Class number problem for imaginary quadratic fields

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In mathematics, the Gauss class number problem (for imaginary quadratic fields), as is usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n. This is a question of effective computation. The basic result that provided assurance that such a list would be finite (and that at most one more such field could exist beyond the nine already known) did not, in the form proved by Hans Heilbronn and Linfoot in 1934, allow such a calculation even in principle. See effective results in number theory.

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 important 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 fields with class number one is Q(√(−d)) with −d one of -1,-2,-3,-7,-11,-19,-43,-67,-163.

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 [1].

The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but hlog ε, where ε is a fundamental unit. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.