Stark–Heegner theorem

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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.

Contents

Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q(√d) is a quadratic extension of Q. The class number of Q(√d) is one if and only if the ring of integers of Q(√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(√d) is one if and only if

These are known as the Heegner numbers.

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

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.

Notes

  1. ↑ Elkies (1999) calls this the Heegner theorem (cognate to Heegner points as in page xiii of Darmon (2004)) but omitting Baker's name[ inconsistent ] is atypical. Chowla (1970) gratuitously adds Deuring and Siegel in his paper's title.
  2. ↑ Elkies (1999), p. 93.
  3. ↑ Heegner (1952)
  4. ↑ Stark (1967)
  5. ↑ Stark (2011) page 42
  6. ↑ Goldfeld (1985).
  7. ↑ Stark (1969a)
  8. ↑ Birch (2004)
  9. ↑ Stark (1969b)
  10. ↑ Birch (2004)
  11. ↑ Stark (1967)
  12. ↑ Deuring (1968)
  13. ↑ Siegel (1968)
  14. ↑ Chowla (1970)
  15. ↑ Kenku (1985).
  16. ↑ Chen (1999)
  17. ↑ Goldfeld (1985)

References

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