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The elliptic threefold y^2=x^3+16s^6+16t^6-32(t^3s^3+t^3+s^3)+16

Remke Kloosterman

Abstract

We present a method to calculate the rank of $E(\oQ(s,t))$ for the elliptic curve mentioned in the title. This method uses a generalization of a method from Van Geemen and Werner to calculate $h^4(Y)$ for nodal hypersurfaces $Y$.

The elliptic threefold y^2=x^3+16s^6+16t^6-32(t^3s^3+t^3+s^3)+16

Abstract

We present a method to calculate the rank of for the elliptic curve mentioned in the title. This method uses a generalization of a method from Van Geemen and Werner to calculate for nodal hypersurfaces .

Paper Structure

This paper contains 3 sections, 11 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The hypersurface $Y$
  3. Calculating $h^i(Y)$ and $\mathop{\mathrm{rank}}\nolimits E({\mathbf{Q}}(s,t))$

Key Result

Lemma 2.1

Suppose $K$ is algebraically closed and not of characteristic $2,3$. Let $\omega$ be a primitive third root of unity. Then the hypersurface $Y_{\mathop{\mathrm{sing}}\nolimits}$ consists of the $9$ points for $i,j,k\in \{0,1,2\}$, and at each of these points $Y$ has a $D_4$-singularity.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • Lemma 3.1
  • ...and 14 more