Unramified sections of the Legendre scheme and modular forms
type: Item
instance of: scholarly article
title: Unramified sections of the Legendre scheme and modular forms
zbMATH Open document ID: 1497.11147
author: Umberto Zannier
author: Pietro Corvaja
published in: Journal of Geometry and Physics
publication date: 2021-06-14T00:00:00Z
review text: Let \(\lambda\) be a complex number \(\not\in\{0,1\}\), and consider the Legendre elliptic scheme given by its affine model \(y^2=x(x-1)(x-\lambda)\). Each torsion section of unrestricted order is defined by algebraic functions of \(\lambda\) unramified outside \(\{0,1,\infty\}\), and by Shioda's result, it is shown that every algebraic section unramified outside \(\{0,1,\infty\}\) is torsion. As one of the main results of the paper under review, the authors present an alternative proof of Shioda's result, in terms of unramified coverings of \(\mathbb P_1 - \{0,1,\infty\}\), associated to finite index subgroups of its monodromy group. Another main result of the paper can be formulated as follows where \(\pi_1\) denotes a fundamental group: Let \(\mathcal E \to B\) be a non-isotrivial elliptic scheme where \(B\) is a smooth affine complex curve. If a rational section \(\sigma : B \to \mathcal E\) is non-torsion, then the associated homomorphism: \(\pi_1(B,b_0) \to \pi_1(\mathcal E,\sigma(b_0))\) does not factor through the mapping class group. Stated above is a topological version of their theorem, and they introduce different versions as well, one of which presents information on the monodromy of the Picard-Fuchs differential equation satisfied by the periods and the logarithm of the section. The authors also point out that the theorem has applications to arbitrary non-isotrivial elliptic schemes.
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 7358398
zbMATH Keywords: elliptic schemes
zbMATH Keywords: torsion sections
zbMATH Keywords: monodromy
zbMATH Keywords: Gauss-Manin connection
MaRDI profile type: Publication
full work available at URL: j.geomphys.2021.104266
OpenAlex ID: W3157457177
cites work: Q4009141
cites work: The Betti map associated to a section of an abelian scheme
cites work: Q4175362
cites work: Q3358852
cites work: Invarianten linearer Differentialgleichungen
cites work: The 1-2-3 of modular forms. Lectures at a summer school in Nordfjordeid, Norway, June 2004
cites work: Q3695591
cites work: On the torsion values for sections of an elliptic scheme
cites work: On the zeros of the Weierstrass \(\wp\)-function
cites work: Selmer groups and Mordell–Weil groups of elliptic curves over towers of function fields
cites work: On compact analytic surfaces. II
cites work: Q2762139
cites work: Q4723876
cites work: Q5568135
cites work: Q5550858
cites work: Q4000367
cites work: The Group of Rational Solutions of y 2 = x(x - 1)(x - t 2 - c)
cites work: Q4875376
cites work: On elliptic modular surfaces
cites work: An interesting elliptic surface over an elliptic curve
cites work: Mordell-Weil groups of generic Abelian varieties
cites work: The rank of elliptic surfaces in unramified abelian towers
cites work: Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields
DOI: 10.1016/J.GEOMPHYS.2021.104266
description: scientific article; zbMATH DE number 7358398