Class Fields of Complex Multiplication: translation of "Die Klassenkörper der komplexen Multiplikation" by Max Deuring
aa r X i v : . [ m a t h . HO ] M a y Class Fields of Complex Multiplication by Max Deuring in G¨ottingen translated from German by
CheeWhye Chin
Contents
A. Function-theoretic basics
1. Modular functions.2. Modular functions for subgroups of M .3. Transformation of modular functions.4. The functions p ϕ S ( ω ) .5. Elliptic functions.6. The Weber function.7. The division values of the Weber function. B. Number-theoretic basics
8. Orders in quadratic number fields.
C. The first main theorem
9. Formulation of the first main theorem.10. Proof of the first main theorem by means of the general theory of abeliannumber fields.11. Overview of the proof of the first main theorem not relying on general classfield theory.12. The class polynomial.13. The singular values of the functions ϕ S ( ω ) . First part.14. Proof of the fundamental congruence (33)15. The isomorphism σ of R R with G Ω R Σ .16. The Galois group G Ω R P .17. The reciprocity law for the ring class field Ω R / Σ .18. The genus field of the order R .19. The correspondence theorem for the ring class field.20. The singular values of functions from P M S .21. The principal ideal theorem for imaginary quadratic number fields.22. The singular values of the functions ϕ S ( ω ) . Second part.23. Congruences for the singular values of functions from P M S .1 . The second main theorem
24. The singular elliptic functions.25. The singular values of the Weber function. Ray class invariants.26. The second main theorem.27. Proof of the second main theorem not relying on general class field theory.28. The fundamental congruence.29. The generation of the ray class field by a single ray class invariant.30. The reciprocity law for the ray class field.31. The ray class polynomial.
E. RemarksMonographs
H. Weber , Lehrbuch der Algebra. III. 3. Aufl., Braunschweig 1908.
R. Fricke , Lehrbuch der Algebra. III. Braunschweig 1928.
R. Fueter , Vorlesungen ¨uber die singl¨aren Moduln und die komplexe Multiplikation derelliptischen Funktionen. I u. II. Leipzig 1924 u. 1928.
A. Function-theoretic basics
1. Modular functions.
Let M ( h ) denote the homogeneous modular group , i.e. the group ofall integer-entry matrices M = a bc d ! with | M | = ad − bc = 1 . Each M ∈ M gives rise toa homogeneous linear transformation ω ω ! → M ω ω ! = ω ′ ω ′ ! of a pair ω , ω of complexvariables; the group of these transformations is isomorphic to M ( h ) and will henceforth also bedenoted by M ( h ) . The fractional linear transformations ω = ω ω → ω ′ = ω ′ ω ′ = a ω + bc ω + d = M ( ω )associated to M = a bc d ! from M ( h ) form the inhomogeneous modular group M ; the kernel ofthe homomorphism M → [ ω → M ( ω ) ]is { E, − E } , and M ∼ = M ( h ) / { E, − E } . We reserve the notation ω for the quotient ω ω ; here, ω will always be restricted to the half-plane I ω > f ( ω ) of the complex variable ω is called a modular form of weight t if:1. f ( ω ) is holomorphic in the half-plane I ω > For the tools from function theory used below, please refer to the literature. f ( M ( ω )) = f ( ω ) ( c ω + d ) − t holds for every M = a bc d ! ∈ M ,3. the Fourier expansion f ( ω ) = X ν a ν q ν , q = e πiω , in a suitable half-plane I ω > α , which ispossible since f ( ω +1) = f ( ω ) , has at most finitely many terms with negative exponents, sothat f ( ω ) as a function of q is regular or polar at q = 0 .A modular form is called entire if it is holomorphic in I ω > M , are called modular functions ; they form a field k M . Here, k alwaysdenotes the field of complex numbers.The existence of modular functions and modular forms can be deduced by means of the existencetheorem in function theory, but for the following we will rely on the explicit formulae from theclassical theory of elliptic functions and modular functions, as these also allow one to read off thearithmetic properties of the fundamental series expansion.For the lattice w of all w = n ω + n ω , where it is understood that n ν = 0 , ± , ± , . . . , andfor any integer m > − m given by(1) G m ( ω ) = ω m X w ∈ w , w =0 w − m . Its Fourier expansion reads as(2) G m ( ω ) = (2 π ) m (2 m )! " B m + 4 m ( − m ∞ X n =1 n m − ∞ X ν =1 q n ν , where B m = m -th Bernoulli number . Set(3) g ( ω ) = 60 G ( ω ) , g ( ω ) = 140 G ( ω );the discriminant (4) ∆( ω ) = g ( ω ) − g ( ω ) is an entire modular form of weight −
12 . The
Dedekind function (5) η ( ω ) = q ∞ Y n =1 (1 − q n )is holomorphic in I ω > ω ) can also be expressed in terms of it as(6) ∆( ω ) = (2 π ) η ( ω ) . The absolute invariant of the modular group (7) j ( ω ) = 2 g ( ω ) ∆( ω ) − is an entire modular function. One has(8) j ( i ) = 2 , j ( e πi ) = 0 , j ( ∞ ) = ∞ . j ( ω ) can be considered as a meromorphic function on the quotient space of the half-plane I ω > ∞ ) mod M , which is a compact Riemann surface F of genus 0; it maps thisquotient space bijectively onto the Riemann sphere, and is normalized by (8). From this it followsthat k M is the field of rational functions in j ( ω ) with complex number coefficients: k M = k ( j ( ω )) .The entire modular functions are the polynomials in j ( ω ) .Two lattices w and w ′ are called equivalent if they differ only by a scaling factor: w ′ = ̺ w , ̺ = 0 . We denote an equivalence class of lattices by k . The classes k correspond in a uniquelyreversible way to the points = ∞ of F , where the class k of a lattice with basis ω , ω is assignedto the residue class of ω = ω ω modulo M . Henceforth we write j ( k ) = j (cid:18) ω ω (cid:19) ; the value j ( k ) iscalled the invariant of k , and it determines the class k uniquely.The Fourier expansions(9) j ( ω ) = q − + c + c q + · · · and(10) ∆( ω ) = (2 π ) q [ 1 + D q + · · · ]can be easily calculated from (2) (that of ∆ of course also from (5) and (6)), and from that followsthe fact that the coefficients c ν and D ν are rational integers, which is fundamental for the number-theoretic properties of the modular functions. First we use it to get the following conclusions:For any number field Λ , we denote by Λ M the field Λ( j ( ω )) of rational functions in j ( ω ) withcoefficients in Λ . Then: Λ M consists of all modular functions f ( ω ) whose Fourier expansion f ( ω ) = P a ν q ν has coefficients a ν in the field Λ . Furthermore: if Λ is an algebraic numberfield, then an entire modular function f ( ω ) = A + A j ( ω ) + · · · + A N j ( ω ) N has integers A ν of Λ as coefficients if and only if the Fourier coefficients a ν of f ( ω ) are integers of Λ . Both followeasily when the Fourier expansion of j ( ω ) is inserted into the expression of f ( ω ) as a rationalfunction (or polynomial) of j ( ω ) ; the first assertion depends only on the fact that the c ν in (9) arerational, while the second depends on the fact that the c ν are rational integers and that the leadingcoefficient in (9) has value 1.
