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Pure extension of the theta divisor over the moduli space of abelian varieties

Ana María Botero, José Ignacio Burgos Gil, David Holmes, Robin de Jong

Abstract

A theta divisor on the universal principally polarised abelian variety can be extended to a compactification either by taking the Zariski closure, or by taking the unique extension which is pure of weight 2. For the latter, following ideas of Yuan and Zhang, we need to pass to the category of adelic- or b-divisors. We show that the two choices of extension differ by a tropicalisation of the Riemann theta function. We prove an extension of Moret-Bailly's ''key formula'' that features the pure weight 2 extension of the theta divisor, and discuss various arithmetic applications, including a ''universal'' formula for the Néron--Tate height of a point. A key technical input is the systematic use of the theory of logarithmic abelian varieties due to Kajiwara, Kato, and Nakayama.

Pure extension of the theta divisor over the moduli space of abelian varieties

Abstract

A theta divisor on the universal principally polarised abelian variety can be extended to a compactification either by taking the Zariski closure, or by taking the unique extension which is pure of weight 2. For the latter, following ideas of Yuan and Zhang, we need to pass to the category of adelic- or b-divisors. We show that the two choices of extension differ by a tropicalisation of the Riemann theta function. We prove an extension of Moret-Bailly's ''key formula'' that features the pure weight 2 extension of the theta divisor, and discuss various arithmetic applications, including a ''universal'' formula for the Néron--Tate height of a point. A key technical input is the systematic use of the theory of logarithmic abelian varieties due to Kajiwara, Kato, and Nakayama.
Paper Structure (61 sections, 52 theorems, 143 equations, 1 figure)

This paper contains 61 sections, 52 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The universal abelian variety
  3. Extending to a compactification
  4. Extending the theta divisor
  5. Extending the universal polarising line bundle
  6. Applications
  7. Main tools
  8. Characteristic 2
  9. Background on logarithmic and tropical geometry
  10. Log schemes
  11. Morphisms of log schemes
  12. Log algebraic spaces and stacks
  13. Cone complexes and cone stacks
  14. Tropicalisation
  15. Subdivisions
  16. ...and 46 more sections

Key Result

Theorem 1.1

(cf. corollary cor:theta_pure) The adelic- or b-divisor is a pure weight 2 extension of the theta divisor $\Theta$ on $\overline N_{g,1}$, up to pullbacks from the base.

Figures (1)

  • Figure 1: Log and tropical moduli in genus one

Theorems & Definitions (181)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • ...and 171 more