Diophantine analysis and modular forms.
(Diophantische Analysis und Modulfunktionen.)

*(German)*Zbl 0049.16202The original review of Bruno Schoeneberg from 1954 reads:

“Die Transformationstheorie der Modulformen und gewisse Ergebnisse Webers über komplexe Multiplikation werden zur Lösung von diophantischen Gleichungen \(f(x)=z^2\), wo \(f(x)\) ein Polynom 3. oder 4. Grades ist, und zur Bestimmung aller imaginär-quadratischen Zahlkörper \(R\left(\sqrt{d}\right)\) mit der Klassenzahl \(h=1\) verwendet. Danach existieren keine anderen \(R\left(\sqrt{d}\right)\) mit \(h=1\) als die seit langem bekannten. Es sei jedoch bemerkt, daß die Beweise dem Ref. an mehreren Stellen unverständlich sind.”

In 2011, when the significance of Heegner’s breakthrough has been well established, it seemed about time to ask S. J. Patterson for a second review: \bigskip Heegner’s paper is in many ways an unusual one. It was published in 1952 when the author was 58 years of age; Kurt Heegner was born in Berlin on the 16th December, 1893. He had worked in electronics for many years but around 1933 began to concentrate on mathematics. From about this point on he seems to have had some sort of contact with mathematical circles in Berlin but he remained an outsider. After the war he had no independent source of income and seems to have lived in considerable poverty. For a few years in the late 1940s he worked for Zentralblatt für Mathematik. Around 1950 he was seriously ill, probably from a stroke. Then in 1952 he published this paper which not only claimed a proof of Gauss’ class-number one problem, a problem which had been the subject of important investigations by Landau, Hecke, Deuring, Mordell, Heilbronn and Siegel in the inter-war period, but also made a major contribution to the congruent number problem. It is perhaps not so surprising that the paper was treated with a certain degree of scepticism. Some, but not all, of this was justified.

The paper is written rather tersely and not all of Heegner’s hints at proofs are that easy to complete. This may well be due to his external circumstances when he was writing but it seems to have always been a characteristic of himself. The reviewers in Zentralblatt (Bruno Schoeneberg) and for Mathematical Reviews (Morgan Ward) clearly made no great efforts to follow the argument in detail. In the Princeton Seminar of 1957/58 (published in 1966 with some revisions made in 1965) there is a brief reference by S. Chowla to Heegner’s work [p. VI-2 in A. Borel, S. Chowla, C.S. Herz, K. Iwasawa and J.-P. Serre, Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957–58. Lecture Notes in Mathematics 21. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0147.03902)], but no attempt is made to go into any detail. Heegner’s contributions to the theory of congruent numbers were ignored more or less completely.

In 1966 the situation changed completely. Harold Stark solved the class-number one problem in his paper [“A complete determination of the complex quadratic fields of class-number one”, Mich. Math. J. 14, 1–27 (1967, Zbl 0148.27802), henceforth referred to as [Stark 1967]] which left absolutely no doubt in the matter. He also referred directly to Heegner’s paper and it is clear that even then he suspected it was fundamentally sound. Unfortunately Heegner had died around the 31st January, 1965. Stark’s proof generated a great deal of interest in Heegner’s paper. Its essential correctness was demonstrated in papers by Birch, Deuring, Meyer and Siegel. Birch and Stark were the two mathematicians who took up Heegner’s ideas with the most energy. Stark’s interest lay in class number problems, Birch’s lay in the theory of elliptic curves. Both Birch and Stark have written about the history of this period [B. Birch, “Heegner points: the beginnings”, H. Darmon (ed.) et al., Heegner points and Rankin \(L\)-series. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 49, 1–10 (2004; Zbl 1073.11001); H. M. Stark, “The Gauss class-number problems”, in: W. Duke (ed.) et al., Analytic number theory. A tribute to Gauss and Dirichlet. Providence, RI: American Mathematical Society. Clay Mathematics Proceedings 7, 247–256 (2007; Zbl 1130.11062), henceforth referred to as [Stark 2007]].

