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ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.

Abhishek Juyal1, Shiv Datt Kumar2, Dustin Moody3

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Summary
This summary is machine-generated.

This study examines Legendre elliptic curves parameterized by triangular numbers. Researchers found these curves have a rank of 1 over the function field but 0 over the rational function field, with some subfamilies showing positive Mordell-Weil rank.

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Area of Science:

  • Number Theory
  • Algebraic Geometry

Background:

  • Elliptic curves are fundamental objects in number theory.
  • The Legendre family of elliptic curves is a well-studied class with interesting properties.

Purpose of the Study:

  • To investigate the rank of the Legendre family of elliptic curves parameterized by triangular numbers.
  • To analyze the rank over different fields, specifically the rational function field and its algebraic closure.

Main Methods:

  • Parametrization of elliptic curves using triangular numbers (Δt = t(t+1)/2).
  • Analysis of the Mordell-Weil rank over the function field Q(t) and its algebraic closure Q̄(t).

Main Results:

  • The rank of the Legendre elliptic curves E_t over the function field Q̄(t) is proven to be 1.
  • The rank of these curves over the rational function field Q(t) is determined to be 0.
  • Infinite subfamilies with positive Mordell-Weil rank were identified, including high-rank curves.

Conclusions:

  • The rank of Legendre elliptic curves exhibits field-dependent behavior.
  • The study provides insights into the arithmetic of elliptic curves over function fields.
  • The findings open avenues for further research into high-rank elliptic curves.