Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Brauer and Neron-Severi groups of surfaces over finite fields

Thomas H. Geisser

Abstract

We give a version of the Artin-Tate formula for surfaces over finite fields not assuming Tate's conjecture. It gives an equality between terms related to the Brauer group on the one hand and terms related to the Neron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to gives sharp estimates on the size of the Brauer group of abelian surfaces depending on the p-rank.

Brauer and Neron-Severi groups of surfaces over finite fields

Abstract

We give a version of the Artin-Tate formula for surfaces over finite fields not assuming Tate's conjecture. It gives an equality between terms related to the Brauer group on the one hand and terms related to the Neron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to gives sharp estimates on the size of the Brauer group of abelian surfaces depending on the p-rank.
Paper Structure (13 sections, 26 theorems, 80 equations)

This paper contains 13 sections, 26 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Estimating $\omega_X$
  4. The invariants of the Brauer group
  5. Estimating Weil-polynomials
  6. Abelian surfaces
  7. $\rho(\bar{A})=6$
  8. $\rho(\bar{A})=4$
  9. $\rho(A)=2, \tau_A=2, \omega_A\in \{1,2,3,4\}$.
  10. $\rho(A)=4, \tau_A=0$.
  11. $\rho(\bar{A})=2$
  12. $p$-rank one
  13. $p$-rank two

Key Result

Theorem 1.1

The map $\beta_X$ has finite kernel and cokernel, and $\omega_X\in {\mathbb N}$. We have

Theorems & Definitions (34)

  • Theorem 1.1: \ref{['mainth']}
  • Corollary 1.2: \ref{['maincor']}
  • Proposition 1.3: \ref{['omegaX']}
  • Theorem 1.4: \ref{['mainbr']}
  • Proposition 1.5: \ref{['g=2']}
  • Proposition 1.6: \ref{['c2']}
  • Proposition 1.7: \ref{['ss']}, \ref{['largest']}, \ref{['largest2']}
  • Proposition 1.8: \ref{['ssp']}
  • Lemma 2.1
  • Theorem 2.2
  • ...and 24 more