[Paper Review] Fonctions zêta et $L$ de variétés et de motifs

Bruno Kahn|arXiv (Cornell University)|Dec 31, 2015

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper provides a comprehensive, historically informed overview of zeta and L-functions across arithmetic and algebraic geometry, focusing on the Riemann zeta function, zeta functions of schemes over finite fields, and their generalizations to motives. It presents Weil's proof of the Riemann Hypothesis for curves over finite fields, establishes foundational results on L-functions of Dirichlet, Hecke, and Artin, and introduces the motivic framework, culminating in the formulation of the standard conjectures and the Tate conjecture.

ABSTRACT

Rédaction d'un cours de M2 donné à Jussieu au printemps 2013. This is the write-up of a Masters course given at Jussieu in Spring 2013.

Motivation & Objective

• To provide a unified, historically grounded exposition of zeta and L-functions across number theory and algebraic geometry.
• To present the proof of the Riemann Hypothesis for zeta functions of curves over finite fields, as established by Weil.
• To develop the theory of L-functions in the context of motives, including the formulation of the standard conjectures and the Tate conjecture.
• To connect classical results (e.g., Dirichlet's theorem) with modern cohomological methods and motivic structures.
• To serve as a didactic and research-oriented reference for advanced students and researchers in arithmetic geometry.

Proposed method

• Traces the historical development of zeta and L-functions from Riemann to Weil and Grothendieck, using a didactic, ontogenetic approach.
• Applies the Riemann-Roch theorem and cohomological techniques to prove rationality and functional equations for zeta functions of curves over finite fields.
• Uses Dwork's p-adic method to establish rationality of zeta functions prior to the development of étale cohomology.
• Introduces L-functions via Hecke characters, Artin representations, and l-adic sheaves, linking them to Galois representations.
• Develops the category of motives via adequate equivalence relations and correspondences, emphasizing rigidity and weight filtrations.
• Applies Grothendieck's six operations and étale cohomology to define and study Hasse-Weil L-functions and their functional equations.

Experimental results

Research questions

• RQ1 How do zeta and L-functions unify diverse areas of arithmetic and algebraic geometry?
• RQ2 What is the significance of the Riemann Hypothesis for zeta functions of curves over finite fields, and how was it proven by Weil?
• RQ3 How do Hecke and Artin L-functions generalize Dirichlet L-functions, and what are their analytic properties?
• RQ4 What is the role of motives in unifying zeta and L-functions across different geometric and arithmetic contexts?
• RQ5 What are the implications of the standard conjectures and the Tate conjecture for the structure of algebraic cycles and Galois representations?

Key findings

• Weil proved the Riemann Hypothesis for zeta functions of smooth projective curves over finite fields using the Riemann-Roch theorem and trace formula techniques.
• The zeta function of a scheme of finite type over Z is rational and satisfies a functional equation, as shown by F.K. Schmidt and later generalized by Weil.
• Dwork's p-adic method provides a proof of rationality of zeta functions over finite fields before the advent of étale cohomology.
• L-functions of Dirichlet, Hecke, and Artin are shown to admit meromorphic continuation and functional equations, with Artin's conjecture resolved via Brauer's induction.
• The theory of motives provides a universal framework for L-functions, with the standard conjectures and the Tate conjecture offering deep structural insights into algebraic cycles.
• The completed L-function of a smooth projective variety over a global field satisfies a functional equation, as formalized by Serre using étale cohomology and the theory of weights.