Stark–Heegner theorem

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In number theory, a branch of mathematics, the Stark–Heegner theorem 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 square-free integer (i.e., a product of distinct primes) other than 1. Then Q(√d) is a finite extension of Q, 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 Stark-Heegner 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
d = − 1, − 2, − 3, − 7, − 11, − 19, − 43, − 67, − 163.

This list is also written[1]:

D = − 3, − 4, − 7, − 8, − 11, − 19, − 43, − 67, − 163,

where D is interpreted as the discriminant (either of the number field or of an elliptic curve with complex multiplication).

Contents

[edit] History

This result was first conjectured by Gauss and essentially proven by Kurt Heegner in 1952. Heegner's proof had some minor gaps and was not accepted until Harold Stark gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner died unrecognized[2]. Stark formally filled in the gap in Heegner's proof in 1969.[3] Alan Baker gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation).

In 1985, Monsur Kenku[4] gave a novel proof using the Klein quartic. Noam Elkies gives an exposition of this result[5].

[edit] 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.

[edit] Notes

  1. ^ Elkies [1999: p. 93]
  2. ^ Goldfeld [1985]
  3. ^ Stark [1969]
  4. ^ Kenku [1985]
  5. ^ Elkies [1999: section 4.3]

[edit] References

Elkies, Noam D. (1999). "The Klein Quartic in Number Theory", in Levy [1999: p.51-101].

Goldfeld, Dorian. "The Gauss Class Number Problem For Imaginary Quadratic Fields".

Goldfeld, Dorian (1985). "Gauss' Class Number Problem for Imaginary Quadratic Fields", Bulletin of the American Mathematical Society, 13: 23-37.

Kenku, M.Q. (1985). “A note on the integral points of a modular curve of level 7”, Mathematika, 32: 45–48.

Levy, Silvio [editor] (1999). The Eightfold Way: The Beauty of Klein's Quartic Curve (Cambridge University Press).

Stark, H.M. (1969). "On the gap in the theorem of Heegner", Journal of Number Theory, 1: 16-27.

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