On the brauer–siegel ratio for abelian varieties over function fields

Research output: Contribution to journalArticle

Abstract

Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

Original languageEnglish (US)
Pages (from-to)1069-1120
Number of pages52
JournalAlgebra and Number Theory
Volume13
Issue number5
DOIs
StatePublished - Jan 1 2019

Fingerprint

Abelian Variety
Function Fields
Regulator
Elliptic Curves
High-dimensional
Analogue
Theorem
Family

Keywords

  • Abelian variety
  • Brauer
  • Function field
  • Height
  • Regulator
  • Shafarevich group
  • Siegel ratio
  • Tate

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On the brauer–siegel ratio for abelian varieties over function fields. / Ulmer, Douglas.

In: Algebra and Number Theory, Vol. 13, No. 5, 01.01.2019, p. 1069-1120.

Research output: Contribution to journalArticle

@article{81e7b673b1344faf94584a6eb9e0f895,
title = "On the brauer–siegel ratio for abelian varieties over function fields",
abstract = "Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.",
keywords = "Abelian variety, Brauer, Function field, Height, Regulator, Shafarevich group, Siegel ratio, Tate",
author = "Douglas Ulmer",
year = "2019",
month = "1",
day = "1",
doi = "10.2140/ant.2019.13.1069",
language = "English (US)",
volume = "13",
pages = "1069--1120",
journal = "Algebra and Number Theory",
issn = "1937-0652",
publisher = "Mathematical Sciences Publishers",
number = "5",

}

TY - JOUR

T1 - On the brauer–siegel ratio for abelian varieties over function fields

AU - Ulmer, Douglas

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

AB - Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

KW - Abelian variety

KW - Brauer

KW - Function field

KW - Height

KW - Regulator

KW - Shafarevich group

KW - Siegel ratio

KW - Tate

UR - http://www.scopus.com/inward/record.url?scp=85071123571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071123571&partnerID=8YFLogxK

U2 - 10.2140/ant.2019.13.1069

DO - 10.2140/ant.2019.13.1069

M3 - Article

AN - SCOPUS:85071123571

VL - 13

SP - 1069

EP - 1120

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 5

ER -