2. Modular functions for subgroups of M . Let N be a subgroup of M of finite index [ M : N ] .A modular form f ( ω ) of weight t for the group N is defined by the following three requirements:1. f ( ω ) is regular in I ω > N = a bc d ! ∈ N , one has f ( N ( ω )) = f ( ω ) ( c ω + d ) − t .3. Let M be any element of M . If ℓ denotes the smallest positive whole number for which ! ℓ lies in M − N M , so that f ( M ( ω )) in condition 2 has period ℓ , then in a suitable4egion I ω > ̺ , f ( M ( ω )) should have a Fourier expansion — q -expansion — of the form f ( M ( ω )) = ∞ X ν = n a ν q νℓ , in which at most finitely many coefficients a ν of negative exponent ν are different from 0.Here (and also always in the following), we set(11) q ℓ = e πiωℓ . If N M ν , ν = 1 , , . . . , [ M : N ] , are the residue classes of M mod N , it suffices that thisrequirement holds for M = M , . . . , M [ M : N ] .Forms of weight 0 are called functions for the group N ; a form (function) for N is called entire if it is regular in I ω > N form a field extension k N of k M . For M ∈ M , an isomorphism λ M from k N to k M − N M is defined by(12) f ( ω ) λ M = f ( M ( ω )) , which leaves k M elementwise fixed. The isomorphism λ M depends only on the residue class N M of N in M to which M belongs. The coefficients of the polynomial(13) F ( X ) = [ M : N ] Y ν =1 ( X − f ( ω ) λ M ν ) = [ M : N ] Y ν =1 ( X − f ( M ν ( ω )))remain unchanged under each modular substitution M , and they also satisfy conditions 1 and 3, sothey lie in k M . This proves that k N is algebraic over k M of degree [ M : N ] . By the existencetheorem in function theory, it follows that(14) [ k N : k M ] = [ M : N ] , so the functions f ( ω ) λ ν = f ( M ν ( ω )) are a full system of conjugates of f ( ω ) over k M , and F ( X )is the principal polynomial of f ( ω ) over k M .Later we will only need the transformation subgroups N = M S of M . In this case, we willexplicitly specify functions of k N of degree [ M : N ] over k M , so that here as well we do not need torely on the function-theoretic existence theorem.Instead of a modular form f ( ω ) of weight t , it is sometimes convenient to compute with theassociated homogeneous modular form f ω ω ! , which is defined by f ω ω ! = f ( ω ) ω t . f ω ω ! is homogeneous in ω , ω of degree t , and the invariance property 2 of f ( ω ) meansfor f ω ω ! simply that f N ω ω !! = f ω ω ! holds for every N ∈ N .
3. Transformation of modular functions.
Let A be a group, and M be a subgroup of A .To each S ∈ M , we assign the subgroup M S = M ∩ S − M S of M ; here, M S depends only on the residue class M S of S modulo M . Further, to each pairof elements S, S ′ of A , we assign the complex M S,S ′ = M ∩ S − M S ′ . Then M S,S ′ is non-empty if and only if S ′ lies in M S M , in which case M S,S ′ is a residue class M S M of M modulo M S . Conversely, each residue class M S M of M modulo M S is also a M S,S ′ , namely M S M = M S,SM . One has M S,S ′′ = M S,S ′ if and only if M S ′ = M S ′′ , whence M S,S ′ ←→ M S ′ is a one-one correspondence from the residue classes of M modulo M S onto those residue classesof A modulo M into which M S M is decomposed.For X in M , one has X − M S,S ′ X = M SX,S ′ X , in particular X − M S X = M SX ;in other words: for S ′ in M S M , M S ′ is conjugate to M S in M , and every group which isconjugate to M S in M is a M S ′ , S ′ ∈ M S M .Now let M be the homogeneous modular group and let A be the group of all two-by-tworational-entry matrices with positive determinants. Since for a rational number r = 0 one alwayshas M rS = M S , we can limit ourselves to integer-entry S with relatively prime elements; such an S is called primitive .To the group-theoretic facts just established, two things are to be be added in the present case:6. For a given primitive S , M S M is the totality A s of all primitive matrices which have thesame determinant s as S ; in other words: For every pair
S, S ′ of elements in A s , thereexist unimodular M S,S ′ , M ∗ S,S ′ such that S M
S,S ′ = M ∗ S,S ′ S ′ . The A s decomposes into a finite number ψ ( s ) of residue classes M S ′ ; a system of repre-sentatives for these residue classes is formed by the triangular matrices a b d ! with a > , a d = s, ( a, b, d ) = 1 , and b running over a full system of residues modulo d ,say b < d. They are denoted by S , . . . , S ψ ( s ) in any order. For a prime number s = p , one has inparticular ψ ( p ) = p +1 , and we will then take P ν = ν p ! ; ν = 1 , , . . . , p ; P p +1 = p
00 1 ! as the system of representatives.The triangular form of the representatives has the consequence that the condition 3 for a modularform f ( ω ) for M S needs to be checked only for M = E .The (conjugates of the) subgroups M S of M are called transformation groups of level s ; we callthe classes M S which are in one-one correspondence with them the equivalence classes of level s .All this also holds when we take M as the inhomogeneous modular group and accordingly replace A by the factor group A / { E, − E } , since { E, − E } is contained in every transformation group M S .Let us investigate the transformations of the field k M S of functions for the group M S . The ψ ( s ) isomorphisms λ M of k M S over k M are λ M S,Sν = λ ν , where λ ν maps k M S onto k M Sν .Functions for the group M S can be constructed as follows: if H ω ω ! and H ω ω ! are twomodular forms (for the group M ) of the same weight, then h S ( ω ) = H S ω ω !! H ω ω ! belongs to M S . In particular, one can choose H = f ( ω ) and H = 1 as modular function, thus h S ( ω ) = f ( S ( ω )) . h S ( ω ) λ ν = h S ν ( ω ) , f ( S ( ω )) λ ν = f ( S ν ( ω )) . If the function f ( ω ) is a generator of k M , i.e. if k M = k ( f ( ω )) , then the ψ ( s ) many functions f ( S ( ω )) λ ν = f ( S ν ( ω )) are distinct from one another; that proves that[ k M S : k M ] = [ M : M S ] = ψ ( s )and k M S = k ( f ( ω ) , f ( S ( ω ))) , and in particular this holds for f ( ω ) = j ( ω ) : k M S = k ( j ( ω ) , j ( S ( ω ))) , from which it then also follows that the h S ν ( ω ) are a full system of conjugates of h S ( ω ) over k M .Of course, h S ( ω ) is entire if H and H − are.The q -expansion of h S ( ω ) can be calculated from the q -expansions of H and H .We will now show that there exists an algebraic function field P M S with the constant field P — where P always denotes the field of rational numbers — from which k M S arises by extension ofscalars from P to k , namely P M S = P ( j ( ω ) , j ( S ( ω ))) . For that, it must be shown that the coefficients A ( s ) ν ( j ( ω )) of the principal polynomial of j ( S ( ω ))over k M , J s ( X, j ( ω )) = ψ ( s ) Y ν =1 ( X − j ( S ν ( ω )))= X ψ ( s ) + A ( s ) ψ ( s ) − ( j ( ω )) X ψ ( s ) − + · · · + A ( s )0 ( j ( ω ))already lie in P M . We derive it from the fact that the q -expansion of j ( S ( ω )) has rational coeffi-cients, and indeed we similarly prove in general: Let f S ( ω ) be a function from k M S , whose conjugate f ( s
00 1 ) has a q -expansion with coefficientsbelonging to a field Λ . Then the absolutely irreducible polynomial F ( X, Y ) = N X ν =1 M X µ =1 c νµ X ν Y ν forwhich F ( j ( S ( ω )) , j ( ω )) = 0 likewise has its coefficients belonging to Λ (up to a free constantfactor = 0 ).Namely, in F (cid:16) f ( s
00 1 )( ω ) , j ( ω ) (cid:17) = 0 , let the q -expansions for j ( ω ) and f ( s
00 1 )( ω ) be inserted;that yields a system of homogeneous linear equations for the c νµ with coefficients in Λ , which (upto a factor of proportionality) is uniquely solvable, because N X ν =1 M X µ =1 c νµ f S ( ω ) ν j ( ω ) ν = 0 is the onlylinear relation over k among the f S ( ω ) ν j ( ω ) µ , ν = 0 , , . . . , N , µ = 0 , , . . . , M . The solutions c νµ are therefore, up to a factor of proportionality, elements of Λ .8 he functions f S ( ω ) from P M S are characterized by the property that the q -expansion of f ( s
00 1 )( ω ) has rational coefficients. That the functions from P M S have this property is clear, because they are rational functionsof j ( ω ) and j ( S ( ω )) with rational coefficients. Conversely, suppose f ( s
00 1 )( ω ) has rational q -coefficients. Since the irreducible equation F ( f S ( ω ) , j ( ω )) = 0 for f S ( ω ) in k M has coefficients in P M , as shown above, it follows that f S ( ω ) is algebraic over P M . The field P M S ( f S ( ω )) = P ( j ( ω ) , j ( S ( ω )) , f S ( ω ))contains no constant other than the rational numbers, because the q -expansion of a function from P (cid:16) j ( ω ) , j ( S ( ω )) , f ( s
00 1 )( ω ) (cid:17) has rational coefficients and the q -expansion of a constant ̺ readsas ̺ = ̺ · q . One thus has the degree equation[ P M S ( f S ( ω )) : P ( j ( ω )) ] = [ k M S ( f S ( ω )) : k ( j ( ω )) ] , but since k M S ( f S ( ω )) = k M S , one has[ P M S ( f S ( ω )) : P ( j ( ω )) ] = [ k M S : k M ] = [ P M S : P M ] , whence P M S ( f S ( ω )) = P M S , f S ( ω ) ∈ P M S . Of course, a function from k M S of the form h S ( ω ) = H S ω ω !! H ω ω ! lies in P M S when the q -expansions H ν ( ω ) = X n h ( ν ) n q n have rational coefficients h ( ν ) n .In any case, the q -expansion of a function f S ( ω ) from P M S , for arbitrary S = a b d ! , hascoefficients in the cyclotomic field P ( ζ s ) , ζ s = e πis , because that holds for j ( S ( ω )) = j (cid:18) a ω + bd (cid:19) = X n c n ζ abns q ad n . A function f S ( ω ) from P M S will be called integral if its conjugates f S ν ( ω ) have q -expansionswhose coefficients are algebraic integers (in P ( ζ s ) ).9he coefficients of the principal polynomial F ( X, j ( ω )) = ψ ( s ) Y ν =1 ( X − f S ν ( ω ))of an integral function f S ( ω ) from P M S have rational integer q -coefficients, and from that, itfollows by § that: The principal polynomial F ( X, j ( ω )) of an entire integral function f S ( ω ) from P M S is a poly-nomial in X and j ( ω ) with rational integer coefficients. In particular, J s ( X, j ( ω )) is a polynomial in X and j ( ω ) with rational integer coefficients. When the level of transformation s = p is a prime number, the principal polynomial of anentire integral function from P M P satisfies a simple congruence modulo p . We derive it first for J s ( X, j ( ω )) . As a polynomial in X and j ( ω ) , J p ( X, j ( ω )) satisfies the congruence J p ( X, j ( ω ) ≡ ( X p − j ( ω )) ( X − j ( ω ) p ) mod p. By § it suffices to show that corresponding coefficients of the expansions in X and q on both sidesare congruent modulo p . One has J p ( X, j ( ω )) = ( X − j ( p ω )) p − Y ν =0 (cid:18) X − j (cid:18) ω + νp (cid:19) (cid:19) . From j ( ω ) = X c n q n , it follows that j (cid:18) ω + νp (cid:19) = X c n ζ νnp q np , thus the coefficient-wise congruence means j (cid:18) ω + νp (cid:19) ≡ j (cid:18) ωp (cid:19) mod (1 − ζ p ) ,J p ( X, j ( ω )) ≡ ( X − j ( p ω )) (cid:18) X p − j (cid:18) ωp (cid:19) p (cid:19) mod (1 − ζ p ) . Further, one has j ( p ω ) = X c n q pn , hence j ( p ω ) ≡ j ( ω ) p mod p and j ( ω ) ≡ j (cid:18) ωp (cid:19) mod p, thus J p ( X, j ( ω )) ≡ ( X − j ( ω ) p ) ( X p − j ( ω )) mod (1 − ζ p );10ut as the coefficients of the expansions in X and q from both sides are rational integers, this alsoholds modulo p .Now let f P ( ω ) be an arbitrary entire integral function from P M P , and let F ( X, j ( ω )) be itsprincipal polynomial over P M .For the q -expansion of f (cid:16) ν p (cid:17) ( ω ) , ν = 0 , , , . . . , p − f (cid:16) ν p (cid:17) ( ω ) ≡ f (cid:16) p (cid:17) ( ω ) mod (1 − ζ p ) , whence F ( X, j ( ω )) ≡ (cid:18) X − f (cid:16) p
