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Uniform lower bound of arithmetic Hilbert--Samuel function of hypersurfaces

Chunhui Liu

Abstract

In this article, we give an explicit and uniform lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which has the optimal dominant term. As an application, we apply this lower bound in the determinant method.

Uniform lower bound of arithmetic Hilbert--Samuel function of hypersurfaces

Abstract

In this article, we give an explicit and uniform lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which has the optimal dominant term. As an application, we apply this lower bound in the determinant method.

Paper Structure

This paper contains 25 sections, 232 equations.

Table of Contents

  1. Introduction
  2. History
  3. Main result
  4. An application in counting rational points
  5. Organization of article
  6. Fundamental settings
  7. Normed vector bundles
  8. Some Arakelov invariants
  9. Arithmetic Hilbert--Samuel function
  10. Arithmetic Hilbert-Samuel function of projective spaces
  11. A naive lower bound
  12. Arithmetic Hilbert-Samuel function of projective spaces
  13. Uniform lower bound of hypersurfaces
  14. Comparison of norms by the multiplication of a section
  15. A uniform lower bound
  16. ...and 10 more sections

Theorems & Definitions (19)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 9 more