View source for Stark–Heegner theorem

{{Short description|Quadratic imaginary number fields with unique factorisation}}
In [[number theory]], the '''Heegner theorem''' or '''Stark-Heegner theorem'''<ref>{{harvtxt|Elkies|1999}} calls this the Heegner theorem (cognate to Heegner points as in page xiii of {{harvtxt|Darmon|2004}}) but omitting Baker's name{{inconsistent|date=April 2025|reason=Baker? Not mentioned before}} is atypical. {{harvtxt|Chowla|1970}} gratuitously adds Deuring and Siegel in his paper's title.</ref> establishes the complete list of the [[quadratic field|quadratic imaginary number fields]] whose [[ring of integers|rings of integers]] are  [[unique factorization domain|principal ideal domains.]]  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]].

Let {{math|'''Q'''}} denote the set of [[rational number]]s, and let {{math|''d''}} be a [[square-free integer]]. The field {{math|'''Q'''({{radic|''d''}})}} is a [[quadratic extension]] of {{math|'''Q'''}}. The [[class number (number theory)|class number]] of {{math|'''Q'''({{radic|''d''}})}} is one [[if and only if]] the ring of integers of {{math|'''Q'''({{radic|''d''}})}} is a [[principal ideal domain]]. The Baker–Heegner–Stark theorem{{inconsistent|date=April 2025}} can then be stated as follows:

:If {{math|''d'' < 0}}, then the class number of {{math|'''Q'''({{radic|''d''}})}} is one 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.

By replacing {{mvar|d}} with the [[Discriminant of an algebraic number field|discriminant]] {{mvar|D}} of {{math|'''Q'''({{radic|''d''}})}} this list is often  written as:<ref>{{harvtxt|Elkies|1999}}, p.&nbsp;93.</ref>
:<math>D \in\{ -3, -4, -7, -8, -11, -19, -43, -67, -163\}.</math>

==History==
This result was first conjectured by [[Carl Friedrich Gauss|Gauss]] in Section 303 of his ''[[Disquisitiones Arithmeticae]]'' (1798).  It was essentially proven by [[Kurt Heegner]] in 1952,<ref>{{harvtxt|Heegner|1952}}</ref> but Heegner's proof was not accepted until an academic mathematician [[Harold Stark]] published a proof<ref>{{harvtxt|Stark|1967}}</ref> in 1967 which had many commonalities to Heegner's work, though Stark considers the proofs to be different.<ref>{{harvtxt|Stark|2011}} page 42</ref> Heegner "died before anyone really understood what he had done".<ref>{{harvtxt|Goldfeld|1985}}.</ref> In {{harv|Stark|1969a}} Stark works through Heegner's proof to highlight what the gap in Heegner’s proof consisted of; other contemporary papers produced various similar proofs using modular functions.<ref>{{harvtxt|Stark|1969a}}</ref> (Heegner's paper dealt mainly with the [[congruent number]] problem, also using modular functions.<ref>{{harvtxt|Birch|2004}}</ref>)

[[Alan Baker (mathematician)|Alan Baker]]'s slightly earlier 1966 proof used completely different principles which reduced the result to a finite amount of computation, with Stark's 1963/4 thesis already providing this computation; Baker won the [[Fields Medal]] for his methods. Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to a statement about only 2 logarithms which was already known from 1949 by Gelfond and Linnik.<ref>{{harvtxt|Stark|1969b}}</ref>

Stark's 1969 paper {{harv|Stark|1969a}} also cited the 1895 text by [[Heinrich Martin Weber|Weber]] and noted that if Weber had "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". [[Bryan Birch]] notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's ''Algebra'' to appreciate Heegner's achievement."<ref>{{harvtxt|Birch|2004}}</ref> Stark's 1969 paper can be seen as a good argument for calling the result '''Heegner's Theorem'''.

In the immediate years after Stark,<ref>{{harvtxt|Stark|1967}}</ref> Deuring,<ref>{{harvtxt|Deuring|1968}}</ref> Siegel,<ref>{{harvtxt|Siegel|1968}}</ref> and Chowla<ref>{{harvtxt|Chowla|1970}}</ref> all gave slightly variant proofs by [[Modular_form#Modular_functions|modular functions]]. Other versions in this genre have also cropped up over the years. For instance, in 1985, Monsur Kenku gave a proof using the [[Klein quartic]] (though again utilizing modular functions).<ref>{{harvtxt|Kenku|1985}}.</ref> And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).<ref>{{harvtxt|Chen|1999}}</ref>

The work of Gross and Zagier (1986) {{harv|Gross|Zagier|1986}} combined with that of Goldfeld (1976) also gives an alternative proof.<ref>{{harvtxt|Goldfeld|1985}}</ref>

==Real case==
On the other hand, it is unknown whether there are infinitely many ''d'' > 0 for which '''Q'''({{radic|''d''}}) has class number 1. Computational results indicate that there are many such fields. [[List of number fields with class number one#Quadratic number fields|Number Fields with class number one]] provides a list of some of these.