00 1 (cid:17) ( ω ) (cid:19) (cid:18) X p − f (cid:16) p (cid:17) ( ω ) p (cid:19) mod (1 − ζ p )in the sense of the q -coefficients. Let us write F ( X, j ( ω )) out as F ( X, j ( ω )) = X p +1 + Q p ( j ( ω )) X p + · · · + Q ( j ( ω )) , from which it follows that F ( X, j ( ω )) ≡ X p +1 + Q p ( j ( ω )) X p + Q ( j ( ω )) X + Q ( j ( ω )) mod p, i.e. the polynomials Q µ ( j ( ω )) of j ( ω ) , 2 µ p − p .In addition to j ( S ( ω )) , we still need another primitive element of P M S / P M , namely the functionformed by H = s ∆ and H = ∆ :(15) ϕ S ( ω ) = s ∆ S ω ω !! ∆ ω ω ! . That ϕ S ( ω ) belongs to P M S follows from the fact that the q -expansion coefficients of ∆ arerational. The q -expansions of the conjugates of ϕ S ( ω ) yield(16) ϕ ( a b d )( ω ) = a ζ a bs q a − dd D ζ a bs q ad + · · · + D n ζ a b ns q ad n + · · · D q + · · · + D n q n + · · · . They show that the ψ ( s ) conjugates ϕ S ν ( ω ) are distinct from one another; whence ϕ S ( ω ) is indeeda primitive element of P M S / P M .Since ∆ and ∆ − are both entire modular forms, and since the q -expansions (16) have obviouslyinteger coefficients, it follows that ϕ S ( ω ) is an entire integral function from P M S , and similarly for j ( S ( ω )) . Its principal polynomialΦ s ( X, j ( ω )) = Y S ( X − ϕ S ( ω ))= X ψ ( s ) + B ( s ) ψ ( s ) − ( j ( ω )) X ψ ( s ) − + · · · + B ( s )0 ( j ( ω ))11s therefore also a polynomial in X and j ( ω ) with rational integer coefficients.The coefficient B ( s )0 ( j ( ω )) can be calculated. As ϕ S ( ω ) obviously has no zero for any finite valueof ω , it follows that B ( s )0 ( j ( ω )) has no zero either, and thus B ( s )0 as a polynomial in j ( ω ) mustbe a constant. Its value is obtained as the product of the leading coefficients of the q -series for all ϕ S ( ω ) : Y S ϕ S ( ω ) = ( − ψ ( s ) B ( s )0 ( j ( ω )) = Y S a ζ abs , and since this must be a rational number,(17) B ( s )0 ( j ( ω )) = ± Y S a . When s = p is a prime number, this becomes(18) B ( s )0 ( j ( ω )) = ( − p +1 p p − Y b =0 ζ bp = p . To express a function f S ( ω ) from k M S in terms of the basis 1 , ϕ S ( ω ) , . . . , ϕ S ( ω ) ψ ( s ) − of P M S / P M , we use the Lagrange interpolation formula in the well-known manner: f S ( ω ) Φ ′ s ( ϕ S ( ω ) , j ( ω )) = Sp k M S /k M (cid:20) f S ( ω ) Φ s ( X, j ( ω )) X − ϕ S ( ω ) (cid:21) X = ϕ S ( ω ) or f S ( ω ) Φ ′ s ( ϕ S ( ω ) , j ( ω )) = a ( j ( ω )) + a ( j ( ω )) ϕ S ( ω ) + · · · + a ψ ( s ) − ( j ( ω )) ϕ S ( ω ) ψ ( s ) − with a ν ( j ( ω )) = ψ ( s ) X µ = ν +1 B µ ( j ( ω )) Sp k M S /k M (cid:2) f S ( ω ) ϕ S ( ω ) µ − ( ν +1) (cid:3) . From this we read off:1. If f S ( ω ) is entire, the a ν ( j ( ω )) are the polynomials. If f S ( ω ) is an integral function from P M S , the q -coefficients of all a ν ( j ( ω )) are integers. and thus furthermore3. If f S ( ω ) is an entire integral function from P M S , the polynomials a ν ( j ( ω )) in j ( ω ) haverational integer coefficients. We apply this to the case when s = p is a prime number to obtain a proposition on the divisibilityof the coefficients of a ( j ( ω )) by p : Let f P ( ω ) be an entire integral function from P M P , where P is of determinant p . Furthermore,suppose the coefficients of the q -expansion of f (cid:16) p
00 1 (cid:17) ( ω ) are divisible by p . Then the coefficients of a ( j ( ω )) are divisible by p . This theorem summarizes an argument given by
Hasse , J. f. Math. , 125–126 (1927). f P ( ω ) Φ ′ p ( ϕ P ( ω ) , j ( ω )) = a ( j ( ω )) + a ( j ( ω )) ϕ P ( ω ) + · · · + a p ( j ( ω )) ϕ P ( ω ) p for P = p
00 1 ! ; since f (cid:16) p
00 1 (cid:17) ( ω ) and ϕ (cid:16) p
00 1 (cid:17) ( ω ) = p q p − D q p + · · · D q + · · · are divisible by p and the a ν ( j ( ω )) have integer q -coefficients, it follows that the q -coefficients of a ( j ( ω )) are divisible by p .
4. The functions p ϕ S ( ω ) . The twenty-fourth root of ∆( ω ) , that is p ∆( ω ) = √ π η ( ω ) , is a function of the variable ω ; it is regular in I ω > M ,but it is well-known how it transforms when a modular substitution is performed on ω . Namely,for M = α βγ δ ! with γ > M , one has η ( M ( ω )) = ε ( M ) p − i ( γ ω + δ ) η ( ω ) , where the root is taken with positive real part and ε ( M ) denotes the following 24-th root of unity:let γ = 2 λ γ with γ odd and γ = 1 in the case when γ = 0 ; then ε α βγ δ ! = (cid:18) αγ (cid:19) ζ ( β δ (1 − γ )+ γ ( α + δ )+3 (1 − γ )+3 α ( γ − γ )+ λ · ( α − ) , ζ = e πi . We now define the 24-th root of ϕ ( s
00 1 )( ω ) by q ϕ ( s
00 1 )( ω ) = vuut s ∆ s ω ω ! vuut ∆ ω ω ! = √ s η ( s ω ) η ( ω ) , √ s > . How does q ϕ ( s
00 1 )( ω ) transform when an element M of M ( s
00 1 ) is applied to ω ? Since s
00 1 ! α βγ δ ! s
00 1 ! − = α s βs − γ δ ! , This transformation formula for the η -function was in essence already stated by Dedekind . M ( s
00 1 ) are the modular substitutions α βγ δ ! with γ ≡ s . Therefore, vuut ϕ ( s
00 1 ) α βγ δ ! ( ω ) ! = √ s η α s βs − γ δ ! ( s ω ) ! η α βγ δ ! ( ω ) ! = ε α s βs − γ δ ! ε α βγ δ ! q ϕ ( s
00 1 )( ω ) . From this it is now easy to conclude: if s = t is a square relatively prime to 6, then for M ∈ M (cid:16) t
00 1 (cid:17) one has: q ϕ (cid:16) t
00 1 (cid:17) ( M ω ) = q ϕ (cid:16) t
00 1 (cid:17) ( ω ) . One has to prove(19) ε α t βt − γ δ ! = ε α βγ δ ! . For both ε -values, λ is the same number, since s is odd. The exponent of ζ in ε α t βt − γ δ ! thus arises from that in ε α βγ δ ! by replacing β by t β , γ by t − γ , and γ by t − γ .Since t ≡ ζ appears in ε α t βt − γ δ ! as in ε α βγ δ ! . Since (cid:18) αt − γ (cid:19) = (cid:18) αγ (cid:19) as well, (19) follows.Still keeping in mind that the q -expansion of η ( ω ) and hence also that of q ϕ (cid:16) t
00 1 (cid:17) ( ω ) haverational coefficients, we thus have: For t relatively prime to 6, q ϕ (cid:16) t
00 1 (cid:17) ( ω ) is a function from P M (cid:18) t
00 1 (cid:19) . For any primitive S of determinant t , we define p ϕ S ( ω ) as the function from the field P M S which is conjugate to q ϕ (cid:16) t
00 1 (cid:17) ( ω ) . The function q ϕ (cid:16) t
00 1 (cid:17) ( ω ) is entire and integral, since thatholds for ϕ (cid:16) t
00 1 (cid:17) ( ω ) .
