Gauss’ class number problem for imaginary quadratic fields. (English) Zbl 0572.12004

Goldfeld, Dorian

Bull. Am. Math. Soc., New Ser. 13, 23-37 (1985).

Gauss’ class number problem for imaginary quadratic fields \(K_ D:=\mathbb Q(\sqrt{D})\), where \(D\in\mathbb Z\), \(D<0\), denotes the discriminant, consists in finding an effective algorithm for determining all such fields \(K_ D\) with given class number \(h_ D=h\). Of course, this task presupposes the truth of Gauss’ conjecture stating that there is only a finite number of those fields \(K_ D\) with fixed class number \(h_ D\).
The author first describes the history of the problem, starting out with the observation of Euler and Legendre that \(x^ 2-x+41\) and \(x^ 2+x+41\) is a prime for \(x=1, 2,\dots, 40\) and \(x=0,1,\dots,39\) respectively, and winding up with the recent solution of the problem due to Goldfeld–Gross–Zagier. In fact, Gauss’ problem is settled by the following theorem of Goldfeld–Gross–Zagier: For every \(\varepsilon >0\), there exists an effectively computable constant \(c>0\) such that \(h_ D>c (\log | D|)^{1-\varepsilon}.\)
Oesterlé (1984) computed the constant in this theorem and Mestre, Oesterlé and Serre, by showing that a certain elliptic curve is modular, made it possible to establish a complete list of all imaginary quadratic fields \(K_ D\) with class number \(h_ D=3\). The corresponding lists for the cases of \(h_ D=1\) and \(h_ D=2\) had been obtained earlier by Heegner-Baker-Stark-Deuring-Siegel and by Baker-Stark, respectively. The author points out that, when combined with the solution of the case of \(h_ D=4\), these results would yield the complete finite list of all integers n admitting a unique representation as a sum of three squares: \(n=x^ 2+y^ 2+z^ 2\) \((x\geq y\geq z\geq 0).\)
Gauss’ conjecture itself turned out to be true already as a consequence of the theorem of Hecke–Deuring–Heilbronn stating that \(h_ D\to \infty\) as \(D\to -\infty\). A theorem of the author (1976) then reduces Gauss’ class number problem via the famous conjecture of Birch and Swinnerton-Dyer to showing that there exists a modular elliptic curve \(E_ 0\) over \(\mathbb Q\) whose Hasse-Weil \(L\)-function \(L(E_ 0,s)\) has a triple zero at \(s=1\). The amazing theorem of Gross-Zagier (1983) implies that this is true, e.g., for the curve \(E_ 0: -139y^ 2=x^ 3+4x^ 2-48x+80\) of conductor \(N=37\cdot (139)^ 2\). The combination of these latter two theorems finally led to the above-cited decisive theorem of Goldfeld–Gross–Zagier and hence to a complete solution of Gauss’ class number problem.
We should like to mention that in the list of contributors to the class-number-one problem given on pp. 32–33, an important paper of C. Meyer [J. Reine Angew. Math. 242, 179–214 (1970; Zbl 0218.12007)] is missing, in which, aside from a detailed description of the history of the problem, a conjecture of Weber is proved. This conjecture played a controversial role in the interpretation of Heegner’s original reasoning in his solution of the class-number-one case.

Reviewer: Horst G. Zimmer (Saarbrücken)

MSC:

11R29	Class numbers, class groups, discriminants
11R11	Quadratic extensions
11Y40	Algebraic number theory computations
11G05	Elliptic curves over global fields
01A50	History of mathematics in the 18th century
11-03	History of number theory
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11F11	Holomorphic modular forms of integral weight

Keywords:

Birch-Swinnerton-Dyer conjecture; sums of squares; class number problem; imaginary quadratic fields; Gauss’ conjecture; modular elliptic curve; Hasse-Weil L-function; class-number-one problem

Citations:

Zbl 0551.12003; Zbl 0355.12005; Zbl 0345.12007; Zbl 0538.14023; Zbl 0218.12007

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