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In [[number theory]], a branch of [[mathematics]], the '''Stark–Heegner theorem''' states precisely which [[quadratic field|quadratic imaginary number fields]] admit [[unique factorization domain|unique factorisation]] in their [[ring of integers]]. It solves a special case of Gauss's [[Class number problem for imaginary quadratic fields|class number problem]] of determining the number of imaginary quadratic fields that have a given fixed [[ideal class group|class number]].
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Let '''Q''' denote the set of [[rational number]]s, and let ''d'' be a square-free [[integer]] (i.e., a product of distinct [[prime number|primes]]) other than 1. Then [[algebraic number field|'''Q'''(√''d'')]] is a [[field extension|finite extension]] of '''Q''', called a quadratic extension. The [[ideal class group|class number]] of '''Q'''(√''d'') is the number of [[equivalence class]]es of [[ring ideal|ideal]]s of the ring of integers of '''Q'''(√''d''), where two ideals ''I'' and ''J'' are equivalent [[if and only if]] there exist [[principal ideal]]s (''a'') and (''b'') such that (''a'')''I'' = (''b'')''J''. Thus, the ring of integers of '''Q'''(√''d'') is a [[principal ideal domain]] (and hence a [[unique factorization domain]]) if and only if the class number of '''Q'''(√''d'') is equal to 1. The Stark–Heegner theorem can then be stated as follows:
 
:If ''d'' < 0, then the class number of '''Q'''(√''d'') is equal to 1 if and only if
 
:: <math>d \in \{\, -1, -2, -3, -7, -11, -19, -43, -67, -163\,\}.</math>
These are known as the [[Heegner number]]s.
 
This list is also written, replacing &minus;1 with &minus;4 and &minus;2 with &minus;8 (which does not change the field), as:<ref>{{harvtxt|Elkies|1999}}, p. 93.</ref>
 
:<math>D = -3, -4, -7, -8, -11, -19, -43, -67, -163,\,</math>
where ''D'' is interpreted as the [[Discriminant of an algebraic number field|discriminant]] (either of the [[number field]] or of an [[elliptic curve]] with [[complex multiplication]]).
 
==History==
This result was first conjectured by [[Carl Friedrich Gauss|Gauss]].  It was essentially proven by [[Kurt Heegner]] in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until [[Harold Stark]] gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done".<ref>{{harvtxt|Goldfeld|1985}}.</ref> Stark formally filled in the gap in Heegner's proof in 1969.<ref>{{harvtxt|Stark|1969}}.</ref> [[Alan Baker (mathematician)|Alan Baker]] gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation). Stark's 1969 paper also cited the widely known 1895 text by [[Heinrich Martin Weber]] and noted that had Weber "only made the observation that the reducibility of [a certain equation] would lead to a [[Diophantine equation]], the class-number one problem would have been solved 60 years ago".
 
In 1985, Monsur Kenku<ref>{{harvtxt|Kenku|1985}}.</ref> gave a novel proof using the [[Klein quartic]]. [[Noam Elkies]] gives an exposition of this result.<ref>{{harvtxt|Elkies|1999}}, section 4.3.</ref>
 
==Real case==
On the other hand, it is unknown whether there are infinitely many ''d'' > 0 for which '''Q'''(√''d'') has class number 1. Computational results indicate that there are many such fields.
 
==Notes==
{{reflist|2}}
 
==References==
 
*{{citation|authorlink=Noam Elkies|last=Elkies|first=Noam D.|year=1999|contribution-url=http://library.msri.org/books/Book35/files/elkies.pdf|contribution=The Klein Quartic in Number Theory|pages=51–101|editor-last=Levy|editor-first=Silvio|url=http://library.msri.org/books/Book35|title=The Eightfold Way: The Beauty of Klein's Quartic Curve|publisher=Cambridge University Press|series=MSRI Publications|volume=35}}.
*{{citation|last=Goldfeld|first=Dorian|authorlink=Dorian M. Goldfeld|title=The Gauss Class Number Problem For Imaginary Quadratic Fields|year=1985|mr=788386|journal=[[Bulletin of the American Mathematical Society]]|volume=13|pages=23–37|doi=10.1090/S0273-0979-1985-15352-2}}.
*{{citation | last=Heegner | first=Kurt | authorlink=Kurt Heegner | doi=10.1007/BF01174749|mr=0053135 | title=Diophantische Analysis und Modulfunktionen | journal=[[Mathematische Zeitschrift]] | volume=56 | year=1952 | pages=227–253}}
*{{citation|last=Kenku|first=M. Q.|year=1985|title=A note on the integral points of a modular curve of level 7|journal=Mathematika|volume=32|pages=45–48|doi=10.1112/S0025579300010846}}.
*{{citation|editor-last=Levy|editor-first=Silvio|year=1999|url=http://library.msri.org/books/Book35/contents.html|title=The Eightfold Way: The Beauty of Klein's Quartic Curve|publisher=Cambridge University Press|series=MSRI Publications|volume=35}}.
*{{citation|last=Stark|first=H. M.|authorlink=Harold Stark|year=1969|url=http://deepblue.lib.umich.edu/bitstream/2027.42/33039/1/0000425.pdf|title=On the gap in the theorem of Heegner|journal=[[Journal of Number Theory]]|volume=1|pages=16&ndash;27|doi=10.1016/0022-314X(69)90023-7}}.
 
{{DEFAULTSORT:Stark-Heegner theorem}}
[[Category:Theorems in algebraic number theory]]

Latest revision as of 21:36, 20 November 2014

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