5. Elliptic functions.
We need the Weierstass ℘ -function ℘ z , ω ω !! = z − + + ∞ X n ,n = −∞ n ,n =0 , [ ( z − n ω − n ω ) − − ( n ω + n ω ) − ]as a function of the three complex variables z, ω , ω , in which we assume that I ( ω ) > ℘ as a function of ω , ω depends only on the lattice w with14he basis ω , ω , we also occasionally write ℘ ( z, w ) . In ℘ z , ω ω !! , ω ω ! is understood as acolumn vector. The function ℘ is homogeneous in z, ω , ω of degree − ℘ as a functionof ω = ω ω has period 1, ℘ has a Fourier expansion in powers of q = e πiω ; with U = e πizω , itreads as ℘ z , ω ω !! = − (cid:16) πω (cid:17) (cid:16) U − U − (cid:17) − ∞ X n,m =1 n q nm + 12 ∞ X n,m =1 n q nm ( U n + U − n ) , valid for I ω > | q | < | q | < U < | q | , i.e. (cid:12)(cid:12)(cid:12)(cid:12) I (cid:18) zω (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) > I ω .
6. The Weber function.
For number-theoretic purposes the ℘ -function must be replaced bya suitably normalized elliptic function which is adapted to an imaginary quadratic number field Σand an order R in it. To the order R is assigned an elliptic function τ R z , ω ω !! , the Weberfunction of the order R , namely(20) τ R z , ω ω !! = g ( e ) ω ω ! ℘ z , ω ω !! e , in which e denotes the number of units of R ; thus e = 2 except when R is the principal order of P ( √−
1) where e = 4 , or when R is the principal order of P ( √−
3) where e = 6 ; and g ( e ) is acertain entire modular form of weight e , namely(21) g (2) ω ω ! = − · g ( ω ) g ( ω )∆( ω ) ω g (4) ω ω ! = 2 · g ( ω )∆( ω ) ω g (6) ω ω ! = − · g ( ω )∆( ω ) ω . The function τ R z , ω ω !! is homogenous of degree 0 as a function of the three complex variables z, ω , ω , and as a function of ω , ω , it is invariant with respect to modular substitutions. The g ( e ) is chosen so that τ R is not identically 0 when a basis of an ideal of R is used for ω , ω ; thenumerical factor in g ( e ) causes the Fourier expansion of τ R to have the form(22) τ R z , ω ω !! = h q e + · · · + t ν q ν + · · · i " U (1 − U ) + 12 ∞ X n,m =1 n q nm ( U n + U − n − e , where the coefficients t ν are rational integers. 15e can also write τ R ( z, w ) instead of τ R z , ω ω !! where w denotes the lattice with thebasis ω , ω , since τ R depends only on w and not on the particular basis.
7. The division values of the Weber function. Let N be a natural number. By the N -thdivision values of τ R z , ω ω !! , we mean the functions τ R x ω + x ω N , ω ω !! = τ R N ( x , x ) ω ω ! , ω ω !! , where x , x are integers with ( x , x ) (0 ,
0) mod N. It only depends on x , x modulo N .If ( x , x , N ) = 1 , then τ R x ω + x ω N , ω ω !! is called a proper N -th division value of τ R .Obviously τ R x ω + x ω N , ω ω !! is a proper N -th division value, for N = N/ ( x , x , N ) . The division values of τ R are regular functions of ω in I ω > . Let M be a modularsubstitution. We apply M to ω ; then τ R x ω + x ω N , ω ω !! becomes τ R N ( x , x ) · M ω ω ! , M ω ω !! = τ R N ( x , x ) M · ω ω ! , ω ω !! . Every M in M thus interchanges the proper N -th division values of τ R with one another.The N -th division values of τ R can be expanded in powers of q N ; we just have to substitute U = e πiN ( x ω + x ) = q x N ζ x N in (22), whereby we choose x in its residue class modulo N satisfying0 x < N so as to fulfill the convergence condition of (22). That gives(23) τ R x ω + x ω N , ω ω !! = h q − e + · · · i ζ x N q x N (cid:16) − ζ x N q x N (cid:17) + 12 ∞ X n,m =1 n q nm (cid:16) ζ x nN q x N n + ζ − x nN q − x N n − (cid:17) e . In this series only finitely many powers of q with negative exponents occur. The coefficientslie in P ( ζ N ) . When we apply the automorphism ζ N → ζ rN , ( r, N ) = 1 , of the field P ( ζ N ) to Hasse , loc. cit.
131 to 136. q -expansion of τ R x ω + x ω N , ω ω !! , it becomes the q -expansion of τ R x ω + r x ω N , ω ω !! .It now follows from the above that: Y x ,x mod N ( x ,x ,N )=1 X − τ R x ω + x ω N , ω ω !! ! is a polynomial T N ( X, j ( ω )) in X and j ( ω ) with rational coefficients. The polynomial T N ( X, j ( ω ))is called the N -th order division polynomial of the function τ R .We further note that the series (23) has integer coefficients when x N ) , whereas itbecomes integral only after multiplication by (1 − ζ x N ) e when x ≡ N ) . Now one has Y x mod N ( x,N )=1 (1 − ζ xN ) = ( , if N is not a prime power ,ℓ, if N is a power of the prime number ℓ. Consequently the coefficients of the divison polynomials T N ( X, j ( ω )) of order N are rationalintegers when N is not a prime power; and if N is a power of the prime number ℓ , then theybecome integers after multiplication by ℓ e . Let p be a prime number which does not divide N . With a matrix P of determinant p , weform the function δ P (( x , x ); ω ) = τ R N ( x , x ) ω ω ! , ω ω !! p − τ R pN ( x , x ) ω ω ! , P ω ω !! . The function δ P depends only on the equivalence class of P and is regular in I ω > τ R pN ( x , x ) ω ω ! , P ω ω !! = τ R N · ( x , x ) p P − · P ω ω ! , P ω ω !! is an N -th division value of τ R z , P ω ω !! .If M is a modular matrix, one has(24) δ P (( x , x ) ; M ( ω )) = δ MP (( x , x ) M ; ω ) . If M in fact belongs to the group M P , then M only interchanges those δ P (( x , x ); ω ) with( x , x , N ) = 1 with one another. We note that δ P (( x , x ); ω ) has a q -expansion with only finitelymany negative exponents of q , because that holds for the N -th division value of τ R z , ω ω !! and also for that of τ R z , P ω ω !! ; hence we have:17 he coefficients D ( ν ) P ( ω ) of the polynomial S P ( X, ω ) = Y x ,x mod N ( x ,x ,N )=1 ( X − δ P (( x , x ); ω ) ) = X ν D ( ν ) P ( ω ) X ν are entire functions from k M P . By the way, as already expressed in the notation, D ( ν ) P µ ( ω ) isconjugate to D ( ν ) P ( ω ) as a function from k M Pµ , for this conjugate can be calculated as D ( ν ) P ( ω ) λ µ = D ( ν ) P ( M P,P µ ( ω )) = D ( ν ) P M
P,Pµ ( ω ) = D ( ν ) M ∗ P,Pµ P µ ( ω ) = D ( ν ) P µ ( ω ) . We now prove: D ( ν ) P ( ω ) in fact lies in P M P . For that we investigate the q -expansion coefficients of δ (cid:16) p
00 1 (cid:17) (( x , x ) ; ω ) = τ R N ( x , x ) ω ω ! , ω ω !! p − τ R N ( x , x p ) p ω ω ! , p ω ω !! . These coefficients obviously lie in the cyclotomic field P ( ζ N ) . We apply to them simultaneouslythe automorphisms ζ N → ζ rN , ( r, N ) = 1 , of P ( ζ N ) ; then the q -expansion of δ (cid:16) p
