Explicit points on the Legendre curve III

Douglas Ulmer

doi:10.2140/ant.2014.8.2471

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arXiv · 1406.6674

Explicit points on the Legendre curve III

Douglas Ulmer

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Abstract

We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d=\mathbf{F}_p(μ_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ 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: $[E(K_d):V_d]^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over $\mathbf{Z}_p[\mathrm{Gal}(K_d/F_p(t))]$.

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Douglas Ulmer. 2017-05-24. Explicit points on the Legendre curve III. https://doi.org/10.2140/ant.2014.8.2471

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