Birch and Swinnerton-Dyer had developed the Birch–Swinnerton-Dyer Conjectures to a publishable state by around 1962/63. Congruent numbers are the square-free representatives of the areas of Pythagorean triangles. One can verify that a number is congruent precisely when the elliptic curve \(y^2=x^3-n^2x\) has non-trivial points. This curve has complex multiplication by \(\mathbb Z [\imath]\) and its \(L\)-function was known. A variant, due to Lucas, is that \(n\) is congruent precisely when \(y^2=x^4-n^2\) has non-trivial solutions. There is a substantial literature about congruent numbers. There were infinite families known, for example due to Gérardin who in 1915 showed that numbers of the form \(2u^4+2v^4\) and \(4u^4+v^4\) are congruent; this was done by finding parametric solutions to Lucas’ equation. Heegner went much further and showed that any prime number \({\equiv 5}\) (mod \(8\)) is congruent. This involved a much more profound understanding of elliptic curves and it was this that attracted Birch’s attention.

What was Heegner’s new idea? He had apparently begun his mathematical investigations by studying the theory of equations and was a very careful reader of the works of Klein and the subsequent works of Klein and Fricke. He also took up a question that had a long history in the theory of abelian functions: When and how does the Jacobian variety of a curve split into a product of abelian varieties of lower dimension? The question was treated in many classical texts. Heegner’s starting point was in the work of Klein and Fricke and his nomenclature was consequently a little old-fashioned at the time he was writing and is very different from current ones. Heegner investigates, in more modern terminology, the Jacobians of certain modular curves. He seems to have been one of the first people to do so with the exception of Fricke who, taking up the baton from Klein, had already investigated special cases.

The central object of study in Heegner’s 1952 paper is a modular curve familiar from the theory of elliptic and modular functions, \(k^4-k^2=1-16/f^{24}\). Here we can interpret \(k^2\) as the Legendre-Jacobi modulus of an elliptic curve in Jacobi normal form

\[ y^2=(1-x^2)(1-k^2x^2) \]

and \(f\) is the Weber function

\[ f(z)= e^{-\pi \imath/24}\frac{ \eta( \frac{z+2}2 ) }{ \eta(z) }. \]

Weber had introduced this function and the two related functions \(f_1\) and \(f_2\) because it was possible to express their evaluations at “complex multiplication” points in the upper half-plane much more simply than is possible with the more canonical function \(j\). Heegner considers the relationship between \(k\) and \(f\) as defining an algebraic curve; he determined the genus as \(21\). We know from the Nachlass (Cod. Ms. K. Heegner 1:5 in the Handschriftenabteilung of the SUB, Göttingen) that he was aware of Weil’s thesis and therefore of the Mordell Conjecture. This would predict that such a curve would have very few points whereas Weber’s results produced a large number of interesting points over fairly small number fields, small that is in terms of their degree and discriminant. He must have been aware of the dialectic tension between these two standpoints.

He determines, very rapidly, in Section 2 of his paper, the reduction of the Jacobian variety into irreducible factors. What he discovers is that there are six factors with complex multiplication by \(\mathbb Z[\omega]\), \(\omega\) a primitive third root of \(1\), and nine factors with complex multiplication by \(\mathbb Z [\imath]\). He claims, but does not prove, that there are two factors with complex multiplication by \(\mathbb Z[ \sqrt{-2}]\). What remains, he seems to indicate, are two Jacobian varieties of genus \(2\). This part of Heegner’s paper has attracted rather less attention than other parts. For elliptic curves one is now mainly interested in the Jacobian of \(X_0(N)\) of which Shimura’s theorem gives a complete description. For other diophantine questions techniques akin to those of Heegner that are important – see, for example, Chapter 14 in [J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus \(2\). London Mathematical Society Lecture Note Series 230. Cambridge: Cambridge Univ. Press (1996; Zbl 0857.14018)].

The complex multiplication points yield then points on these factors. The coordinates will lie in class fields. Heegner gives an algebraic procedure that reduces the field in question to one whose degree over a given base field is a power of \(2\); this argument is repeated in [B. J. Birch, “Elliptic curves and modular functions”, Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, 27–32 (1970; Zbl 0225.14016)]. This particular power is then often known from class-field theory and so one obtains points over a known field. Heegner gives a large number of examples of this kind; see also Birch’s paper just cited.