00 1 (cid:17) (( x , x ); ω )becomes that of δ (cid:16) p
00 1 (cid:17) (( x , r x ); ω ) . The q -expansion coefficients of D ( ν ) (cid:16) p
00 1 (cid:17) ( ω ) are therefore in-variant under all automorphisms of P ( ζ N ) , and so they are rational numbers. But from that theclaim D ( ν ) P ( ω ) ∈ P M P follows by § . The function D ( ν ) P ( ω ) has p -integral q -expansion coefficients, because that holds for the N -thdivision value of τ R . But furthermore, we have: The q -expansion coefficients of D ( ν ) (cid:16) p
00 1 (cid:17) ( ω ) aredivisible by p . Namely, one has δ (cid:16) p
00 1 (cid:17) (( x , x ) ; ω ) = τ R N ( x , x ) ω ω ! , ω ω !! p − τ R N ( x , x p ) p ω ω ! , p ω ω !! , and since the q -expansion of τ R N ( x , x p ) p ω ω ! ; p ω ω !! arises from that of τ R N ( x , x ) ω ω ! ; ω ω !! by replacing q by q p and ζ N by ζ pN , it follows that the p -th power of the q -expansion of τ R N ( x , x ) ω ω ! ; ω ω !! is termwise congruent to it modulo p , whence the q -expansionof δ (cid:16) p
00 1 (cid:17) (( x , x ); ω ) indeed has coefficients divisible by p .18 . Number-theoretic basics
8. Orders in quadratic number fields. Let Σ be a number field; the principal order of Σwill be denoted by R , while R stands for an arbitrary order of Σ . Ideals of R will be denotedby a R , b R , . . . ; an ideal of R is called integral in R if it lies in R . In particular, let F R be the conductor of R , that is to say the ring-quotient R : R , or, what amounts to the same thing, thelargest ideal of R , integral in R , which is at the same time an ideal of R .As usual, two integral ideals a R and b R of R are called coprime if a R + b R = R . An arbitraryideal a R of R is called prime to the integral ideal b R of R if there exists an element β of Σ suchthat β R and β a R are integral in R and are coprime to b R . An ideal a R is called a characteristicideal of R or from the order R if the module-quotient a R : a R equals R , that is to say, if R ismaximal among the orders of Σ of which a R is an ideal. An ideal a R is called invertible if thereexists an ideal a − R such that a R a − R = R ; then a R is necessarily a characteristic ideal of R . Theinvertible ideals of R form a group J R . The ideals of R prime to F R are invertible (and hencecharacteristic in R ); they thus form a subgroup J (0) R of J R . The group J (0) R is a canonical isomorphic image of the group D R of divisors of Σ prime to F R ; we denote the ideal associated to the divisor a by a R . More precisely, we can describe thisisomorphism as follows. Let S R be the ring of all elements αβ where α and β = 0 lie in R and β R is prime to F R . Then for any given divisor a , the associated ideal a R is the collection of theelements of S R divisible by a ; conversely a is the greatest common divisor of all elements of a R .For integral a , the index [ R : a R ] is equal to the norm N a of the divisor; in general one has[ b R : b R a R ] = N a for an invertible ideal b R of R . Let R ′ be an order contained in R ; then one has F R ′ ⊆ F R . Fordivisors a prime to F R ′ (and hence to F R ), one has a R ′ = a R ∩ S R ′ , thus a R ′ = a R ∩ R for integral a and a R = a R ′ R. Let H R be the group of principal ideals α R ( α = 0 from Σ ) of R , and let R R = J R / H R bethe ideal class group of R . In every ideal class of R there exists an ideal prime to F R , i.e. one has J R = J (0) R H R , whence setting J (0) R ∩ H R = H (0) R , one sees that R R is canonically isomorphic with J (0) R / H (0) R . Here, H (0) R is the group of all ideals α R , α ∈ G R , where G R is the group of all quotients βγ of elements β = 0 , γ = 0 , of R with β R , γ R prime to F R . The caonical isomorphism from D R onto J (0) R thus induces a canonical isomorphism from the factor group of D R modulo the group( G R ) of principal divisors ( α ) , α ∈ G R — which shall be called the divisor class group of R —onto the ideal class group of R . The divisor class group of R shall henceforth also be denoted by R R . Dirichlet-Dedekind , Vorlesungen ¨uber Zahlentheorie (Lectures on number theory), XI. Supplement, cf. also thisEncyclopedia I 2, 19. quadratic number field Σ , the following holds:
Any characteristic ideal a R of an order R is invertible. The conductor F R of an order R isof the form F R = f R with f being a natural number; for every natural number f , there existsexactly one order with conductor f R (we say briefly: with conductor f ), namely, the ring R f ofall integral elements of Σ which are congruent modulo f to rational numbers. Then S R f is thering of all f -integral elements of Σ which are congruent modulo f to rational numbers, and G R f consists of all elements of Σ prime to f (as divisor) which are congruent modulo f to rationalnumbers.Having R f ′ ⊆ R f is equivalent to having f | f ′ . If a is a divisor prime to f ′ , and f | f ′ , andif α , α is a basis of a R f while α ′ , α ′ is a basis of a R f ′ , then α ′ α ′ ! = S α α ! for a primitive matrix S of determinant s = f ′ /f .To prove that, we choose β so that β a is integral and prime to f ′ ; thus β a R f ′ + f ′ R = R f ′ , from which it follows, on multiplication by R f , that β a R f ′ + f ′ R = R f , and as f ′ R ⊆ R f ′ , one has β a R f ′ + R f ′ = R f . Thus we have the following relation between additive factor groups: R f / R f ′ = ( β a R f + R f ′ ) / R f ′ ∼ = β a R f / β a R f ∩ R f ′ = β a R f / β a R f ′ ∼ = a R f / a R f ′ ;but it is easy to see that R f ′ has index s = f ′ /f in R f , whence α ′ α ′ ! = S α α ! for an integer-entry matrix S of determinant s . The fact that S is primitive arises from the following fact: If α , α is a basis of a characteristic ideal of R f , and S is a primitive matrix of determinant s ,then S α α ! is a basis of a characteristic ideal b R f ′ of an order R f ′ whose conductor f ′ satisfies: f ′ | f s and f s − | f ′ . As S and s S − are primitive and of determinant s , it suffices to prove that f ′ | f s , i.e. thatfor any ξ ∈ R fs , one always has b R f ′ ξ ⊆ b R f ′ , or that one has S α α ! ξ = D ′ ξ S α α ! for an integer -entry matrix D ′ ξ . Now if ξ = r + f sη with r being a rational integer and η beingan integer in Σ , then f η is in R f , and α α ! f η = D fη α α ! with an integer-entry D fη , thus S α α ! ξ = ( r E + S D fη s S − ) S α α ! , and D ′ ξ = ( r E + S D fη s S − ) has integer-entries.20hen p is a prime number, one has the sharper result: If α , α is a basis of a characteristic ideal a