Explicit points on the Legendre curve III

Douglas Ulmer

doi:10.2140/ant.2014.8.2471

Explicit points on the Legendre curve III

Douglas Ulmer

Mathematics

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3 Scopus citations

Abstract

We continue our study of the Legendre elliptic curve y²x(x+1)(x+t) over function fields K_d=F_p(μ_d,t^1/d). When d = p^f+1, we have previously exhibited explicit points generating a subgroup V_d ⊂ E (K_d) of rank d - 2 and of finite, p-power index. We also proved the finiteness of III(E/K_d) and a class number formula: (equation found). In this paper, we compute E(K_d)/V_dand III(E/K_d) explicitly as modules over Z_p[Gal(K_d/F_p(t))].

Original language	English (US)
Pages (from-to)	2471-2522
Number of pages	52
Journal	Algebra and Number Theory
Volume	8
Issue number	10
DOIs	https://doi.org/10.2140/ant.2014.8.2471
State	Published - 2014

Keywords

Elliptic curves
Function fields
Tate-Shafarevich group

ASJC Scopus subject areas

Algebra and Number Theory

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10.2140/ant.2014.8.2471

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abstract = "We continue our study of the Legendre elliptic curve y2x(x+1)(x+t) over function fields Kd=Fp(μd,t1/d). When d = pf+1, we have previously exhibited explicit points generating a subgroup Vd ⊂ E (Kd) of rank d - 2 and of finite, p-power index. We also proved the finiteness of III(E/Kd) and a class number formula: (equation found). In this paper, we compute E(Kd)/Vdand III(E/Kd) explicitly as modules over Zp[Gal(Kd/Fp(t))].",

keywords = "Elliptic curves, Function fields, Tate-Shafarevich group",

author = "Douglas Ulmer",

note = "Publisher Copyright: {\textcopyright} 2014 Mathematical Sciences Publishers.",

year = "2014",

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N2 - We continue our study of the Legendre elliptic curve y2x(x+1)(x+t) over function fields Kd=Fp(μd,t1/d). When d = pf+1, we have previously exhibited explicit points generating a subgroup Vd ⊂ E (Kd) of rank d - 2 and of finite, p-power index. We also proved the finiteness of III(E/Kd) and a class number formula: (equation found). In this paper, we compute E(Kd)/Vdand III(E/Kd) explicitly as modules over Zp[Gal(Kd/Fp(t))].

AB - We continue our study of the Legendre elliptic curve y2x(x+1)(x+t) over function fields Kd=Fp(μd,t1/d). When d = pf+1, we have previously exhibited explicit points generating a subgroup Vd ⊂ E (Kd) of rank d - 2 and of finite, p-power index. We also proved the finiteness of III(E/Kd) and a class number formula: (equation found). In this paper, we compute E(Kd)/Vdand III(E/Kd) explicitly as modules over Zp[Gal(Kd/Fp(t))].

KW - Elliptic curves

KW - Function fields

KW - Tate-Shafarevich group

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JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 10

ER -

Explicit points on the Legendre curve III

Abstract

Keywords

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