Torsion and Tamagawa numbers

Dino Lorenzini[1]

  • [1] University of Georgia Department of mathematics Athens, GA 30602 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 1995-2037
  • ISSN: 0373-0956

Abstract

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Let K be a number field, and let A / K be an abelian variety. Let c denote the product of the Tamagawa numbers of A / K , and let A ( K ) tors denote the finite torsion subgroup of A ( K ) . The quotient c / | A ( K ) tors | is a factor appearing in the leading term of the L -function of A / K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over ℚ or quadratic extensions K / ℚ , and for abelian surfaces A / ℚ . The smallest possible ratio c / | E ( ℚ ) tors | for elliptic curves over ℚ is 1 / 5 , achieved only by the modular curve X 1 ( 11 ) .

How to cite

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Lorenzini, Dino. "Torsion and Tamagawa numbers." Annales de l’institut Fourier 61.5 (2011): 1995-2037. <http://eudml.org/doc/219806>.

@article{Lorenzini2011,
abstract = {Let $K$ be a number field, and let $A/K$ be an abelian variety. Let $c$ denote the product of the Tamagawa numbers of $A/K$, and let $A(K)_\{\textrm\{tors\}\}$ denote the finite torsion subgroup of $A(K)$. The quotient $c/ |A(K)_\{\textrm\{tors\}\}|$ is a factor appearing in the leading term of the $L$-function of $A/K$ in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over $\mathbb\{Q\}$ or quadratic extensions $K/\mathbb\{Q\}$, and for abelian surfaces $A/\mathbb\{Q\}$. The smallest possible ratio $c/ |E(\mathbb\{Q\})_\{\textrm\{tors\}\}|$ for elliptic curves over $\mathbb\{Q\}$ is $1/5$, achieved only by the modular curve $X_1(11)$.},
affiliation = {University of Georgia Department of mathematics Athens, GA 30602 (USA)},
author = {Lorenzini, Dino},
journal = {Annales de l’institut Fourier},
keywords = {Abelian variety over a global field; torsion subgroup; Tamagawa number; elliptic curve; abelian surface; dual abelian variety; Weil restriction},
language = {eng},
number = {5},
pages = {1995-2037},
publisher = {Association des Annales de l’institut Fourier},
title = {Torsion and Tamagawa numbers},
url = {http://eudml.org/doc/219806},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Lorenzini, Dino
TI - Torsion and Tamagawa numbers
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 1995
EP - 2037
AB - Let $K$ be a number field, and let $A/K$ be an abelian variety. Let $c$ denote the product of the Tamagawa numbers of $A/K$, and let $A(K)_{\textrm{tors}}$ denote the finite torsion subgroup of $A(K)$. The quotient $c/ |A(K)_{\textrm{tors}}|$ is a factor appearing in the leading term of the $L$-function of $A/K$ in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over $\mathbb{Q}$ or quadratic extensions $K/\mathbb{Q}$, and for abelian surfaces $A/\mathbb{Q}$. The smallest possible ratio $c/ |E(\mathbb{Q})_{\textrm{tors}}|$ for elliptic curves over $\mathbb{Q}$ is $1/5$, achieved only by the modular curve $X_1(11)$.
LA - eng
KW - Abelian variety over a global field; torsion subgroup; Tamagawa number; elliptic curve; abelian surface; dual abelian variety; Weil restriction
UR - http://eudml.org/doc/219806
ER -

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