An explicit algebraic family of genus-one curves violating the Hasse principle

Bjorn Poonen

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 263-274
  • ISSN: 1246-7405

Abstract

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We prove that for any t ∈ 𝐐 , the curve 5 x 3 + 9 y 3 + 10 z 3 + 12 t 2 + 82 t 2 + 22 3 ( x + y + z ) 3 = 0 in 𝐏 2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its jacobian E t is given. The Shafarevich-Tate group of each E t contains a subgroup isomorphic to 𝐙 / 3 × 𝐙 / 3 .

How to cite

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Poonen, Bjorn. "An explicit algebraic family of genus-one curves violating the Hasse principle." Journal de théorie des nombres de Bordeaux 13.1 (2001): 263-274. <http://eudml.org/doc/248722>.

@article{Poonen2001,
abstract = {We prove that for any $t \in \mathbf \{Q\}$, the curve\begin\{equation*\} 5x^3 + 9y^3+ 10z^3 + 12 \left(\frac\{t^2 + 82\}\{t^2 + 22\}\right)^3 (x + y + z)^3 = 0 \end\{equation*\}in $\mathbf \{P\}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $\mathbf \{Z\}/3 \times \mathbf \{Z\}/3$.},
author = {Poonen, Bjorn},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {263-274},
publisher = {Université Bordeaux I},
title = {An explicit algebraic family of genus-one curves violating the Hasse principle},
url = {http://eudml.org/doc/248722},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Poonen, Bjorn
TI - An explicit algebraic family of genus-one curves violating the Hasse principle
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 263
EP - 274
AB - We prove that for any $t \in \mathbf {Q}$, the curve\begin{equation*} 5x^3 + 9y^3+ 10z^3 + 12 \left(\frac{t^2 + 82}{t^2 + 22}\right)^3 (x + y + z)^3 = 0 \end{equation*}in $\mathbf {P}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $\mathbf {Z}/3 \times \mathbf {Z}/3$.
LA - eng
UR - http://eudml.org/doc/248722
ER -

References

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