Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

DOI10.1016/j.jnt.2004.01.006zbMath1083.11036OpenAlexW2007608286MaRDI QIDQ1434337

John Brillhart, Patrick Morton

Publication date: 4 August 2004

Published in: Journal of Number Theory (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jnt.2004.01.006

zbMATH Keywords

class numberelliptic curvequadratic fieldHasse invariantsupersingularLegendre formmeromorphism

Mathematics Subject Classification ID

Quadratic extensions (11R11) Elliptic curves over global fields (11G05) Arithmetic ground fields for curves (14H25) Class numbers, class groups, discriminants (11R29) Arithmetic ground fields for abelian varieties (14K15)

Related Items (30)

Improved supersingularity testing of elliptic curves using Legendre form ⋮ On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials ⋮ Legendre polynomials roots and the \(F\)-pure threshold of bivariate forms ⋮ Congruences involving \({2k\choose k}^2{3k\choose k}\) ⋮ The Hasse invariant of the Tate normal form \(E_7\) and the supersingular polynomial for the Fricke group \(\Gamma_0^*(7)\) ⋮ Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials ⋮ Congruence formulas for Legendre modular polynomials ⋮ An elementary proof for the number of supersingular elliptic curves ⋮ Formal groups, supersingular abelian varieties and tame ramification ⋮ Explicit congruences for class equations ⋮ On the Galois groups of Legendre polynomials ⋮ On the ranks of the Jacobians of curves defined by Jacobi polynomials ⋮ HYPERELLIPTIC CURVES, CARTIER — MANIN MATRICES AND LEGENDRE POLYNOMIALS ⋮ On the \(j\)-invariants of CM-elliptic curves defined over \(\mathbb{Z}_p\) ⋮ Evaluation of some \(q\)-integrals in terms of the Dedekind eta function ⋮ Explicit identities for invariants of elliptic curves ⋮ Counting points on hyperelliptic curves of type \(y^2=x^{2g+1}+ax^{g+1}+bx\) ⋮ On the Hasse invariants of the Tate normal forms \(E_5\) and \(E_7\) ⋮ The cubic Fermat equation and complex multiplication on the Deuring normal form ⋮ Periodic points of algebraic functions and Deuring's class number formula ⋮ ON ASYMPTOTICALLY OPTIMAL TOWERS OVER QUADRATIC FIELDS RELATED TO GAUSS HYPERGEOMETRIC FUNCTIONS ⋮ Legendre polynomials and complex multiplication. I. ⋮ An elementary computation of the \(F\)-pure threshold of an elliptic curve ⋮ The Hasse invariant of the Tate normal form \(E_5\) and the class number of \(\mathbb{Q}(\sqrt{-5l})\) ⋮ Supersingular parameters of the Deuring normal form ⋮ Tame Galois realizations of \(\text{GL}_2(\mathbb F_{\ell})\) over \(\mathbb Q\) ⋮ On the \(p\)-th division polynomial ⋮ Minimal resolution of Atkin-Lehner quotients of \(X_0(N)\) ⋮ The number of linear factors of supersingular polynomials and sporadic simple groups ⋮ Algebraic properties of a family of Jacobi polynomials

Cites Work

• Supersingular j-invariants as singular moduli mod p
• The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\).
• Elliptic curves and modular forms in algebraic topology. Proceedings of a conference held at the Institute for Advanced Study, Princeton, NJ, Sept. 15-17, 1986
• Die Anzahl der Typen von Maximalordnungen einer definiten Quaternionenalgebra mit primer Grundzahl
• New cases of irreducibility for Legendre polynomials
• The coefficients of singular elliptic functions
• Congruence properties of special elliptic functions
• Some arithmetic properties of the Legendre polynomials
• On the Problem of Runs
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