Hilbert, in his 9th problem, posed the problem of finding the general reciprocity law for any number field. The class field theory developed in the first half of the 20th century was successful in answering this question for finite abelian extensions of . As an easy consequence of class field theory, one can reproduce the classical Kronecker-Weber theorem, that is, every finite abelian extension of is a subfield of some cyclotomic extension of . Hilbert's 12th problem also asked how to extend the Kronecker-Weber theorem to an arbitrary ground number field, in other words, how to describe abelian extensions of a number field more explicitly.
Motivated by the case of where all abelian extensions are obtained by adjoining the values of the exponential function, Kronecker conjectured, while he was studying elliptic functions, that all abelian extensions of an imaginary quadratic field should also arise in such a manner. This was achieved by the beautiful theory of complex multiplication, which allows one to describe all abelian extensions of an imaginary quadratic field via the values of the modular -function and the Weber function, or in the language of elliptic curves, via the -invariant and (certain powers of) -coordinates of all torsion points of the corresponding elliptic curve. This problem for general number fields, known as Kronecker's Jugendtraum (dream of youth), is still largely open and is at the heart of current research in number theory.
In this chapter, we will start by establishing the one-one correspondence between , the ideal class group of an order in an imaginary quadratic field , and , the set of all isomorphism classes of elliptic curves with complex multiplication by . Next, we will define the ring class field of and then prove that it is obtained by adjoining the value for any proper fractional ideal of , using the method of reduction of elliptic curves. Moreover, the ray class fields of can be obtained by adjoining the values of the Weber function. Next, we will discuss the modular equation and use it to prove the integrality of . The values of , i.e., the values of at the imaginary quadratic argument , are called singular moduli as they correspond to the -invariants of singular elliptic curves. We will use Weber's method to explicitly compute singular moduli for orders of class number 1. These singular moduli turn out to be highly divisible as predicted by a remarkable theorem of Gross and Zagier. Gross and Zagier's theorem on singular moduli completely determines the prime factorization of the norm of the difference between two singular moduli. Our last section will be devoted to the algebraic proof of this theorem, relying on Deuring's results on the endomorphism rings of elliptic curves.
As we saw in the first chapter, for a complex elliptic curve , the endomorphism ring can be identified with . When has complex multiplication, is an order in an imaginary quadratic extension of . Though complex conjugation will give us two isomorphisms between and , we do have a canonical way to identify with . Namely, for every , we identify it with the endomorphism of induced by the multiplication . In other words, the effect of on the invariant differential is given by . We call this identification the normalized identification.
Fix an order in . We are interested in studying all complex elliptic curves with endomorphism ring . We denote the set of all these isomorphism classes by . As we will see in Theorem 1, it turns out that is in bijection with a purely algebraic object constructed from , namely the ideal class group of .
In other words, the above correspondence gives a bijection .
Conversely, suppose is a proper fractional ideal, then , hence . ¡õ
We will soon see that is a finite group. The order of , denoted by , is called the class number of . So by Theorem 1, there are exactly isomorphism classes of elliptic curves with complex multiplication by and each of them corresponds to an ideal class of . Let be an elliptic curve with , then for any automorphism of , is also an elliptic curve with . So by the finiteness of the ideal class group and the above correspondence between and , we know has finitely many values, hence is an algebraic number of degree at most .
An amazing fact is that for an elliptic curve with complex multiplication by , is actually an algebraic integer of degree (Theorem 8). In particular, an elliptic curve with complex multiplication has a rational -invariant if and only . We will calculate these -invariants later.
Let be an imaginary quadratic extension of of discriminant . The ring of integers of is equal to where . Every order of is a free -module of rank 2, hence is of the form , where the positive integer is called the conductor of . When , is the maximal order and is a Dedekind domain. But when , is not integrally closed and hence is not a Dedekind domain. So it is not necessary for the unique factorization to hold for ideals in . For example, is an order of conductor 2 of , and the ideal has two different prime factorizations . However, we will see that the situation becomes better when we restrict our attention to the ideals prime to .