He also shows that for every quadratic imaginary field whose ring of integers has class-number one there is an integral solution to the equation \(y^2=2x(x^3+1)\). He can find all the integral points on this elliptic curve and verifies in this way that the list of such fields was complete. In doing this he makes use of four modular functions \(\alpha, \beta,\vartheta,\zeta\) which are defined by algebraic relations. First \(\zeta \) is determined by \(-4\zeta(\zeta^3+1)=j^{1/3}\). There are four functions so determined with the usual normalisation of \(j^{1/3}\). This is not a problem. From \(\zeta\) one determines \(\alpha\), \(\beta\) with the equations \(\zeta=2(\beta - \alpha^2)\) and \(\zeta^2=2(\beta^2+2\alpha)\). These parametrize an elliptic curve, \((\beta-2\alpha^2)^2=2\alpha(\alpha^3+1)\), which is the curve above. Heegner simply claims that \(\alpha\) and \(\beta\) take integral values at the points corresponding to the rings of integers of the fields of class-number one. He seems to indicate that this happens because of the general theory but the functions are not at all easy to handle, for example \(\zeta=-f^4+\sqrt{-(f^8-8f^{-4})}\) which has coefficients in its expansion at \(\infty\) in \(\mathbb Z [\imath]\). The last function Heegner introduces is \(\vartheta=\zeta/\beta\); he points out that \(\alpha^3\), \(\beta^3\) and \(\zeta^3\) can be expressed as rational functions in \(\vartheta\). He writes that \(\zeta\) is a Hauptmodul of level 12 and index 12 and that \(\vartheta\) is a Hauptmodul of level 8. The most systematic analysis of the functions used here is given by Birch (with some help from Atkin) in [B. J. Birch, “Weber’s class invariants”, Mathematika, Lond. 16, 283–294 (1969; Zbl 0226.12005)]. He does not identify Heegner’s functions further. Heegner also states that \(\alpha\) and \(\beta\) are of level 24 and index 48.

Despite the difficulty readers have had in understanding Heegner’s functions it seems clear that he regarded such questions as routine. They were the main obstacle to understanding Heegner’s solution of the class-number one problem. Once Stark had presented his proof considerable interest arose in Heegner’s paper and in particular B. J. Birch [“Diophantine analysis and modular functions”, Algebr. Geom., Bombay Colloquium 1968, 35–42 (1969; see Zbl 0246.10017 for a review of the Russian translation)], M. Deuring [“Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins”, Invent. Math. 5, 169–179 (1968; Zbl 0155.38001)] and H. M. Stark[“On the ‘gap’ in a theorem of Heegner”, J. Number Theory 1, 16–27 (1969; Zbl 0198.37702), henceforth referred to as [Stark 1969]] showed that one could avoid, by a field theoretic argument, any analysis of the modular functions \(\alpha, \beta,\vartheta,\zeta\). Already in 1966 Stark [Stark 1967] was clearly not convinced that there was a gap in Heegner’s proof and later confirmed this opinion in [Stark 1969, Stark 2007]. Of the papers of Birch, Deuring and Stark just mentioned, Stark’s [Stark 1969] is the most elementary and discusses Weber’s results at some length. Both Birch and Deuring use later results. In his paper [Stark 1967] Stark uses no modular functions, although they are implicit, and develops all that he needs from the Kronecker Limit Formula. C. L. Siegel’s paper [“Zum Beweise des Starkschen Satzes”, Invent. Math. 5, 180–191 (1968; Zbl 0175.33602)] on Stark’s proof gives an alternative which is further from Heegner’s proof; like Stark he starts from a consequence from the Kronecker Limit Formula and applies it in the context of modular functions of level 5. Unlike Stark Siegel does identify all the functions occurring with known modular functions. C. Meyer [“Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins”, J. Reine Angew. Math. 242, 179–214 (1970; Zbl 0218.12007)] follows the same path as Birch and Deuring but gives a much more complete analysis of the theory of complex multiplication. From today’s standpoint one could use Shimura’s reciprocity law instead of the ad hoc arguments of Birch, Deuring and Meyer.

