[Paper Review] Weil-etale Cohomology and Special Values of L-functions

Minh-Hoang Tran|arXiv (Cornell University)|Nov 6, 2016

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper constructs Weil-étale cohomology and Euler characteristics for strongly ℤ-constructible sheaves on open subschemes of the ring of integers of a number field. It proves that the special value of an Artin L-function of toric type at s=0 equals the Weil-étale Euler characteristic of an associated sheaf up to sign, providing a formula analogous to Ono’s Tamagawa number formula for algebraic tori.

ABSTRACT

We construct the Weil-étale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on an open subscheme of the spectrum of the ring of integers of a number field. Then we show that the special value of an Artin L-function of toric type at zero is given by the Weil-étale Euler characteristic of an appropriate $\mathbb{Z}$-constructible sheaf up to signs. As applications of our result, we will prove a formula for the special value of the L-function of an algebraic torus at zero which is similar to Ono's Tamagawa Number Formula.

Motivation & Objective

• To develop a Weil-étale cohomology theory for a subclass of ℤ-constructible sheaves on the spectrum of the ring of integers of a number field.
• To define and study the Weil-étale Euler characteristic for strongly ℤ-constructible sheaves with key properties: normalization, pushforward compatibility, and multiplicativity.
• To establish a precise relationship between the special value of Artin L-functions of toric type at s=0 and the Weil-étale Euler characteristic of associated sheaves.
• To derive a formula for the special value of the L-function of an algebraic torus at s=0, analogous to Ono’s Tamagawa number formula.
• To provide evidence for Lichtenbaum’s conjecture that special values of L-functions are encoded in Weil-étale Euler characteristics.

Proposed method

• Constructs the Weil-étale cohomology complex using ideas from Lichtenbaum’s work, adapted to handle non-totally imaginary number fields.
• Defines strongly ℤ-constructible sheaves as a subclass of ℤ-constructible sheaves with finitely generated abelian groups as stalks and compatible Galois actions.
• Uses the regulator pairing and cohomology with compact support to relate arithmetic invariants to Euler characteristics.
• Applies results on determinants of exact sequences and orders of torsion subgroups (from the appendix) to compute Euler characteristics precisely.
• Employs the Artin-Verdier duality and étale cohomology machinery to relate L-function special values to cohomological invariants.
• Establishes the main result via a comparison of the Weil-étale Euler characteristic with the leading term of the L-function using analytic and algebraic techniques.

Experimental results

Research questions

• RQ1 How can Weil-étale cohomology be systematically constructed for ℤ-constructible sheaves on number fields?
• RQ2 What properties must the Weil-étale Euler characteristic satisfy to be a meaningful invariant of L-functions?
• RQ3 Is the special value of an Artin L-function of toric type at s=0 equal to the Weil-étale Euler characteristic of an associated sheaf?
• RQ4 Can a Tamagawa number-type formula be derived for the L-function of an algebraic torus using Weil-étale cohomology?
• RQ5 How does the Weil-étale Euler characteristic relate to classical invariants like class numbers, regulators, and Tate-Shafarevich groups?

Key findings

• The special value $ L_S^*(M,0) $ of an Artin L-function of toric type at $ s=0 $ is equal to the Weil-étale Euler characteristic $ ho_U(j_*M) $ up to sign.
• The order of vanishing of $ L_S(M,s) $ at $ s=0 $ is equal to the $ \mathbb{Z} $-rank of $ \mathrm{Hom}_U(j_*M, \mathbb{G}_m) $.
• For an algebraic torus $ T $ over a number field $ K $, the special value $ L_S^*(\hat{T},0) $ is given by a formula involving the $ S $-class number $ h_{T,S} $, $ S $-regulator $ R_{T,S} $, number of roots of unity $ w_T $, and orders of Tate-Shafarevich and Galois cohomology groups.
• The formula for $ L_S^*(\hat{T},0) $ matches the structure of Ono’s Tamagawa number formula, with the Weil-étale Euler characteristic playing the role of the Tamagawa number.
• The Weil-étale Euler characteristic is multiplicative for certain short exact sequences of strongly ℤ-constructible sheaves.
• The construction is compatible with finite pushforwards: $ \chi_V(\mathcal{F}) = \chi_U(\pi'_*\mathcal{F}) $ for finite morphisms $ \pi': V \to U $.