Next, let be multiplication by . Then if and only if is surjective. But is an finite abelian group of order , so is surjective if and only if is prime to . ¡õ
Therefore the ideals prime to are in and are closed under multiplication. They generate a subgroup of fractional ideals . Similarly define . Given any integer , every ideal class in contains an ideal prime to ([3, 7.17]). So the natural inclusion induces a surjective map . Moreover, the kernel of this surjective map is , so we have an isomorphism ([3, 7.19]).
Viewing as a cycle of , we know that and that is a subgroup of containing (we use the symbols for generalized ideal groups as in [4, VI]). Hence, by class field theory, corresponds to a finite abelian extension of .
Combining the isomorphism and the second part of Proposition 2, we know that , and composing with the Artin map we obtain an isomorphism . In particular, is a finite group. Again, when , i.e., is the maximal order , the ring class field of is just the usual Hilbert class field of . So we may regard the ring class field as a generalization of the Hilbert class field.
The first main result of this section is the ``First Main Theorem'' of complex multiplication, which says the ring class field of can be obtained by adjoining the value of -function at any proper ideal of . The key step of the proof is to establish the so-called Hasse congruence (Theorem 2), which expresses the Frobenius action on the value of the -function via the action on the argument of the -function. There are several different approaches to do so: the complex analytic method using the modular equation, or the algebraic method we choose here using the reduction of elliptic curves. We follow mainly the exposition of  and .
First, by using Chebotarev's density theorem, one can show the following characterization of field extensions by the primes which split completely ([3, 8.20]). We will utilize this Lemma 1 twice to characterize the ring class field in Lemma 2.
From now on, let us fix an order of conductor in an imaginary quadratic field . Let () be the representatives of the ideal class group and be the corresponding elliptic curves with complex multiplication by via Theorem 1. Let .
Next let us show that . Let be the Galois closure of . Suppose is a prime which splits completely in . Again by Lemma 1, it suffices to show that for all but finitely such , splits completely in . Since splits completely in , must be a principal ideal of and then . So by assumption, for all but finitely many such and any prime of above . Now suppose further that does not divide the index , then we have for any , hence has degree 1 for any above . Therefore splits completely in . ¡õ
Now since the 's have good reduction at , reducing modulo we get an isogeny . But , so where is the invariant differential of . Therefore is inseparable. As the reduction does not change the degree of an isogeny, is prime to , hence is separable. We conclude that is inseparable. But , hence is purely inseparable. So is the composition of the -Frobenius and an isomorphism . Hence and the claim follows. ¡õ
From Theorem 2 and Lemma 2, we already know that is actually the ring class field of . Further more, we can use the limited information of the Hasse congruence to compute the Galois action on the -values.
Now we are in a position to prove the First Main Theorem of complex multiplication.
As a consequence of the First Main Theorem 4, all everywhere unramified extensions of can be obtained as a subfield of . Moreover, one can show that an abelian extension of is generalized dihedral over if and only if it is contained in the ring class field of some order in ([3, 9.18]), so the First Main Theorem also tells us how to construct generalized dihedral extensions explicitly. Now, it is natural to ask how to generate all abelian extensions of , in other words, how to give an explicit description of the ray class fields of . This is the content of the ``Second Main Theorem'' of complex multiplication.
Now let us establish an analog of the Hasse congruence for the Weber function.
After the congruence in Theorem 5 is established, a similar argument as for ring class fields will allow us to construct all the ray class fields of . We state this Second Main Theorem and omit the details of the proof here. Roughly speaking, the maximal abelian extension of is generated by and the -coordinates of all torsion points of the corresponding elliptic curve with complex multiplication by . See [3, 11.39] and [6, II.5] for more.
We have seen that is an algebraic number of degree from Theorem 3. But in fact more is true: it is an algebraic integer. There are several possible proofs of this fact: the complex analytic proof using the modular equation, the good reduction proof due to Serre and Tate, and the bad reduction proof due to Serre ([6, II.6]). We have not talked much about the analytic aspect of -function so far, so we will choose the first approach here.
