Class Number Problem - Modern Developments

Modern Developments

In 1934, Hans Heilbronn proved the Gauss Conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.

Also in 1934, Heilbronn and Edward Linfoot showed that there were at most 10 imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further). The result was ineffective (see effective results in number theory): it did not allow bounds on the size of the remaining field.

In later developments, the case n = 1 was first discussed by Kurt Heegner, using modular forms and modular equations to show that no further such field could exist. This work was not initially accepted; only with later work of Harold Stark and Bryan Birch was the position clarified, and Heegner's work understood. See Stark–Heegner theorem, Heegner number. Practically simultaneously, Alan Baker proved an Baker's theorem on linear forms in logarithms of algebraic numbers which resolved the problem by a completely different method. The case n = 2 was tackled shortly afterwards, at least in principle, as an application of Baker's work.

The complete list of imaginary quadratic fields with class number one is with k one of

The general case awaited the discovery of Dorian Goldfeld that the class number problem could be connected to the L-functions of elliptic curves. This reduced the question, in principle, of effective determination, to one about establishing the existence of a multiple zero of such an L-function. This could be done on the basis of the later Gross-Zagier theorem. So at that point one could specify a finite calculation, the result of which would be a complete list for a given class number. In fact in practice such lists that are probably complete can be made by relatively simple methods; what is at issue is certainty. The cases up to n = 100 have now (2004) been done: see Watkins (2004).