One can ask whether the “mathematical community” was right or wrong to be sceptical of Heegner’s proof. The fact that the paper is written in quite sophisticated German with a fair number of obsolescent or antiquated termini (such as “äquiharmonisch” meaning “with complex multiplication by the third roots of unity”) meant that the wider community was dependent on German mathematicians. Here the responsibility falls on a small group, the Hasse school, Deuring and Siegel. All of these (Meyer had been a student of Hasse) commented on Heegner’s work after 1966. Before that there had been a serious problem. Heegner’s paper was the first paper to use the full force of the theory of complex multiplication to solve an open number-theoretical problem. Earlier applications, such as Kronecker’s proof of his class number formulae used relatively little and the conclusion could be verified by other means. In his approach to the Riemann Hypothesis for elliptic curves over a finite field Hasse had clearly profited from his experience in reworking of Weber’s results but he did not need them. Also Eichler’s identification of the zeta-function of \(X_0(11)\) with the \(L\)-function of the cusp form of weight 2 in [M. Eichler, “Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion”, Arch. Math.\ 5, 355–366 (1954; Zbl 0059.03804)] needed only the congruence relation and not the fine detail of class-field theory and singular moduli. Up to this point singular moduli had to a large extent been used to prove striking formulae; here the work of Ramanujan and his successors is particularly noteworthy.

In giving two applications of the theory of complex multiplication to classical number theoretic problems Heegner was breaking new ground. Those involved had to consider whether they could really rely on the results in the literature. It seems as if they had doubts; that these doubts were justified has been documented by N. Schappacher in [“On the history of Hilbert’s twelfth problem: a comedy of errors”, Material on the history of mathematics in the 20th century. Marseille: Société Mathématique de France. Sémin. Congr. 3, 243–273 (1998; Zbl 1044.01530)]. The papers of Stark and those following him avoided as far as possible more delicate results from the theory of complex multiplication. The Kronecker Limit Formula could be proved ab ovo and was therefore reliable. In the theory of complex multiplication one has learnt that the more general the results were formulated the better the chance that they were accurate. Shimura’s reciprocity law is such that it is only possible to state and prove when one has found the correct formulation. It is noteworthy that Birch used results of H. Schöngen. This was from his doctoral thesis written under Artin in 1935 and it contains a formulation of Hasse’s results that go some distance towards Shimura’s law. It was only with the adoption of idelic language that Shimura’s version could be completed. In [G. Shimura, Automorphic functions and number theory. Lecture Notes in Mathematics 54. Berlin-Heidelberg-New York: Springer Verlag (1968; Zbl 0183.25402)], dealing with such matters but written before the final version of the reciprocity law had been established, Shimura writes on page 13, “It may be said that the world of mathematics is built with a great harmony but not always in the form we expect before unveiling it. This certainly applies to our present question.” It applied likewise to Weber and others. In developing both class-field theory and the theory of complex multiplication they were dealing with very difficult material where there was little to guide them. They made mistakes; even the proof of the Kronecker-Weber theorem was the result of a difficult struggle – see [O. Neumann, “Two proofs of the Kronecker-Weber theorem ‘according to Kronecker, and Weber”’, J. Reine Angew. Math. 323, 105–126 (1981; Zbl 0471.12001)]. Definitions tended to vary subtly and the reader often has the impression of consulting Proteus. Also the notion of the “compositum” of two fields was often a source of confusion.

In view of the difficulty of the theory it is hardly surprising that the founders had to struggle, but this state of affairs does explain why there were doubts as to whether this bridge would hold in times of need. Beyond this it is perhaps the case that the leading mathematicians of the day in Germany were less convinced that it was worth putting effort into the work of an eccentric outsider than they might have been had Heegner been an established figure. One cannot rewrite history and it fell to Stark to vindicate Heegner’s work on Gauss’ problem and to Birch to realize the significance of Heegner’s constructions of points on elliptic curves for number theory generally.

