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Noncommutative geometry of elliptic surfaces

Igor Nikolaev

Abstract

We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate the Picard numbers of elliptic surfaces with fibers having complex multiplication.

Noncommutative geometry of elliptic surfaces

Abstract

We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate the Picard numbers of elliptic surfaces with fibers having complex multiplication.

Paper Structure

This paper contains 12 sections, 4 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continued fractions
  4. Elliptic surfaces
  5. Surfaces
  6. Blow-ups
  7. Minimal models
  8. Non-commutative tori
  9. Proofs
  10. Proof of theorem \ref{['thm1.1']}
  11. Proof of theorem \ref{['thm1.3']}
  12. Complex multiplication

Key Result

Theorem \oldthetheorem

For each elliptic surface $\mathscr{E}_{\mathbf{Q}(t)}$ there exists a section $\mathscr{P}\to$$U_{b_1,\dots,b_N; a_1,\dots, a_k}$ of a sub-bundle $(U_{b_1,\dots,b_N; a_1,\dots, a_k}, \mathscr{P}, \pi')$ of the fiber bundle $(V_{N,k}(\mathbf{C}), \mathscr{P}, \pi)$, such that:

Figures (1)

  • Figure 1:

Theorems & Definitions (19)

  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • ...and 9 more