Stark–Heegner theorem
In
Let Q denote the set of
- 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, but Heegner's proof was not accepted until an establishment mathematician Harold Stark rewrote the 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 paraphrases Heegner's proof in 1969 (other contemporary papers produced various similar proofs by modular functions.[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 (
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
- ^ Elkies (1999) calls this the Heegner theorem (cognate to 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.
- ^ Elkies (1999), p. 93.
- ^ Stark (2011) page 42
- ^ Goldfeld (1985).
- ^ Stark (1969a)
- ^ Stark (1969b)
- ^ Birch (2004)
- ^ Chowla (1970)
- ^ Kenku (1985).
- ^ Chen (1999)
- ^ 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,
- Chowla, S. (1970), "The Heegner–Stark–Baker–Deuring–Siegel Theorem",
- Darmon, Henri (2004), "Preface to Heegner Points and Rankin L-Series" (PDF), MSRI Publications, 49: ix–xiii
- MR 1722413
- MR 0788386
- S2CID 125716869.
- S2CID 120109035
- Kenku, M. Q. (1985), "A note on the integral points of a modular curve of level 7", MR 0817106
- Levy, Silvio, ed. (1999), The Eightfold Way: The Beauty of Klein's Quartic Curve, MSRI Publications, vol. 35, Cambridge University Press
- hdl:2027.42/33039
- Stark, H. M. (2011), The Origin of the "Stark" conjectures, vol. appearing in Arithmetic of L-functions