Counterexamples to the Hasse principle among the twists of the Klein quartic
type: Item
instance of: scholarly article
title: Counterexamples to the Hasse principle among the twists of the Klein quartic
DOI: 10.1016/j.indag.2023.08.007
author: Elisa Lorenzo García
author: Michaël Vullers
published in: Indagationes Mathematicae. New Series
publication date: 2024-08-09T00:00:00Z
review text: The Klein quartic is isomorphic to the modular curve \(X(7)\) over the cyclotomic field \(\mathbb Q(\zeta_7)\), and it is given by \(x^3y + y^3z + z^3x =0\). It is the curve with the smallest genus that attains the Hurwitz upper bound \(84(g-1)\) on the number of automorphisms for a curve of a genus \(g\) in characteristic \(0\). The authors prove that there are twists of the Klein quartic that are counterexamples to the Hasse principle. Such twists are constructed from the existence of an irreducible polynomial \(f(x)=x^3+bx+c\in \mathbb Z[x]\) such that \(2\nmid c\), \(3\nmid b\), \(7\mid b\), \(7|| c\), \(p\mid b\) and \(p|| c\) for all primes \(p\ne 2,3,7\) dividing \(4b^3+27c^2\), and the splitting field of \(f(x)\) is not isomorphic to \(\mathbb Q(\zeta_7)\) or to the field numbered 6.0.214375.1. by The \(L\)-function and Modular Forms Database.
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11G18
zbMATH DE Number: 7894509
zbMATH Keywords: local-global principle
zbMATH Keywords: modular curves
zbMATH Keywords: twists
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7894509