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Three Aspects of the Theory of Complex Multiplication

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Three Aspects of the Theory of Complex Multiplication Takase Masahito Introduction In the letter addressed to Dedekind dated March 15, 1880, Kronecker wrote: - the Abelian equations with square roots of rational numbers are ex­ hausted by the transformation equations of elliptic functions with sin­ gular modules, just as the Abelian equations with integral coefficients are exhausted by the cyclotomic equations. ([Kronecker 6] p. 455) This is a quite impressive conjecture as to the construction of Abelian equations over an imaginary quadratic number fields, which Kronecker called "my favorite youthful dream" (ibid.). The aim of the theory of complex multiplication is to solve Kronecker's youthful dream; however, it is not the history of formation of the theory of complex multiplication but Kronecker's youthful dream itself that I would like to consider in the present paper. I will consider Kronecker's youthful dream from three aspects: its history, its theoretical character, and its essential meaning in mathematics. First of all, I will refer to the history and theoretical character of the youthful dream (Part I). Abel modeled the division theory of elliptic functions after the division theory of a circle by Gauss and discovered the concept of an Abelian equation through the investigation of the algebraically solvable conditions of the division equations of elliptic functions. This discovery is the starting point of the path to Kronecker's youthful dream. If we follow the path from Abel to Kronecker, then it will soon become clear that Kronecker's youthful dream has the theoretical character to be called the inverse problem of Abel. Next, I will consider what the solution of Kronecker's youthful dream really means, that is, its essential meaning in mathematics (Part III). From the purely theoretical viewpoint, Abel's theory of Abelian equations is only one origin of the youthful dream; however, the essential meaning of the youthful dream can be found not in Abel's theory but in Eisenstein's theory, the latter of which gives us a proof of the biquadratic reciprocity law in the Gaussian number field, based on the division theory of the lemniscatic function. Today's theory of class fields teaches us that, if we can know all the relative Abelian number fields over the Gaussian 91 C. Sasaki et al. (eds.), The Intersection of History and Mathematics © Birkhäuser Verlag 1994 92 Takase Masahito number field, then we will be able to obtain a proof of the biquadratic reciprocity law; so we must recall here another statement of Kronecker's, made before his youthful dream, by which we can know that every relative Abelian number field over the Gaussian number field is generated by a special value of the lemniscatic function. This statement is very important, for it clearly shows that Kronecker was deeply interested in the reason why Eisenstein's theory succeeded. Kronecker wanted further to know why the lemniscatic function was enough to generate all the relative Abelian number fields over the Gaussian number field and he proposed the youthful dream with the aim of answering this question; therefore we will be able to say that the youthful dream shows why there can be a transcendental proof of the biquadratic reciprocity law. This is the essential meaning of Kronecker's youthful dream, or we can say, if we like, this is the essential meaning of the theory of complex multiplication. Thus there is an intimate internal relation between the youthful dream and the reciprocity law; so, before the consideration of the essential meaning of the youthful dream, I would like to review the history of the reciprocity law from Gauss to Artin and observe how the difficulties that Kummer had faced were overcome by Hilbert's theory of class fields(Part II). The class field theory is the ultimate key to the investigation of the theory of reciprocity law and its prototype was given by Kronecker's youthful dream. This situation is never accidental. I firmly believe it will back up my opinion that the essential meaning of the youthful dream can be found in Eisenstein's theory. I The Way to Kronecker's Youthful Dream -From Abel to Kronecker- 1 Discovery of Abelian Equations (1) Gauss's Message In 1801, Gauss developed the division theory of a circle in his great work Dis­ quisitiones Arithmeticae, Chap. VII. At the beginning of this chapter he left a mysterious message to his successors. This message is the birthplace of the con­ cept of an Abelian equation and so I would like to begin by reviewing it. Gauss wrote: The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions, e.g. to