Heights

Definition

In the context of elliptic curves, heights are numerical measures that quantify the complexity or 'size' of points on an elliptic curve. They are essential for understanding the distribution of rational points and play a critical role in the Birch and Swinnerton-Dyer conjecture, which relates the number of rational points on an elliptic curve to its L-function.

5 Must Know Facts For Your Next Test

1. Heights provide a way to measure how 'large' a point is in relation to the coordinates of the elliptic curve, with smaller heights indicating more 'special' or 'important' points.
2. In the Birch and Swinnerton-Dyer conjecture, heights are used to relate the rank of an elliptic curve to the behavior of its L-function at specific points.
3. There are various definitions of heights, including naive height and canonical height, each offering insights into different properties of elliptic curves.
4. The canonical height is particularly significant as it is designed to be a quadratic form that is invariant under isogenies, making it useful for studying rational points.
5. Height functions help in identifying rational points on curves and are instrumental when analyzing the finiteness or infiniteness of these points.

Review Questions

• How do heights contribute to our understanding of rational points on elliptic curves?
  • Heights serve as a measurement tool for assessing the complexity and significance of rational points on elliptic curves. By evaluating these heights, mathematicians can determine which points are considered special and how they relate to the overall structure of the curve. This understanding is crucial for deeper insights into the distribution of these points and forms a foundation for conjectures like Birch and Swinnerton-Dyer.
• Discuss the role of heights in the Birch and Swinnerton-Dyer conjecture and how they connect to the L-function.
  • In the Birch and Swinnerton-Dyer conjecture, heights are integral in establishing a connection between the rank of an elliptic curve and its associated L-function. Specifically, the conjecture posits that there is a correlation between the number of rational points on the curve (as reflected by their heights) and specific values of this L-function. This relationship suggests that studying heights could provide valuable insights into predicting the nature of rational points.
• Evaluate how different definitions of heights, such as naive and canonical height, affect our analysis of elliptic curves in relation to their rational points.
  • The choice between naive height and canonical height significantly impacts our analysis of elliptic curves. Naive height provides a straightforward measure based solely on coordinates, while canonical height offers a more nuanced approach that remains invariant under isogenies. This invariance allows for more robust conclusions about rational points, especially when considering their distribution across different elliptic curves. Ultimately, using canonical heights can yield deeper insights into finiteness or infiniteness of rational points as they relate to conjectures like Birch and Swinnerton-Dyer.

Related terms

Mordell-Weil Theorem: A theorem stating that the group of rational points on an elliptic curve over a number field is finitely generated.

L-function: A complex analytic function associated with an elliptic curve, which encodes information about the curve's rational points.

Tamagawa Number: A number that measures the size of the group of rational points of an elliptic curve at various primes, playing a role in calculating heights.