Editing Class number problem (section)

==Modern developments==
In 1934, [[Hans Heilbronn]] proved the Gauss conjecture.<ref name="GaussClassNumber" /> Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.

Also in 1934, Heilbronn and [[Edward Linfoot]] showed that there were at most 10<ref name="HeilbronnLinfoot">{{cite journal | last=Heilbronn | first=H. | last2=Linfoot | first2=E. H. | title=ON THE IMAGINARY QUADRATIC CORPORA OF CLASS-NUMBER ONE | journal=The Quarterly Journal of Mathematics | volume=os-5 | issue=1 | date=1934 | issn=0033-5606 | doi=10.1093/qmath/os-5.1.293 | pages=293–301 | url=https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-5.1.293 | access-date=2025-04-21| url-access=subscription }}</ref> imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).
The result was ineffective (see [[effective results in number theory]]): it did not give bounds on the size of the remaining field.

In later developments, the case ''n'' = 1 was first discussed by [[Kurt Heegner]], using [[modular form]]s and [[modular equation]]s to show that no further such field could exist. This work was not initially accepted; only with later work of [[Harold Stark]] and [[Bryan Birch]] (e.g. on the [[Stark–Heegner theorem]] and [[Heegner number]]) was the position clarified and Heegner's work understood. Practically simultaneously, [[Alan Baker (mathematician)|Alan Baker]] proved what we now know as [[Baker's theorem]] on [[linear forms in logarithms]] of [[algebraic number]]s, which resolved the problem by a completely different method.  The case ''n'' = 2 was tackled shortly afterwards, at least in principle, as an application of Baker's work.<ref name=Baker>{{harvtxt|Baker|1990}}</ref>

The complete list of imaginary quadratic fields with class number 1 is <math>\mathbf{Q}(\sqrt{d})</math> where ''d'' is one of
:<math>-1, -2, -3, -7, -11, -19, -43, -67, -163.</math>

The general case awaited the discovery of [[Dorian Goldfeld]] in 1976 that the class number problem could be connected to the [[L-function|''L''-function]]s of [[elliptic curve]]s.<ref name=Goldfeld>{{harvtxt|Goldfeld|1985}}</ref> This effectively reduced the question of effective determination to one about establishing the existence of a multiple zero of such an ''L''-function.<ref name=Goldfeld/> With the proof of the [[Gross–Zagier theorem]] in 1986, a complete list of imaginary quadratic fields with a given class number could be specified by a finite calculation. All cases up to ''n'' = 100 were computed by Watkins in 2004.<ref name=watkins/> The class number of <math>\mathbf{Q}(\sqrt{-d})</math> for ''d'' = 1, 2, 3, ... is

:<math>1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 4, 2, 1, 4, 1, 1, 2, 4, 2, 3, 2, 1, 6, 1, 1, 6, 4, 3, 1, ...</math> {{OEIS|A202084}}.