R f of R f , where f is divisible by p , then for aprimitive matrix P of determinant p , P α α ! is a basis of a characteristic ideal b R ′ in R ′ = R fp − or in R ′ = R fp but not in R f . Proof. One has p a R f ⊂ b R ′ ⊂ a R f with b R ′ = a R f or p a R f ; thus p R f ⊂ b R ′ a − R f ⊂ R f , b R ′ a − R f = R f or p R f , [ R f : b R ′ a − R f ] = [ b R ′ a − R f : p R f ] = p because [ R f : p R f ] = p .For the module-quotient [ R f : R fp − ] = q (the conductor of R f with respect to R fp − ) one hascorrespondingly(25) p R f ⊂ q ⊂ R f , [ R f : q ] = [ q : p R f ] = p. This is because one has q = 1 · q ⊆ R fp − · q ⊆ R f , and p R f · R fp − = p R fp − ⊆ R f , i.e. p R f ⊆ q , since for ξ ∈ R fp − , there exists a rational number r with ξ ≡ r mod f p − , so p ξ ≡ p r mod f ,which proves that p ξ ∈ R f . Here q is an ideal of R fp − , consequently = R f or p R f , because R fp − · R fp − q = R fp − q ⊆ R f proves that R fp − q = q . Now suppose b R ′ and thus also b R ′ a − R f are characteristic in R f ; then one must have q = b R ′ a − R f , and since q is prime by (25), onehas q + b R ′ a − R f = R f , and consequently q b R ′ a − R f = q ∩ b R ′ a − R f = p R f , whence it follows that: q b R ′ a − R f = R fp − q R ′ a − R f = p R f p − , which is to say R f = R fp − , a contradiction. Let α , α be a basis of a characteristic ideal a R f of R , and let S be primitive of determinant s such that S α α ! is a basis of a characteristic ideal b R fs of R fs . Then one has b R fs R f = a R f .If a second primitive matrix S ′ of determinant s also makes S ′ α α ! a basis of a characteristicideal b ′ R fs of R fs , and if b ′ R fs is equivalent to b R fs , then S ′ is equivalent to S D ε , where D ε isthe representing matrix of a unit ε of R f for the basis α , α : one has α α ! ε = D ε α α ! . Proof. Let β in Σ be chosen so that β b R fs is integral in R fs and prime to f s R , so that β b R fs + f s R = R fs . Since it follows by multiplication with R f that β b R fs R f + f s R = R f , and since β b R fs R f ⊆ β a R f ∩ R f ⊆ R f , one has( β a R f ∩ R f ) + f s R = R f , β a R f ∩ R f is integral in R f and prime to f s R . By the above proof, the canonically associatedideal β a R f ∩ R f ∩ R fs = β a R f ∩ R fs of R fs thus has index s in β a R f ∩ R f , and because β b R fs ⊆ β a R f ∩ R fs ⊆ β a R f ∩ R f ⊆ β a R f and by assumption one has [ β a R f : β b R fs ] = s , it follows that β a R f ∩ R fs = β b R fs , β a R f ∩ R f = β a R f , and hence one must have β b R fs R f = β a R f , b R fs R f = a R f , as claimed.The fact that b ′ R fs is equivalent to b R fs means that there exists an element ε of Σ with b ′ R fs = b R fs ε . Multiplication by R f yields a R f = a R f ε ; thus ε is a unit in R f .Then S ′ α α ! and S α α ! ε = S D ε α α ! are bases of b ′ R fs = b R fs ε ; consequently S ′ and S D ε are equivalent. Suppose the conductor f is divisible by s . If α , α is a basis of a characteristic ideal a R f of R f , then there exists exactly one class of primitive matrices S of determinant s for which S α α ! is a basis of a characteristic ideal of R fs − . When a R f is prime to f , this ideal of R fs − is equalto s a R fs − . Proof. Since the result does not depend on a factor of a R f , one can assume a R f to be prime to f . A basis of a R f canonically associated to the ideal a R fs − of R fs − is of the form S − α α ! ,where S is primitive, | S | = s . Then S = s S − is primitive of determinant s and S α α ! is abasis of the characteristic ideal s a R fs − of R fs − .Let S ′ be any primitive matrix of determinant s , for which S ′ α α ! is a basis of a characteristicideal b R fs − of R fs − . The primitive matrix s S ′− of determinant s sends the basis S ′ α α ! of b R fs − to the basis s α , s α of s a R f ; thus one has s a R f R fs − = b R fs − , and for S ′ = S this becomes s a R f R fs − = s a R fs − , whence b R fs − = s a R fs − , so S ′ α α ! and S α α ! are bases of s a R fs − , and therefore S ′ and S are equivalent. 22 . The first main theorem
9. Formulation of the first main theorem.
Let Σ be an imaginary quadratic number field,and let R = R f be the order of conductor f in Σ . The ideals in a given ideal class k R of R are lattices in an equivalence class in the sense of § . Let j ( k R ) denote the invariant of this latticeclass; thus j ( k R ) = j (cid:18) α α (cid:19) where α , α is a basis of an ideal a R from k R with I (cid:18) α α (cid:19) > invariant of the ideal class k R determines k R uniquely, because if k ′ R ′ is an ideal class in afield Σ ′ and j ( k ′ R ′ ) = j ( k R ) , then with α ′ , α ′ being a basis of an ideal a ′ R ′ from k ′ R ′ , we have α ′ α ′ ! = M α α ! β with β = 0 , M ∈ M , so (cid:18) α ′ α ′ (cid:19) = M (cid:18) α α (cid:19) , therefore Σ ′ = Σ , β ∈ Σ , and a ′ R ′ = a R β lies in k R , whence k ′ R ′ = k R .Let h R be the ideal class number of R , i.e. the order of the divisor class group D R / ( G R ) of R ,and let k (1) R , . . . , k ( h R ) R be the ideal classes of R . First main theorem. The class invariants j ( k (1) R ) , . . . , j ( k ( h R ) R ) of R are algebraic numbers. Thenumber field Ω R = Σ( j ( k R )) generated over Σ by one of the invariants j ( k R ) is the class field of Σ for the ring divisor class group D R / ( G R ) = R R . The field Ω R is called the ring class field of Σ forthe order R . We give two proofs of this theorem. The first uses the general theory of abelian number fieldsand is accordingly short. The central idea of the second, after the class invariants are identified asalgebraic numbers, is to determine the structure of the field Ω R by means of the theory of modularfunctions, i.e. in essence, to see that the Galois group of Ω R / Σ is isomorphic to the divisor classgroup of R and to prove the reciprocity law for Ω R / Σ , and in the process to use only the elementaryparts of the theory of algebraic numbers.We can view the statement of the first main theorem in a different way.If f ( ω ) denotes a modular function associated to the full modular group or to one of its subgroupsof finite index, we call the complex numbers f ( ξ ) for imaginary quadratic argument ξ (with positiveimaginary part) the singular values of the function f ( ω ) . The first main theorem identifies thenumber field generated by the singular values of the function j ( ω ) (the determination of P ( j ( ξ ))inside Σ( j ( ξ )) is done in § ), because the lattice with basis ξ,
