The cuspidal torsion packet on hyperelliptic Fermat quotients

David Grant[1]; Delphy Shaulis[1]

  • [1] Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 577-585
  • ISSN: 1246-7405

Abstract

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Let ℓ ≥ 7 be a prime, C be the non-singular projective curve defined over ℚ by the affine model y ( 1 - y ) = x ℓ , ∞ the point of C at infinity on this model, J the Jacobian of C , and φ : C → J the albanese embedding with ∞ as base point. Let ℚ ¯ be an algebraic closure of ℚ . Taking care of a case not covered in [12], we show that φ ( C ) ∩ J tors ( ℚ ¯ ) consists only of the image under φ of the Weierstrass points of C and the points ( x , y ) = ( 0 , 0 ) and ( 0 , 1 ) , where J tors denotes the torsion points of J .

How to cite

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Grant, David, and Shaulis, Delphy. "The cuspidal torsion packet on hyperelliptic Fermat quotients." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 577-585. <http://eudml.org/doc/249268>.

@article{Grant2004,
abstract = {Let $\ell \ge 7$ be a prime, $C$ be the non-singular projective curve defined over $\mathbb\{Q\}$ by the affine model $y(1-y)=x^\ell $, $\infty $ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$, and $\phi : C\rightarrow J$ the albanese embedding with $\infty $ as base point. Let $\overline\{\mathbb\{Q\}\}$ be an algebraic closure of $\mathbb\{Q\}$. Taking care of a case not covered in [12], we show that $\phi (C)\cap J_\{\operatorname\{tors\}\}(\overline\{\mathbb\{Q\}\})$ consists only of the image under $\phi $ of the Weierstrass points of $C$ and the points $(x,y)=(0,0)$ and $(0,1)$, where $J_\{\operatorname\{tors\}\}$ denotes the torsion points of $J$.},
affiliation = {Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA; Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA},
author = {Grant, David, Shaulis, Delphy},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {torsion point; cuspidal torsion packet; Fermat quotient curve; Jacobian; hyperelliptic curve},
language = {eng},
number = {3},
pages = {577-585},
publisher = {Université Bordeaux 1},
title = {The cuspidal torsion packet on hyperelliptic Fermat quotients},
url = {http://eudml.org/doc/249268},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Grant, David
AU - Shaulis, Delphy
TI - The cuspidal torsion packet on hyperelliptic Fermat quotients
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 577
EP - 585
AB - Let $\ell \ge 7$ be a prime, $C$ be the non-singular projective curve defined over $\mathbb{Q}$ by the affine model $y(1-y)=x^\ell $, $\infty $ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$, and $\phi : C\rightarrow J$ the albanese embedding with $\infty $ as base point. Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Taking care of a case not covered in [12], we show that $\phi (C)\cap J_{\operatorname{tors}}(\overline{\mathbb{Q}})$ consists only of the image under $\phi $ of the Weierstrass points of $C$ and the points $(x,y)=(0,0)$ and $(0,1)$, where $J_{\operatorname{tors}}$ denotes the torsion points of $J$.
LA - eng
KW - torsion point; cuspidal torsion packet; Fermat quotient curve; Jacobian; hyperelliptic curve
UR - http://eudml.org/doc/249268
ER -

References

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