“Die Transformationstheorie der Modulformen und gewisse Ergebnisse Webers über komplexe Multiplikation werden zur Lösung von diophantischen Gleichungen \(f(x)=z^2\), wo \(f(x)\) ein Polynom 3. oder 4. Grades ist, und zur Bestimmung aller imaginär-quadratischen Zahlkörper \(R\left(\sqrt{d}\right)\) mit der Klassenzahl \(h=1\) verwendet. Danach existieren keine anderen \(R\left(\sqrt{d}\right)\) mit \(h=1\) als die seit langem bekannten. Es sei jedoch bemerkt, daß die Beweise dem Ref. an mehreren Stellen unverständlich sind.”

In 2011, when the significance of Heegner’s breakthrough has been well established, it seemed about time to ask S. J. Patterson for a second review: \bigskip Heegner’s paper is in many ways an unusual one. It was published in 1952 when the author was 58 years of age; Kurt Heegner was born in Berlin on the 16th December, 1893. He had worked in electronics for many years but around 1933 began to concentrate on mathematics. From about this point on he seems to have had some sort of contact with mathematical circles in Berlin but he remained an outsider. After the war he had no independent source of income and seems to have lived in considerable poverty. For a few years in the late 1940s he worked for Zentralblatt für Mathematik. Around 1950 he was seriously ill, probably from a stroke. Then in 1952 he published this paper which not only claimed a proof of Gauss’ class-number one problem, a problem which had been the subject of important investigations by Landau, Hecke, Deuring, Mordell, Heilbronn and Siegel in the inter-war period, but also made a major contribution to the congruent number problem. It is perhaps not so surprising that the paper was treated with a certain degree of scepticism. Some, but not all, of this was justified.

The paper is written rather tersely and not all of Heegner’s hints at proofs are that easy to complete. This may well be due to his external circumstances when he was writing but it seems to have always been a characteristic of himself. The reviewers in Zentralblatt (Bruno Schoeneberg) and for Mathematical Reviews (Morgan Ward) clearly made no great efforts to follow the argument in detail. In the Princeton Seminar of 1957/58 (published in 1966 with some revisions made in 1965) there is a brief reference by S. Chowla to Heegner’s work [p. VI-2 in A. Borel, S. Chowla, C.S. Herz, K. Iwasawa and J.-P. Serre, Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957–58. Lecture Notes in Mathematics 21. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0147.03902)], but no attempt is made to go into any detail. Heegner’s contributions to the theory of congruent numbers were ignored more or less completely.

In 1966 the situation changed completely. Harold Stark solved the class-number one problem in his paper [“A complete determination of the complex quadratic fields of class-number one”, Mich. Math. J. 14, 1–27 (1967, Zbl 0148.27802), henceforth referred to as [Stark 1967]] which left absolutely no doubt in the matter. He also referred directly to Heegner’s paper and it is clear that even then he suspected it was fundamentally sound. Unfortunately Heegner had died around the 31st January, 1965. Stark’s proof generated a great deal of interest in Heegner’s paper. Its essential correctness was demonstrated in papers by Birch, Deuring, Meyer and Siegel. Birch and Stark were the two mathematicians who took up Heegner’s ideas with the most energy. Stark’s interest lay in class number problems, Birch’s lay in the theory of elliptic curves. Both Birch and Stark have written about the history of this period [B. Birch, “Heegner points: the beginnings”, H. Darmon (ed.) et al., Heegner points and Rankin \(L\)-series. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 49, 1–10 (2004; Zbl 1073.11001); H. M. Stark, “The Gauss class-number problems”, in: W. Duke (ed.) et al., Analytic number theory. A tribute to Gauss and Dirichlet. Providence, RI: American Mathematical Society. Clay Mathematics Proceedings 7, 247–256 (2007; Zbl 1130.11062), henceforth referred to as [Stark 2007]].