Let us first recall some facts about the modular curve , which plays an important role in modern number theory. The modular curve is a compact Riemann surface constructed by compactifying , the quotient of upper half plane by the congruence group . It is the compactification of the moduli space of elliptic curves along with the level structure of a cyclic subgroup of order . Viewing as a complex algebraic curve, the function field of is equal to . So has a planar model defined by some complex polynomial satisfying , called the modular equation of level ().
An unexpected result is that the modular equation in fact has rational, or even better, integer coefficients. Therefore, can be defined as an algebraic curve over without reference to the complex numbers and it has a planar model over defined by the modular equation. The goal of this section is to prove this unexpected fact and deduce the integrality of as a consequence.
To define the modular equation, we need the following lemma.
Since , it follows immediately that . Also, it is an easy computation ([5, 4.5]) to see that the set of orbit representatives can be chosen as
Now suppose is not a perfect square. The leading coefficient of is the same as the leading coefficient of the -expansion of , so let us show that the latter is . Now begins with and begins with , so since is not a perfect square and , we know that and cannot cancel out, hence the leading coefficient of is a root of unity. Multiplying them together, we know that the leading coefficient of is a root of unity. But we already know it is an integer, hence it must be . ¡õ
Now we are in a position to prove the integrality of the singular moduli.
As an application of Proposition 2, we get an exact sequence Hence the class numbers of and are related by One can show that there is an exact sequence ([3, Exercise 7.30]) Therefore we are able to reproduce the following formula due to Gauss for the class number of an order in ([3, 7.24]).
Our goal in this section is to compute the for the orders of class number 1. By Theorem 8, all these singular moduli are rational integers. A famous result about the Gauss class number problem, now known as the Stark-Heegner theorem, says that there only 9 imaginary quadratic fields having class number 1. Moreover, using Theorem 9, we can conclude that there are only four more cases for non-maximal orders of class number 1. More precisely, when , only and can occur; when (i.e., ), only can occur; when (i.e., ), only can occur. Since an order is uniquely determined by its discriminant , we may summarize the results as the following theorem.
To compute these singular moduli, one may proceed by plugging into the -expansion of , This -expansion can be computed via the -expansions of and , So nowadays we can handle this task using a computer. The numerical method will work pretty well for our purpose since we know a priori that these values are integers. All the 13 singular moduli of integer values are listed in Table 1.
By Theorem 10, we easily obtain a quick method of detecting complex multiplication for elliptic curves over : if appears in Table 1 , then has complex multiplication with the corresponding order ; otherwise, does not have complex multiplication.
However, we are not fully satisfied with this direct computation. For example, it does not explain why most singular moduli in Table 1 (except the two boxed ones) are cubes. Nor does it explain the observation that all these singular moduli are highly divisible. We will try to explain the reasons for these phenomena in the last two sections of this chapter (see Theorem 11, Corollary 1).
Now let us introduce Weber's method for computing the singular moduli of integer values. The main tool of Weber's computation is a class of Weber functions , , and (not to be confused with the Weber function defined earlier).
The Weber function satisfies the following transformation property Then it is straightforward to check that is a modular function with respect to ([3, 12.3]). So . Moreover, when the order has discriminant prime to 3, we have the following even better relationship between and ([3, 12.2]).
By the last part of Theorem 11, we know that has the same degree as when . In particular, when , is an integer since is so. Therefore is a cube when , which coincides with the result listed in Table 1 . The two boxed exceptions , as expected, are divisible by 3.
We may summarize the most important relationship and transformation properties of these Weber functions as follows [3, 12.17, 12.19]. These properties are crucial to Weber's computation of singular moduli.
Now let us use Theorem 12 to compute for those orders with discriminants prime to 3. As a consequence, we will be able to compute the singular moduli for those orders easily by raising the corresponding to the third power.
