeta Functions

• Étale cohomology can be used to study the rational points of varieties over finite fields and to prove results about their distribution and density
  • The Grothendieck-Lefschetz trace formula relates the number of rational points on a variety over a finite field to the traces of the Frobenius action on the étale cohomology groups
• The étale cohomology of a variety over a finite field is closely related to its zeta function and can be used to compute its L-function
  • The zeta function of a variety over a finite field can be expressed as an alternating product of the characteristic polynomials of the Frobenius action on the étale cohomology groups
  • This expression allows for the study of the zeros and poles of the zeta function using the properties of étale cohomology

Étale Cohomology, L-functions, and the Riemann Hypothesis

L-functions from Étale Cohomology

• L-functions associated to algebraic varieties over finite fields can be defined using étale cohomology
• These L-functions encode important arithmetic information about the variety, such as the number of rational points over finite extensions of the base field
  • The L-function of a variety over a finite field can be expressed as an Euler product, where each factor corresponds to the zeta function of the variety over a finite extension of the base field
• The study of L-functions and their properties using étale cohomology has led to further developments in arithmetic geometry, such as the Langlands program and the theory of motives
  • The Langlands program seeks to relate L-functions arising from different sources, such as Galois representations and automorphic forms, and to prove reciprocity laws between them

The Riemann Hypothesis for Varieties over Finite Fields

• The Riemann hypothesis for varieties over finite fields, which is part of the Weil conjectures, states that the zeros of the L-function associated to a smooth projective variety lie on a specific vertical line in the complex plane (the line $\text{Re}(s) = \frac{1}{2}$)
• The proof of the Riemann hypothesis for varieties over finite fields using étale cohomology has important consequences for the distribution of rational points and the behavior of the zeta function
  • It implies that the number of rational points on a smooth projective variety over a finite field is close to the expected value predicted by the Weil conjectures
  • It also implies that the zeros and poles of the zeta function of a variety over a finite field satisfy a precise symmetry, which is a manifestation of the functional equation
• The Riemann hypothesis for varieties over finite fields is a powerful tool for studying the arithmetic of varieties and has led to significant advances in areas such as coding theory and cryptography (for example, the construction of good error-correcting codes and the analysis of the security of certain cryptographic protocols)