Supersingular primes for points on X0(p)/wp

2004/08/04 by David Jao
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT #msc:11G05

paper · pdf · doi:10.1016/j.jnt.2004.09.002

published as Journal of Number Theory, Volume 113, Issue 2, August 2005, pp. 208-225

arxiv created 2004/08/04 · openalex publication_date 2004/12/10 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For small odd primes p, we prove that most of the rational points on the modular curve X₀(p)/w_p parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for which the infinitude of supersingular primes is known. We give concrete examples illustrating how these techniques can be explicitly used to construct supersingular primes for such elliptic curves. Finally, we discuss generalizations to points defined over larger number fields and indicate the types of obstructions that arise for higher level modular curves.

Related

• A bound on the μ-invariants of supersingular elliptic curves
• On the modularity of supersingular elliptic curves over certain totally real number fields
• Infinitely many supersingular primes for some Mumford's abelian fourfolds
• Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves
• Ramification in Division Fields and Sporadic Points on Modular Curves