### Abstract

The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g. The bound is given in terms of the genus of the curve and the number of F_{p}-points on the reduced curve, for all primes p of good reduction such that p > 2g. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus (p - 1) /2 (and rank at least (p - 1) /2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

Original language | French |
---|---|

Pages (from-to) | 459-463 |

Number of pages | 5 |

Journal | Comptes Rendus de l'Academie des Sciences - Series I: Mathematics |

Volume | 329 |

Issue number | 6 |

State | Published - Sep 15 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Comptes Rendus de l'Academie des Sciences - Series I: Mathematics*,

*329*(6), 459-463.

**Sur la borne de Coleman-Chabauty.** / Joshi, Kirti N; Tzermias, Pavlos.

Research output: Contribution to journal › Article

*Comptes Rendus de l'Academie des Sciences - Series I: Mathematics*, vol. 329, no. 6, pp. 459-463.

}

TY - JOUR

T1 - Sur la borne de Coleman-Chabauty

AU - Joshi, Kirti N

AU - Tzermias, Pavlos

PY - 1999/9/15

Y1 - 1999/9/15

N2 - The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g. The bound is given in terms of the genus of the curve and the number of Fp-points on the reduced curve, for all primes p of good reduction such that p > 2g. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus (p - 1) /2 (and rank at least (p - 1) /2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

AB - The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g. The bound is given in terms of the genus of the curve and the number of Fp-points on the reduced curve, for all primes p of good reduction such that p > 2g. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus (p - 1) /2 (and rank at least (p - 1) /2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

UR - http://www.scopus.com/inward/record.url?scp=0033567769&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033567769&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033567769

VL - 329

SP - 459

EP - 463

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 6

ER -