Stark–Heegner theorem

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

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

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:^[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).

History

This result was first conjectured by Gauss. 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 Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done".^[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). Stark's 1969 paper also cited the widely known 1895 text by Heinrich Martin Weber and noted that had Weber "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".

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

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.

Notes

1. Elkies (1999), p. 93.
2. Goldfeld (1985).
3. Stark (1969).
4. Kenku (1985).
5. Elkies (1999), section 4.3.

References

• Elkies, Noam D. (1999), "The Klein Quartic in Number Theory" (PDF), in Levy, Silvio, 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 788386
• Heegner, Kurt (1952), "Diophantische Analysis und Modulfunktionen" [Diophantine Analysis and Modular Functions], Mathematische Zeitschrift (in German) 56: 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. (1969), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory 1: 16–27, doi:10.1016/0022-314X(69)90023-7