Birch and Swinnerton-Dyer had developed the Birch–Swinnerton-Dyer Conjectures to a publishable state by around 1962/63. Congruent numbers are the square-free representatives of the areas of Pythagorean triangles. One can verify that a number is congruent precisely when the elliptic curve \(y^2=x^3-n^2x\) has non-trivial points. This curve has complex multiplication by \(\mathbb Z [\imath]\) and its \(L\)-function was known. A variant, due to Lucas, is that \(n\) is congruent precisely when \(y^2=x^4-n^2\) has non-trivial solutions. There is a substantial literature about congruent numbers. There were infinite families known, for example due to Gérardin who in 1915 showed that numbers of the form \(2u^4+2v^4\) and \(4u^4+v^4\) are congruent; this was done by finding parametric solutions to Lucas’ equation. Heegner went much further and showed that any prime number \({\equiv 5}\) (mod \(8\)) is congruent. This involved a much more profound understanding of elliptic curves and it was this that attracted Birch’s attention.

What was Heegner’s new idea? He had apparently begun his mathematical investigations by studying the theory of equations and was a very careful reader of the works of Klein and the subsequent works of Klein and Fricke. He also took up a question that had a long history in the theory of abelian functions: When and how does the Jacobian variety of a curve split into a product of abelian varieties of lower dimension? The question was treated in many classical texts. Heegner’s starting point was in the work of Klein and Fricke and his nomenclature was consequently a little old-fashioned at the time he was writing and is very different from current ones. Heegner investigates, in more modern terminology, the Jacobians of certain modular curves. He seems to have been one of the first people to do so with the exception of Fricke who, taking up the baton from Klein, had already investigated special cases.

The central object of study in Heegner’s 1952 paper is a modular curve familiar from the theory of elliptic and modular functions, \(k^4-k^2=1-16/f^{24}\). Here we can interpret \(k^2\) as the Legendre-Jacobi modulus of an elliptic curve in Jacobi normal form

\[ y^2=(1-x^2)(1-k^2x^2) \]

and \(f\) is the Weber function

\[ f(z)= e^{-\pi \imath/24}\frac{ \eta( \frac{z+2}2 ) }{ \eta(z) }. \]

Weber had introduced this function and the two related functions \(f_1\) and \(f_2\) because it was possible to express their evaluations at “complex multiplication” points in the upper half-plane much more simply than is possible with the more canonical function \(j\). Heegner considers the relationship between \(k\) and \(f\) as defining an algebraic curve; he determined the genus as \(21\). We know from the Nachlass (Cod. Ms. K. Heegner 1:5 in the Handschriftenabteilung of the SUB, Göttingen) that he was aware of Weil’s thesis and therefore of the Mordell Conjecture. This would predict that such a curve would have very few points whereas Weber’s results produced a large number of interesting points over fairly small number fields, small that is in terms of their degree and discriminant. He must have been aware of the dialectic tension between these two standpoints.

He determines, very rapidly, in Section 2 of his paper, the reduction of the Jacobian variety into irreducible factors. What he discovers is that there are six factors with complex multiplication by \(\mathbb Z[\omega]\), \(\omega\) a primitive third root of \(1\), and nine factors with complex multiplication by \(\mathbb Z [\imath]\). He claims, but does not prove, that there are two factors with complex multiplication by \(\mathbb Z[ \sqrt{-2}]\). What remains, he seems to indicate, are two Jacobian varieties of genus \(2\). This part of Heegner’s paper has attracted rather less attention than other parts. For elliptic curves one is now mainly interested in the Jacobian of \(X_0(N)\) of which Shimura’s theorem gives a complete description. For other diophantine questions techniques akin to those of Heegner that are important – see, for example, Chapter 14 in [J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus \(2\). London Mathematical Society Lecture Note Series 230. Cambridge: Cambridge Univ. Press (1996; Zbl 0857.14018)].

The complex multiplication points yield then points on these factors. The coordinates will lie in class fields. Heegner gives an algebraic procedure that reduces the field in question to one whose degree over a given base field is a power of \(2\); this argument is repeated in [B. J. Birch, “Elliptic curves and modular functions”, Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, 27–32 (1970; Zbl 0225.14016)]. This particular power is then often known from class-field theory and so one obtains points over a known field. Heegner gives a large number of examples of this kind; see also Birch’s paper just cited.

