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

These are known as the Heegner numbers.

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

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

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 in 1967 which had many commonalities to Heegner's work, though Stark considers the proofs to be different.^[4] Heegner "died before anyone really understood what he had done".^[5] Stark formally paraphrases Heegner's proof in 1969; other contemporary papers produced various similar proofs using modular functions.^[6] (Heegner's paper dealt mainly with the congruent number problem, also using modular functions.^[7])

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.^[8]

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."^[9] Stark's 1969 paper can be seen as a good argument for calling the result Heegner's Theorem.

In the immediate years after Stark,^[10] Deuring,^[11] Siegel,^[12] and Chowla all gave slightly variant proofs by modular functions.^[13] 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).^[14] And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).^[15]

The work of Gross and Zagier (1986) (Gross & Zagier 1986) combined with that of Goldfeld (1976) also gives an alternative proof.^[16]

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 (2011) page 42
5. Goldfeld (1985).
6. Stark (1969a)
7. Birch (2004)
8. Stark (1969b)
9. Birch (2004)
10. Chowla (1970)
11. Deuring (1968)
12. Siegel (1968)
13. Chowla (1970)
14. Kenku (1985).
15. Chen (1999)
16. Goldfeld (1985)

References

• Birch, Bryan (2004), "Heegner Points: The Beginnings" (PDF), MSRI Publications, 49: 1–10
• Chen, Imin (1999), "On Siegel's Modular Curve of Level 5 and the Class Number One Problem", Journal of Number Theory, 74 (2): 278–297, doi:10.1006/jnth.1998.2320
• Chowla, S. (1970), "The Heegner–Stark–Baker–Deuring–Siegel Theorem", Journal für die reine und angewandte Mathematik, 241: 47–48, doi:10.1515/crll.1970.241.47
• Darmon, Henri (2004), "Preface to Heegner Points and Rankin L-Series" (PDF), MSRI Publications, 49: ix–xiii
• Deuring, Max (1968). "Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins". Inventiones Mathematicae (in German). 5 (3): 169–179. doi:10.1007/BF01425548. ISSN 0020-9910. Retrieved 2025-08-24.
• Elkies, Noam D. (1999), "The Klein Quartic in Number Theory" (PDF), in Levy, Silvio (ed.), The Eightfold Way: The Beauty of Klein's Quartic Curve, MSRI Publications, vol. 35, Cambridge University Press, pp. 51–101, MR 1722413
• Goldfeld, Dorian (1985), "Gauss's class number problem for imaginary quadratic fields", Bulletin of the American Mathematical Society, 13: 23–37, doi:10.1090/S0273-0979-1985-15352-2, MR 0788386
• Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series", Inventiones Mathematicae, 84 (2): 225–320, Bibcode:1986InMat..84..225G, doi:10.1007/BF01388809, MR 0833192, S2CID 125716869.
• Heegner, Kurt (1952), "Diophantische Analysis und Modulfunktionen" [Diophantine Analysis and Modular Functions], Mathematische Zeitschrift (in German), 56 (3): 227–253, doi:10.1007/BF01174749, MR 0053135, S2CID 120109035
• Kenku, M. Q. (1985), "A note on the integral points of a modular curve of level 7", Mathematika, 32: 45–48, doi:10.1112/S0025579300010846, MR 0817106
• Levy, Silvio, ed. (1999), The Eightfold Way: The Beauty of Klein's Quartic Curve, MSRI Publications, vol. 35, Cambridge University Press
• Siegel, Carl Ludwig (1968). "Zum Beweise des Starkschen Satzes". Inventiones Mathematicae (in German). 5 (3): 180–191. doi:10.1007/BF01425549. ISSN 0020-9910. Retrieved 2025-08-25.
• Stark, H. M. (1969a), "On the Gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1 (1): 16–27, Bibcode:1969JNT.....1...16S, doi:10.1016/0022-314X(69)90023-7, hdl:2027.42/33039
• Stark, H. M. (1969b), "A historical note on complex quadratic fields with class-number one.", Proceedings of the American Mathematical Society, 21: 254–255, doi:10.1090/S0002-9939-1969-0237461-X
• Stark, H. M. (2011), The Origin of the "Stark" conjectures, vol. appearing in Arithmetic of L-functions