• Corpus ID: 14292849

MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM

@inproceedings{Booher2014MODULARCA,
  title={MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM},
  author={Jeremy Booher},
  year={2014}
}
There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner’s paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner’s proof was essentially correct and in fact equivalent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new… 
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References

SHOWING 1-10 OF 16 REFERENCES
ELLIPTIC FUNCTIONS
The first systematic account of the theory of elliptic functions and the state of the art around the turn of the century. Preceding general class field theory and therefore incomplete. Contains a
Noordsij Primes of the form x 2 + ny 2
This essay will address the question of which primes can be represented as x + ny, with x, y ∈ Z, for different values of n ∈ N. One aim of this essay is to classify all such primes p using simple
Weber's class invariants
Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular
The Arithmetic of Elliptic Curves
TLDR
This research focuses on 9 specific elliptic curves E over Q, each with complex multiplication by the maximal order in an imaginary quadratic field, defined by the generators ω1, ω2 ∈ C of the period lattice.
Lehrbuch der Algebra
  • G. M.
  • Mathematics
    Nature
  • 1896
A “Treatise on Algebra” is rarely found to fulfil the promise of its title. It is too often a mere collection of problems and examples, thrown together without much regard to order or method; such
On Siegel's Modular Curve of Level 5 and the Class Number One Problem
Abstract Another derivation of an explicit parametrisation of Siegel's modular curve of level 5 is obtained with applications to the class number one problem.
A First Course in Modular Forms
Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves
Complex Multiplication
  • G. Frey, T. Lange
  • Computer Science
    Handbook of Elliptic and Hyperelliptic Curve Cryptography
  • 2005
...
...