He also shows that for every quadratic imaginary field whose ring of integers has class-number one there is an integral solution to the equation \(y^2=2x(x^3+1)\). He can find all the integral points on this elliptic curve and verifies in this way that the list of such fields was complete. In doing this he makes use of four modular functions \(\alpha, \beta,\vartheta,\zeta\) which are defined by algebraic relations. First \(\zeta \) is determined by \(-4\zeta(\zeta^3+1)=j^{1/3}\). There are four functions so determined with the usual normalisation of \(j^{1/3}\). This is not a problem. From \(\zeta\) one determines \(\alpha\), \(\beta\) with the equations \(\zeta=2(\beta - \alpha^2)\) and \(\zeta^2=2(\beta^2+2\alpha)\). These parametrize an elliptic curve, \((\beta-2\alpha^2)^2=2\alpha(\alpha^3+1)\), which is the curve above. Heegner simply claims that \(\alpha\) and \(\beta\) take integral values at the points corresponding to the rings of integers of the fields of class-number one. He seems to indicate that this happens because of the general theory but the functions are not at all easy to handle, for example \(\zeta=-f^4+\sqrt{-(f^8-8f^{-4})}\) which has coefficients in its expansion at \(\infty\) in \(\mathbb Z [\imath]\). The last function Heegner introduces is \(\vartheta=\zeta/\beta\); he points out that \(\alpha^3\), \(\beta^3\) and \(\zeta^3\) can be expressed as rational functions in \(\vartheta\). He writes that \(\zeta\) is a Hauptmodul of level 12 and index 12 and that \(\vartheta\) is a Hauptmodul of level 8. The most systematic analysis of the functions used here is given by Birch (with some help from Atkin) in [B. J. Birch, “Weber’s class invariants”, Mathematika, Lond. 16, 283–294 (1969; Zbl 0226.12005)]. He does not identify Heegner’s functions further. Heegner also states that \(\alpha\) and \(\beta\) are of level 24 and index 48.

Despite the difficulty readers have had in understanding Heegner’s functions it seems clear that he regarded such questions as routine. They were the main obstacle to understanding Heegner’s solution of the class-number one problem. Once Stark had presented his proof considerable interest arose in Heegner’s paper and in particular B. J. Birch [“Diophantine analysis and modular functions”, Algebr. Geom., Bombay Colloquium 1968, 35–42 (1969; see Zbl 0246.10017 for a review of the Russian translation)], M. Deuring [“Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins”, Invent. Math. 5, 169–179 (1968; Zbl 0155.38001)] and H. M. Stark[“On the ‘gap’ in a theorem of Heegner”, J. Number Theory 1, 16–27 (1969; Zbl 0198.37702), henceforth referred to as [Stark 1969]] showed that one could avoid, by a field theoretic argument, any analysis of the modular functions \(\alpha, \beta,\vartheta,\zeta\). Already in 1966 Stark [Stark 1967] was clearly not convinced that there was a gap in Heegner’s proof and later confirmed this opinion in [Stark 1969, Stark 2007]. Of the papers of Birch, Deuring and Stark just mentioned, Stark’s [Stark 1969] is the most elementary and discusses Weber’s results at some length. Both Birch and Deuring use later results. In his paper [Stark 1967] Stark uses no modular functions, although they are implicit, and develops all that he needs from the Kronecker Limit Formula. C. L. Siegel’s paper [“Zum Beweise des Starkschen Satzes”, Invent. Math. 5, 180–191 (1968; Zbl 0175.33602)] on Stark’s proof gives an alternative which is further from Heegner’s proof; like Stark he starts from a consequence from the Kronecker Limit Formula and applies it in the context of modular functions of level 5. Unlike Stark Siegel does identify all the functions occurring with known modular functions. C. Meyer [“Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins”, J. Reine Angew. Math. 242, 179–214 (1970; Zbl 0218.12007)] follows the same path as Birch and Deuring but gives a much more complete analysis of the theory of complex multiplication. From today’s standpoint one could use Shimura’s reciprocity law instead of the ad hoc arguments of Birch, Deuring and Meyer.

