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Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$

Jean Gillibert, Emmanuel Hallouin, Aaron Levin

Abstract

We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable assumptions on the factorization of $f$, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When $f$ has degree at most $6$, we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the $k$-curves corresponding to the $2$ and $3$-torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation.

Integral points on elliptic curves with $j$-invariant $0$ over $k(t)$

Abstract

We consider elliptic curves defined by an equation of the form , where has coefficients in a perfect field of characteristic not or . By performing and -descent, we obtain, under suitable assumptions on the factorization of , bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When has degree at most , we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the -curves corresponding to the and -torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation.
Paper Structure (14 sections, 32 theorems, 174 equations, 2 tables)

This paper contains 14 sections, 32 theorems, 174 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Extending Picard groups
  3. Heights
  4. The $2$-descent map
  5. The $3$-descent maps
  6. Upper bounds for the number of integral points
  7. Davenport's inequality
  8. Tools from $2$-descent
  9. Tools from $3$-descent
  10. Integral points of small height
  11. Arguments using $2$-descent
  12. Arguments using $3$-descent
  13. An application
  14. Proof of Theorem \ref{['corSh2']}

Key Result

Theorem 1.1

Let $k$ be a perfect field of characteristic not $2$ or $3$, and let $E$ be an elliptic curve over $k(t)$ defined by the Weierstrass equation $y^2=x^3+f(t)$, for some $f\in k[t]$. Assume that $f=f_1f_2^2f_3^3f_4^4f_5^5$ where the $f_i$ are pairwise coprime, separable polynomials in $k[t]$, and $f_1$

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Corollary 3.2
  • proof : Proof of Theorem \ref{['thh']}
  • Lemma 3.3
  • proof
  • Proposition 4.1
  • Remark 4.2
  • Corollary 4.3
  • ...and 59 more