Néron-Tate Height

Definition

The Néron-Tate height is a specific type of height function used in the study of rational points on algebraic varieties, particularly abelian varieties. It plays a crucial role in measuring the arithmetic complexity of these points, providing a way to analyze their distribution and relationships, especially in relation to the Mordell-Weil theorem and the heights of points in dynamics, polarizations, and arithmetic surfaces.

5 Must Know Facts For Your Next Test

1. The Néron-Tate height can be defined for points on an abelian variety over a number field and helps understand the distribution of rational points.
2. It is an important invariant in the context of the Mordell-Weil theorem, where it aids in determining whether a point is 'small' or 'large' in terms of its arithmetic properties.
3. The height can also be related to the concept of canonical heights, which are normalized versions that adjust for certain symmetries in the variety.
4. In dynamics, the Néron-Tate height function can be employed to study the orbits of points under iterated maps on abelian varieties.
5. The Néron-Tate height is also significant when analyzing arithmetic surfaces, as it assists in understanding how heights behave under various morphisms and embeddings.

Review Questions

• How does the Néron-Tate height relate to the Mordell-Weil theorem and why is this connection important?
  • The Néron-Tate height is fundamentally linked to the Mordell-Weil theorem as it provides a means to measure rational points on abelian varieties. Specifically, this height function helps determine the rank of the group of rational points by distinguishing between 'small' and 'large' points. This connection is essential because it allows mathematicians to analyze and predict the structure of these groups, giving insight into their behavior under addition.
• Discuss how height functions like the Néron-Tate height influence our understanding of polarizations on abelian varieties.
  • Height functions, particularly the Néron-Tate height, are closely tied to polarizations because they help quantify geometric features of abelian varieties. A polarization allows one to define an associated height function that respects certain symmetries and structures within the variety. This interaction reveals deep connections between arithmetic properties and geometric characteristics, allowing for a richer understanding of both fields.
• Evaluate the impact of Néron-Tate heights on dynamics involving iterated maps on abelian varieties.
  • The Néron-Tate height significantly impacts the study of dynamical systems involving iterated maps on abelian varieties by providing a way to measure how points evolve over time. By analyzing how heights change under iterations, one can gain insights into stability, periodicity, and chaotic behavior within these systems. This evaluation bridges arithmetic geometry and dynamical systems, showing how algebraic structures influence dynamical properties.

Related terms

Mordell-Weil Theorem:

A fundamental result in arithmetic geometry stating that the group of rational points on an abelian variety over a number field is finitely generated.

Height Function:

A mathematical tool used to measure the complexity of rational points on algebraic varieties, often related to the growth of the coordinates of these points.

Polarization: A form on an abelian variety that provides a geometric interpretation of its structure, often linked to its Néron-Tate height through various properties.