Rank gain of Jacobian varieties over finite Galois extensions
type: Item
instance of: scholarly article
title: Rank gain of Jacobian varieties over finite Galois extensions
zbMATH Open document ID: 1391.14055
author: Bo-Hae Im
author: Erik Wallace
published in: Journal of Number Theory
publication date: 2017-11-21T00:00:00Z
review text: The Klein quartic is given by the homogeneous polynomial \(x^3 y + y^3 z + z^3 x = 0\), and in [\textit{N. D. Elkies}, Math. Sci. Res. Inst. Publ. 35, 51--101 (1999; Zbl 0991.11032)], various interesting properties of the Klein quartic are reviewed. The curve is isomorphic to the modular curve \(X(7)\) admitting a map down to the elliptic curve \(E :y^2=4x^3 + 21x^2 + 28x\) whose conductor is \(49\), and especially, the elliptic curve is obtained as a quotient of \(X(7)\) by a subgroup of the automorphism group of \(X(7)\). Inspired by this map on the Klein quartic, the authors of the paper under review consider a general setting, and introduce results toward the Mordell-Weil ranks of Jacobian varieties of quotient curves over certain field extensions. Let \(A\) be an abelian variety over a number field \(K\), and let \(L_i\) for \(i=1,2,\dots\) be field extensions of \(K\). The abelian variety \(A\) is said to gain rank over the extensions \(L_i\) if the Mordell-Weil rank of \(A(L_1\cdots L_n)\) is an increasing function of \(n\) where \(L_1\cdots L_n\) is the compositum of the extensions \(L_1,\dots, L_n\). Let \(p\) be a prime, and let \(\mathcal X\) be a Riemann surface of genus \(g>1\) such that there is a Galois covering \(\mathcal X \to \mathbb P_K^1\) that has degree \(p\), and is ramified only at \(0\), \(1\), and \(\infty\). The authors prove that if \(p\equiv 1\mod 6\), then there is an automorphism \(\sigma\) of \(\mathcal X\) and field extensions \(L_i/K\) of degree \(p\) such that the Jacobian of \(\mathcal X/\langle \sigma \rangle\) gains rank over the extensions \(L_i\). It is relatively easy to find any field extensions over which the Jacobian gains rank, but it becomes more difficult to find Galois extensions with rank gain, as it is essentially related to the Inverse Galois problem. The hypothesis that a curve \(\mathcal X\) admits a Galois covering of \(\mathbb P_K^1\) implies that \(K\) contains the \(p\)-th root of unity in the complex numbers,…
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 12E25
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 6810544
zbMATH Keywords: Jacobian varieties
zbMATH Keywords: elliptic curves
zbMATH Keywords: rank
MaRDI profile type: Publication
full work available at URL: j.jnt.2017.08.010
OpenAlex ID: W2760390494
cites work: The Klein quartic in number theory
cites work: Vanishing and non-vanishing Dirichlet twists of L -functions of elliptic curves
cites work: Topics from the theory of numbers
cites work: Q4787521
cites work: On ranks of Jacobian varieties in prime degree extensions
cites work: Q3741712
cites work: Q3856855
cites work: Q4003835
cites work: Integer Points on Curves of Genus 1
DOI: 10.1016/J.JNT.2017.08.010
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description: scientific article; zbMATH DE number 6810544