One can ask whether the “mathematical community” was right or wrong to be sceptical of Heegner’s proof. The fact that the paper is written in quite sophisticated German with a fair number of obsolescent or antiquated termini (such as “äquiharmonisch” meaning “with complex multiplication by the third roots of unity”) meant that the wider community was dependent on German mathematicians. Here the responsibility falls on a small group, the Hasse school, Deuring and Siegel. All of these (Meyer had been a student of Hasse) commented on Heegner’s work after 1966. Before that there had been a serious problem. Heegner’s paper was the first paper to use the full force of the theory of complex multiplication to solve an open number-theoretical problem. Earlier applications, such as Kronecker’s proof of his class number formulae used relatively little and the conclusion could be verified by other means. In his approach to the Riemann Hypothesis for elliptic curves over a finite field Hasse had clearly profited from his experience in reworking of Weber’s results but he did not need them. Also Eichler’s identification of the zeta-function of \(X_0(11)\) with the \(L\)-function of the cusp form of weight 2 in [M. Eichler, “Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion”, Arch. Math.\ 5, 355–366 (1954; Zbl 0059.03804)] needed only the congruence relation and not the fine detail of class-field theory and singular moduli. Up to this point singular moduli had to a large extent been used to prove striking formulae; here the work of Ramanujan and his successors is particularly noteworthy.

In giving two applications of the theory of complex multiplication to classical number theoretic problems Heegner was breaking new ground. Those involved had to consider whether they could really rely on the results in the literature. It seems as if they had doubts; that these doubts were justified has been documented by N. Schappacher in [“On the history of Hilbert’s twelfth problem: a comedy of errors”, Material on the history of mathematics in the 20th century. Marseille: Société Mathématique de France. Sémin. Congr. 3, 243–273 (1998; Zbl 1044.01530)]. The papers of Stark and those following him avoided as far as possible more delicate results from the theory of complex multiplication. The Kronecker Limit Formula could be proved ab ovo and was therefore reliable. In the theory of complex multiplication one has learnt that the more general the results were formulated the better the chance that they were accurate. Shimura’s reciprocity law is such that it is only possible to state and prove when one has found the correct formulation. It is noteworthy that Birch used results of H. Schöngen. This was from his doctoral thesis written under Artin in 1935 and it contains a formulation of Hasse’s results that go some distance towards Shimura’s law. It was only with the adoption of idelic language that Shimura’s version could be completed. In [G. Shimura, Automorphic functions and number theory. Lecture Notes in Mathematics 54. Berlin-Heidelberg-New York: Springer Verlag (1968; Zbl 0183.25402)], dealing with such matters but written before the final version of the reciprocity law had been established, Shimura writes on page 13, “It may be said that the world of mathematics is built with a great harmony but not always in the form we expect before unveiling it. This certainly applies to our present question.” It applied likewise to Weber and others. In developing both class-field theory and the theory of complex multiplication they were dealing with very difficult material where there was little to guide them. They made mistakes; even the proof of the Kronecker-Weber theorem was the result of a difficult struggle – see [O. Neumann, “Two proofs of the Kronecker-Weber theorem ‘according to Kronecker, and Weber”’, J. Reine Angew. Math. 323, 105–126 (1981; Zbl 0471.12001)]. Definitions tended to vary subtly and the reader often has the impression of consulting Proteus. Also the notion of the “compositum” of two fields was often a source of confusion.

In view of the difficulty of the theory it is hardly surprising that the founders had to struggle, but this state of affairs does explain why there were doubts as to whether this bridge would hold in times of need. Beyond this it is perhaps the case that the leading mathematicians of the day in Germany were less convinced that it was worth putting effort into the work of an eccentric outsider than they might have been had Heegner been an established figure. One cannot rewrite history and it fell to Stark to vindicate Heegner’s work on Gauss’ problem and to Birch to realize the significance of Heegner’s constructions of points on elliptic curves for number theory generally.

##### MSC:

11G05 | Elliptic curves over global fields |

11G15 | Complex multiplication and moduli of abelian varieties |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11D25 | Cubic and quartic Diophantine equations |

11-03 | History of number theory |

01A60 | History of mathematics in the 20th century |