==Notes==
{{reflist|2}}

==References==
*{{citation | last=Birch|first=Bryan|title=Heegner Points: The Beginnings | pages = 1–10 | volume = 49 | year = 2004 | journal =  MSRI Publications | url=https://library.slmath.org/books/Book49/files/01birch.pdf}}
*{{citation | last=Chen|first=Imin | title = On Siegel's Modular Curve of Level 5 and the Class Number One Problem | year = 1999 | journal = Journal of Number Theory | volume = 74|issue=2 | pages = 278–297 | doi = 10.1006/jnth.1998.2320| doi-access = free }}
*{{citation | last=Chowla|first = S.|title = The Heegner–Stark–Baker–Deuring–Siegel Theorem | journal = [[Journal für die reine und angewandte Mathematik]] | year = 1970 | volume = 241 | pages = 47–48 | doi=10.1515/crll.1970.241.47 | url=http://eudml.org/doc/150996}}
*{{citation | last=Darmon | first = Henri | title=Preface to ''Heegner Points and Rankin L-Series''| pages = ix–xiii | volume = 49 | year = 2004 | journal =  MSRI Publications | url=http://library.msri.org/books/Book49/files/00pref.pdf}}
*{{cite journal | last=Deuring | first=Max | title=Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins | journal=Inventiones Mathematicae | volume=5 | issue=3 | date=1968 | issn=0020-9910 | doi=10.1007/BF01425548 | pages=169–179 | url=http://link.springer.com/10.1007/BF01425548 | language=de | access-date=2025-08-24| url-access=subscription }}
*{{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|mr=1722413}}
*{{citation|last=Goldfeld|first=Dorian|authorlink=Dorian M. Goldfeld|title=Gauss's 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|doi-access=free}}
*{{citation|last1=Gross|first1=Benedict H.|author1-link=Benedict Gross|last2=Zagier|first2=Don B.|author2-link=Don Zagier|doi=10.1007/BF01388809|mr=0833192|title=Heegner points and derivatives of L-series|journal=[[Inventiones Mathematicae]]|volume=84|year=1986|issue=2|pages=225–320|bibcode=1986InMat..84..225G|s2cid=125716869}}.
*{{citation | last=Heegner | first=Kurt | authorlink=Kurt Heegner | doi=10.1007/BF01174749|mr=0053135 | title=Diophantische Analysis und Modulfunktionen | journal=[[Mathematische Zeitschrift]] | volume=56 | issue=3 | year=1952 | pages=227–253 | s2cid=120109035 | language=German | trans-title=Diophantine Analysis and Modular Functions}}
*{{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|mr=0817106}}
*{{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}}
*{{cite journal | last=Siegel | first=Carl Ludwig | title=Zum Beweise des Starkschen Satzes | journal=Inventiones Mathematicae | volume=5 | issue=3 | date=1968 | issn=0020-9910 | doi=10.1007/BF01425549 | pages=180–191 | url=http://link.springer.com/10.1007/BF01425549 | language=de | access-date=2025-08-25| url-access=subscription }}
*{{cite journal | last=Stark | first=H. M. |authorlink=Harold Stark | title=A complete determination of the complex quadratic fields of class-number one. | journal=Michigan Mathematical Journal | volume=14 | issue=1 | year=1967 | issn=0026-2285 | doi=10.1307/mmj/1028999653 | url=https://projecteuclid.org/journals/michigan-mathematical-journal/volume-14/issue-1/A-complete-determination-of-the-complex-quadratic-fields-of-class/10.1307/mmj/1028999653.pdf | access-date=2025-08-27}}
*{{citation|last=Stark|first=H. M.|authorlink=Harold Stark |year=1969a|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|issue=1|pages=16&ndash;27|doi=10.1016/0022-314X(69)90023-7|bibcode=1969JNT.....1...16S|hdl=2027.42/33039|hdl-access=free}}
*{{citation |last=Stark|first=H. M.|authorlink=Harold Stark | title = A historical note on complex quadratic fields with class-number one. | journal = [[Proceedings of the American Mathematical Society]] | year = 1969b | volume = 21 | pages = 254–255 | doi = 10.1090/S0002-9939-1969-0237461-X| doi-access = free }}
*{{citation | title = The Origin of the "Stark" conjectures | last=Stark|first=H. M.|authorlink=Harold Stark | year = 2011 | volume = appearing in ''Arithmetic of L-functions'' | url=https://books.google.com/books?isbn=0821886983}}

{{DEFAULTSORT:Stark-Heegner theorem}}
[[Category:Theorems in algebraic number theory]]