On the supersingular reduction of elliptic curves
type: Item
instance of: scholarly article
title: On the supersingular reduction of elliptic curves
zbMATH Open document ID: 0654.14018
DOI: 10.1007/BF02837827
author: M. Ram Murty
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1987-01-01T00:00:00Z
review text: Let E be an elliptic curve over \({\mathbb{Q}}\), and let p be a prime of supersingular reduction for E. The author shows that the 2-complement of \(E({\mathbb{F}}_ p)\) is cyclic. In particular, if \(E_ a\) is the curve \(y^ 2=(x^ 2+1)(x+a)\) (a\(\in {\mathbb{Q}})\) the author combines the above result with Elkies' theorem (that there are infinitely many primes of supersingular reduction for \(non-C.M.\quad curves)\) to show that there are infinitely many primes p for which \(E_ a({\mathbb{F}}_ p)\) is cyclic. This was previously known assuming the generalized Riemann hypothesis.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4067091
zbMATH Keywords: elliptic curve
zbMATH Keywords: prime of supersingular reduction
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
cites work: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
cites work: The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\).
cites work: Q3731746
cites work: Primitive points on elliptic curves
cites work: On Artin's conjecture
description: scientific article; zbMATH DE number 4067091