those which depend on the integral J[1/ J(1- x4 )]dx and also to various types of congruences. ([Gauss 1] p. 407) The inverse function x = cPa of the lemniscatic integral x dx a- -o~l Three Aspects of the Theory of Complex Multiplication 93 will be called the lemniscatic function; so, according to Gauss, there will be the same theory for the lemniscatic function as for circular functions on the basis of common principles. As far as I know, there are only two young mathematicians, Abel and Eisenstein, who furthered the essence of Gauss's message. Let's first survey Abel's investigation. When we observe Gauss's division theory of a circle from the viewpoint of the theory of algebraic equations, we can find there two important things; one is the establishment of the fact that every division equation of a circle is algebraically solvable, and the other is the complete knowledge of all cases in which a division equation of a circle is solvable using just square roots. This is the algebraic aspect of Gauss's theory that Abel succeeded. In a letter addressed to Holmboe written in December, 1826, Abel vividly wrote: I found that with a ruler and a compass we can divide the lemniscate into 2n + 1 equal parts, when this number is prime. The division de­ pends on an equation of degree (2n + 1)2 -l. But I found the complete solution by the help of square roots. This made me penetrate at the same time the mystery which enveloped Gauss's theory on the division of a circle. ([Abel 5] p. 261) What Abel stated here is nothing but the division theory of a lemniscate; but I think such a statement will be possible only when it is supported with a general theory. Abel lifted the veil of mystery which covered Gauss's theory and saw the essence of algebraic solvability. The ultimate principle which controls the algebraic solvability of an algebraic equation is some specific relation among the roots. On this basic recognition, Abel first discovered the concept of a cyclic equation and then that of an Abelian equation (See [Abel 4]); thus he succeeded in throwing light on one of the principles of Gauss's division theory of a circle. 2 Discovery of Abelian Equations (2) Division Theory of a Lemniscate The division theory of a lemniscate by Abel was developed in his monumental paper Recherches sur les fonctions elliptiques (See [Abell]). Let ¢ia again be the lemniscatic function and 4// + 1 a prime number. Following Abel, I would like to review the method of dividing the periods of ¢ia into 4//+ 1 equal parts in order to see how Abel discovered the concept of an Abelian equation (See [Abell] pp. 352- 362). By a theorem of Fermat's we can express 4// + 1 as a sum of two squared integers in this way and so we can further express it as a product of two odd prime Gaussian integers in the form a 2 + (32 = 4// + 1 = (a + (3i)(a - (3i). 94 Takase Masahito In general, let a + (3i be an arbitrary Gaussian integer; then we can make the division equation of the periods of the lemniscatic function concerning a + (3i by using two fundamental properties of the lemniscatic function. One is that the lem­ niscatic function satisfies the addition theorem; the other is that it has complex multiplication in the Gaussian number field. The latter means the complex multi­ pliers of the lemniscatic function are all in the Gaussian number field. Realistically speaking, this means we have the functional equation ¢(ai) = i . ¢a. (1) By using these two properties, we can express y = ¢( a + (3i)/5 as a rational function of x = ¢/5 in the form T y= S' (2) where T and S denote polynomials of X4 = (¢/5)4 over the Gaussian number field. Then the equation (3) is the division equation of the periods of the lemniscatic function concerning the Gaussian integer a + (3i. If I may say so, this is an imaginary division equation. Under these situations, Abel showed that when a + (3i is an odd prime number, this equation is cyclic just like an equation defining the division of a circle into an odd prime number of equal parts and so algebraically solvable over the Gaussian number field. This is the main point of Abel's theory. Now back to the former case. According to Abel, the two imaginary division equations of the periods of the lemniscatic function concerning a + (3i and a - (3i are both algebraically solvable. By this, we can see that the equation defining the division of the periods of the lemniscatic function into 4v + 1 equal parts is also algebraically solvable. This is a rough sketch of the division theory of a lemniscate by AbeL The idea of introducing an imaginary division equation is very creative and I want to emphasize here that this is the first step towards the discovery of the concept of an Abelian equation.
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