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100% of elliptic curves with a marked point have positive rank

Jun-Yong Park, Tristan Phillips

Abstract

As a consequence of their work on average Selmer ranks of elliptic curves with marked points, Bhargava and Ho proved that $100\%$ of elliptic curves over $\mathbb{Q}$ with an additional marked point have positive rank. In this note we provide an alternate proof which extends the result to global fields of characteristic not two or three.

100% of elliptic curves with a marked point have positive rank

Abstract

As a consequence of their work on average Selmer ranks of elliptic curves with marked points, Bhargava and Ho proved that of elliptic curves over with an additional marked point have positive rank. In this note we provide an alternate proof which extends the result to global fields of characteristic not two or three.

Paper Structure

This paper contains 1 theorem, 3 equations.

Key Result

Theorem 1

Let $K$ be a global field of characteristic not equal to $2$ or $3$. When ordered by height, $100\%$ of elliptic curves defined over $K$ with an additional marked point have positive rank.

Theorems & Definitions (2)

  • Theorem 1
  • proof