Editing Stark–Heegner theorem (section)

==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, but Heegner's proof was not accepted until an academic mathematician [[Harold Stark]] published a proof 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> Stark formally paraphrases Heegner's proof in 1969; other contemporary papers produced various similar proofs using modular functions.<ref>{{harvtxt|Stark|1969a}}</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; he 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]] 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>

Deuring, Siegel, and Chowla all gave slightly variant proofs by [[Modular_form#Modular_functions|modular functions]] in the immediate years after Stark.<ref>{{harvtxt|Chowla|1970}}</ref> 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>