Stark–Heegner theorem

In number theory, the Baker–Heegner–Stark theorem[1] states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. 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 non-square integer. Then Q(√d) is a finite extension of Q of degree 2, called a quadratic extension. The class number of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if there exist principal ideals (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain (and hence a unique factorization domain) if and only if the class number of Q(√d) is equal to 1. The Baker–Heegner–Stark theorem can then be stated as follows:

If d < 0, then the class number of Q(√d) is equal to 1 if and only if

These are known as the Heegner numbers.

This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:[2]

where D is interpreted as the discriminant (either of the number field or of an elliptic curve with complex multiplication). This is more standard, as the D are then fundamental discriminants.

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, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967, which had many commonalities to Heegner's work, but sufficiently many differences that Stark considers the proofs to be different.[3] Heegner "died before anyone really understood what he had done".[4] Stark formally filled in the gap in Heegner's proof in 1969 (other contemporary papers produced various similar proofs by modular functions, but Stark concentrated on explicitly filling Heegner's gap).[5]

Alan Baker gave a completely different proof slightly earlier (1966) than Stark's work (or more precisely Baker reduced the result to a finite amount of computation, with Stark's work in his 1963/4 thesis already providing this computation), and 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 only 2 logarithms, when the result was already known from 1949 by Gelfond and Linnik.[6]

Stark's 1969 paper (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."[7]

Deuring, Siegel, and Chowla all gave slightly variant proofs by modular functions in the immediate years after Stark.[8] 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).[9] And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).[10]

The work of Gross and Zagier (1986) (Gross & Zagier 1986) combined with that of Goldfeld (1976) also gives an alternative proof.[11]

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 Stark–Heegner theorem (cognate to Stark–Heegner points as in page xiii of Darmon (2004)) but omitting Baker's name is atypical. Chowla (1970) gratuitously adds Deuring and Siegel in his paper's title.
  2. Elkies (1999), p. 93.
  3. Stark (2011) page 42
  4. Goldfeld (1985).
  5. Stark (1969a)
  6. Stark (1969b)
  7. Birch (2004)
  8. Chowla (1970)
  9. Kenku (1985).
  10. Chen (1999)
  11. Goldfeld (1985)

References
  • Birch, Bryan (2004), "Heegner Points: The Beginnings", MSRI Publications, 49: 1–10
  • Chen, Imin (1999), "On Siegel's Modular Curve of Level 5 and the Class Number One Problem", J. Number Theory, 74 (2): 278–297, doi:10.1006/jnth.1998.2320
  • Chowla, S. (1970), "The Heegner–Stark–Baker–Deuring–Siegel Theorem", Crelle, 241: 47–48
  • Darmon, Henri (2004), "Preface to Heegner Points and Rankin L-Series", MSRI Publications, 49: ix–xiii
  • 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, 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, doi:10.1007/BF01388809, MR 0833192.
  • 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
  • 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, 35, Cambridge University Press
  • Stark, H. M. (1969a), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1: 16–27, doi:10.1016/0022-314X(69)90023-7
  • Stark, H. M. (1969b), "A historical note on complex quadratic fields with class-number one.", Proc. Amer. Math. Soc., 21: 254–255, doi:10.1090/S0002-9939-1969-0237461-X
  • Stark, H. M. (2011), The Origin of the "Stark" conjectures, appearing in Arithmetic of L-functions
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