• Corpus ID: 59408183

# The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer

@article{Goldfeld1976TheCN,
title={The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer},
author={Dorian Goldfeld},
journal={Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze},
year={1976},
volume={3},
pages={623-663}
}
• D. Goldfeld
• Published 1976
• Mathematics
• Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze
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95 Citations
The conjectures of Birch and Swinnerton-Dyer and the class numbers of quadratic fields
© Société mathématique de France, 1977, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les
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This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory"
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We aim to re-prove a theorem conjectured by Gauss, namely there are exactly 9 imaginary quadratic fields Q( √ −q) with class number 1: specifically the list is q ∈ {3, 4, 7, 8, 11, 19, 43, 67, 163}.
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## References

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An asymptotic formula relating the Siegel zero and the class number of quadratic fields
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions
A Simple Proof of Siegel's Theorem.
• D. Goldfeld
• Mathematics, Medicine
Proceedings of the National Academy of Sciences of the United States of America
• 1974
A brief and simple proof of Siegel's celebrated theorem that h(d) >> d(1/2-[unk]), as d --> infinity, is given. Here h(d) denotes the class number of the quadratic field Q([unk]-d). Simple proofs
ON FUNCTIONAL EQUATIONS OF DIRICHLET FUNCTIONS
The paper adduces a theorem of a general form on functional equations of the Dirichlet functions, as well as certain of its corollaries.