Stark–Heegner theorem

In number theory, the Heegner theorem or Stark-Heegner theorem^[1] 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 $Q$ denote the set of rational numbers, and let $d$ be a square-free integer. The field $Q (\sqrt d)$ is a quadratic extension of $Q$ . The class number of $Q (\sqrt d)$ is one if and only if the ring of integers of $Q (\sqrt d)$ is a principal ideal domain. The Baker–Heegner–Stark theorem^{[inconsistent]} can then be stated as follows:

If

d < 0

, then the class number of

Q (\sqrt d)

is one if and only if

d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.

These are known as the Heegner numbers.

By replacing $d$ with the discriminant $D$ of $Q (\sqrt d)$ this list is often written as:^[2]

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

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,^[3] but Heegner's proof was not accepted until an academic mathematician Harold Stark published a proof^[4] in 1967 which had many commonalities to Heegner's work, though Stark considers the proofs to be different.^[5] Heegner "died before anyone really understood what he had done".^[6] In (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.^[7] (Heegner's paper dealt mainly with the congruent number problem, also using modular functions.^[8])

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.^[9]

Stark's 1969 paper (Stark 1969a) 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."^[10] Stark's 1969 paper can be seen as a good argument for calling the result Heegner's Theorem.

In the immediate years after Stark,^[11] Deuring,^[12] Siegel,^[13] and Chowla^[14] 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).^[15] And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).^[16]

The work of Gross and Zagier (1986) (Gross & Zagier 1986) combined with that of Goldfeld (1976) also gives an alternative proof.^[17]

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. Number Fields with class number one provides a list of some of these.