Stark–Heegner theorem

Template:Short description In number theory, the Heegner theorem or Stark-Heegner theorem<ref>Template:Harvtxt calls this the Heegner theorem (cognate to Heegner points as in page xiii of Template:Harvtxt) but omitting Baker's nameTemplate:Inconsistent is atypical. Template:Harvtxt gratuitously adds Deuring and Siegel in his paper's title.</ref> establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.

Let Template:Math denote the set of rational numbers, and let Template:Math be a square-free integer. The field Template:Math is a quadratic extension of Template:Math. The class number of Template:Math is one if and only if the ring of integers of Template:Math is a principal ideal domain. The Baker–Heegner–Stark theoremTemplate:Inconsistent can then be stated as follows:

If Template:Math, then the class number of Template:Math 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 numbers.

By replacing Template:Mvar with the discriminant Template:Mvar of Template:Math this list is often written as:<ref>Template:Harvtxt, p. 93.</ref>

<math>D \in\{ -3, -4, -7, -8, -11, -19, -43, -67, -163\}.</math>

History

This result was first conjectured by Gauss in Section 303 of his Disquisitiones Arithmeticae (1798). It was essentially proven by Kurt Heegner in 1952,<ref>Template:Harvtxt</ref> but Heegner's proof was not accepted until an academic mathematician Harold Stark published a proof<ref>Template:Harvtxt</ref> in 1967 which had many commonalities to Heegner's work, though Stark considers the proofs to be different.<ref>Template:Harvtxt page 42</ref> Heegner "died before anyone really understood what he had done".<ref>Template:Harvtxt.</ref> In Template:Harv 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>Template:Harvtxt</ref> (Heegner's paper dealt mainly with the congruent number problem, also using modular functions.<ref>Template:Harvtxt</ref>)

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>Template:Harvtxt</ref>

Stark's 1969 paper Template:Harv also cited the 1895 text by 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>Template:Harvtxt</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>Template:Harvtxt</ref> Deuring,<ref>Template:Harvtxt</ref> Siegel,<ref>Template:Harvtxt</ref> and Chowla<ref>Template:Harvtxt</ref> all gave slightly variant proofs by 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>Template:Harvtxt.</ref> And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).<ref>Template:Harvtxt</ref>

The work of Gross and Zagier (1986) Template:Harv combined with that of Goldfeld (1976) also gives an alternative proof.<ref>Template:Harvtxt</ref>

Real case

On the other hand, it is unknown whether there are infinitely many d > 0 for which Q(Template:Radic) has class number 1. Computational results indicate that there are many such fields. Number Fields with class number one provides a list of some of these.

References

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