Next let us consider the case of odd discriminant. The above computation fails, since is now negative. However, we can translate to as follows. By Theorem 12 (b) and (d), we know that and Therefore Now a similar argument shows the desired result . ¡õ
Gross and Zagier  proved a result which completely determines the prime factorization of the norm of the difference between two singular moduli, which in turn justified many classical conjectures on the congruences of singular moduli proposed by Berwick . They provide two proofs of different natures: The first proof, an algebraic proof, is based on Deuring's work on endomorphism rings of elliptic curves mentioned in Chapter 2. The second analytic proof relies on the calculation of the Fourier coefficients of the restriction to the diagonal of an Eisenstein series of the Hilbert modular group of . As the authors remarked, these two methods can be viewed as the special case of the theory of local heights of Heegner points on , which generalizes to the groundbreaking Gross-Zagier formula .
In this section, we will first state Gross-Zagier's theorem, then use it to compute several examples and derive some consequences. At the end, we will discuss a bit of the algebraic proof of Gross-Zagier's theorem.
Now consider two orders with discriminants and satisfying . Let be the numbers of their units and be their class numbers. Let and be the representatives of their ideal class groups. Define Notice that when (e.g., ), is just the norm of any of the differences . In general, is a certain power of this norm and is always an integer.
To state Gross-Zagier's theorem, let us introduce some notation. Let . For a prime , define This is well-defined whenever . More generally, if has the prime factorization with , we define Finally, set This is well-defined whenever all primes dividing satisfy . Now the main theorem is as follows.
As we have noticed, the prime factor of is always a factor of . In fact, we have the following interesting result concerning the function ([3, Exercise 13.15, 13.16]).
Now we can explain the phenomenon we observed in Table 1 . Suppose is a prime dividing an integer singular modulus of discriminant , or equivalently, dividing , then by Corollary 1, we have . So these singular moduli have relatively small prime factors, though their own values can be fairly huge.
Finally, let us come to the algebraic proof of Gross-Zagier's theorem. The proof proceeds locally. As the first step, Gross and Zagier relate the valuation of the difference of two -values to the geometry of elliptic curves and reduce it to a counting problem of isomorphisms between elliptic curves. Next, a generalization of Deuring's lifting theorem will allow one to reduce the problem to counting certain subrings of the endomorphism ring of a supersingular elliptic curve. To complete the proof, Gross and Zagier give a convenient description of a maximal order and its subrings in the rational quaternion algebra ramified at and a prime for explicit computation.
The first step can be viewed as an interesting geometrical interpretation of the difference of -values.
Let us consider the case when for simplicity. Change models for with simplified Weierstrass equations
By definition, we have if and only if we can solve the congruences simultaneously for some unit . In this case , and at least one of and is a unit in since has good reduction mod .
If is a unit in , then is also a unit. By changing models we may assume that . Then On the other hand, the congruences become We may possibly modify by so that is maximal. Then We get So the theorem holds in this case.
If is a unit in , then is also a unit. Similarly by changing models we may assume that . Then where is a primitive cube root of unity in . On the other hand, the congruences become We may possibly modify by or so that is maximal. Then We get This completes the proof. ¡õ
For simplicity, we will assume is a prime from now on (for the general case, see ). Let be the ring of integers of . Let be an elliptic curve over with complex multiplication by and with -invariant . For our purpose, we need to calculate where is an elliptic curve over with complex multiplication by some ring of discriminant . We can rewrite in a manner which only depends on . Suppose , then is an endomorphism of mod , which has the same norm, trace and action on tangent space as . Namely, belongs to the set Conversely, every element of is of the form for some unique ensured by the following lifting theorem, which is a refinement of Deuring's lifting theorem ([12, 14.14]).
Now by Theorem 16, we reduce to the counting problem of .
When , splits in , so has ordinary reduction mod and ([12, 13.12]). But contains no elements of discriminant , so is empty for all . (Another way to say this: if two elliptic curves and with complex multiplication have the isomorphic reduction , then the reduction must be supersingular, since two different orders and have to embed into simultaneously.)
So we only need to consider the case and has supersingular reduction. Then is a maximal order in the rational quaternion algebra ramified at and by Theorem 8. The algebra can be desribed explicitly as a subring of , The subrings can also be desribed explicitly. Using these descriptions, it turns out that in many cases equals to times the number of the solutions (under certain conditions on ) of the equation where we assume is a prime . The more precise result is the following.
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