10. Proof of the first main theorem by means of the general theory of abelian numberfields.
It suffices to prove the following:
The complex number j ( k R ) is an algebraic number, and the degree-one prime divisors p of Σ which split completely in Σ( j ( k R )) are, apart from finitely many exceptions, exactly those for which p R is a principal ideal. p , the transformation polynomial J p ( X, j ( ω )) of j ( ω ) has rational integer coefficients andsatisfies the congruence (cf. § )(26) J p ( X, j ( ω )) ≡ ( X − j ( ω ) p ) ( X p − j ( ω )) mod p. Let p be a degree-one prime divisor of Σ with N Σ / P p = p . Then one has(27) J p ( j ( k R ) , j ( k R p − R )) = 0 , because if α , α is a basis of an ideal a R of k R p − R , so that j ( k R p − R ) = j (cid:18) α α (cid:19) , then there existsan integer-entry matrix P of determinant p such that P α α ! is a basis of a R p R ∈ k R , whence j ( k R ) = j (cid:18) P (cid:18) α α (cid:19)(cid:19) .From (26) it follows that J p ( X, X ) ≡ − ( X p − X ) mod p , whence(28) J p ( X, X ) = 0 . Let us choose p so that p R is a principal ideal; then it follows from (26) that J p ( j ( k R ) , j ( k R )) = 0 , which shows that j ( k R ) is an algebraic number.If we exclude the finitely many p which are not prime to the denominators of all j ( k R ) , it followsfrom (26) that(29) (cid:0) j ( k R ) − j ( k R p − R ) p (cid:1) (cid:0) j ( k R ) p − j ( k R p − R ) (cid:1) ≡ p and thus for each prime divisor P of p in Σ( j ( k R )) ,(30) j ( k R ) p ≡ j ( k R p − R ) or j ( k R ) ≡ j ( k R p − R ) p mod P . As p R is a principal ideal, this simplifies to j ( k R ) p ≡ j ( k R ) mod P ;thus if we now further suppose that p is prime to the discriminant of j ( k R ) over Σ , it follows thatin fact α p ≡ α mod P for all p -integers α of Σ( j ( k R )) ; therefore p splits completely in Σ( j ( k R )) .Conversely, suppose p splits completely in Σ( j ( k R )) . Then one has j ( k R ) p ≡ j ( k R ) mod P , whence by (30), it follows that j ( k R ) ≡ j ( k R p − R ) mod P . We now exclude the finitely many p which are not prime to all the differences j ( k R ) − j ( k ′ R ) , k R = k ′ R ,and it follows that j ( k R ) = j ( k R p − R ) ; thus p R is a principal ideal.24
1. Overview of the proof of the first main theorem not relying on general class fieldtheory.
Deviating from § , we define the field Ω R byΩ R = Σ (cid:0) j ( k (1) R ) , . . . , j ( k ( h R ) R ) (cid:1) . The proof splits into several steps.First, it will be shown that the class invariants j ( k R ) are integral algebraic numbers, and to thatend the proof carried out in § for the algebraicity of j ( k R ) will be expanded; j ( k R ) will be seen asa zero of a suitable J s ( X, X ) .Second, a closer examination on the multiplicity of j ( k R ) as a zero of J s ( X, X ) shows: all(algebraic) numbers which are conjugate over P to one of the class invariants j ( k R ) of the order R are in fact class invariants j ( k ′ R ) of the same order R ; in other words: the class polynomial of theorder R (31) H R ( X ) = h R Y ν =1 ( X − j ( k ( ν ) R )has (integral) rational coefficients. From this it follows that Ω R is normal over P (all the more soover Σ ).Third, an isomorphism k R → σ ( k R ) from the class group R R of R into the Galois group G Ω R Σ of Ω R over Σ will be established.This isomorphism σ has the property that(32) j ( h R ) σ ( k R ) = j ( h R k − R );the j ( h R ) are thus a full system of (algebraic) numbers conjugate over Σ : the class polynomial H R ( X ) is irreducible over Σ . The basis for that, aside from the theory of decomposition group, inparticular the Frobenius -congruence, is a sharpening of the statement (30), namely the congruence(33) j ( k R ) p ≡ j ( k R p − R ) mod P for any degree-one prime divisor p of norm p over Σ and its prime factor P in Ω R ; for its proofthe function ϕ S ( ω ) is needed.Fourth, it will be shown that σ ( R R ) is equal to G Ω R Σ (and not only to a subgroup of G Ω R Σ ).In other words:One has Ω R = Σ( j ( k R )) for every class k R of the order R (whence the deviating definition ofΩ R is justified), or also: Σ( j ( k R )) is already normal over Σ . For that we have to compute the Galois group G Ω R P of Ω R over P , which is done again by means of (33).Fifth: σ is the canonical isomorphism between the class group R R and the Galois group G Ω R Σ asserted by the Artin reciprocity law. Namely, it will be shown that the decomposition assertion ofthe reciprocity law holds: 25 ( e R p R ) is the Frobenius automorphism of the prime divisor p of Σ in Ω R . For the degree-one prime divisors (except finitely many) this will follow directly from the definition of σ ; for thedegree-two prime divisors it follows from the structure of G Ω R P .
12. The class polynomial.
By the class polynomial H R ( X ) of the order R , we mean thepolynomial with the roots j ( k ( ν ) R ) ; thus H R ( X ) = h R Y ν =1 ( X − j ( k ( ν ) R )) . The invariant j ( k R ) can be written in the form j ( k R ) = j ( α ) , α = α α , where α , α is a basis of an ideal from k R . Other expressions for j ( k R ) follow from that. For any µ = 0 in R , one has α α ! µ = D µ α α ! for some rational-integer-entry representation matrix D µ , and hence j ( k R ) = j ( D µ ( α )) . The greatest common divisor of the entries of D µ is at the same time the largest rational part of µ in R , which is the largest rational integer t for which µ t − still lies in R . Hence D µ is primitiveexactly for those µ whose largest rational part in R is equal to 1 ; these µ are called primitive elements of R . Since N µ = | D µ | , one has for primitive µ (34) J Nµ ( j ( k R ) , j ( k R )) = 0 , or also(35) H R ( X ) is a divisor of J Nµ ( X, X ) . Now in J s ( X, X ) for non-square s , the highest power of X in J s ( X, X ) has coefficient ± q -expansion of the function J s ( j ( ω ) , j ( ω )) . Bymeans of the system of representatives a b d ! of the equivalence classes of level s , it amounts to J s ( j ( ω ) , j ( ω )) = Y (cid:18) j (cid:16) aω + bd (cid:17) − j ( ω ) (cid:19) = Y (cid:16) ζ − abs q − ad + · · · − q − + · · · (cid:17) . The leading coefficient of the series in the brackets is either ζ − abs or − a > d or a < d ; the case a = d is not possible, because s = ad should not be a square. The leadingcoefficient of J s ( j ( ω ) , j ( ω )) is consequently a root of unity, which as a rational number must beequal to ± N µ is not a square. A primitive element µ with26on-square norm N µ in the order R = R f ( √− d ) of conductor f in the field Σ = P ( √− d ) with d square-free is µ = f √− d, N µ = d f for d > ,µ = f (1+ √− d ) , N µ = 2 f for d = 1 .H R ( X ) has rational integer coefficients. It suffices to see that the coefficients of H R ( X ) are rational numbers; that they are integers followsfrom the fact that the roots of H R ( X ) are integral.For a given s , we first ask for all the zeros of J s ( X, X ) and their multiplicities; as J ( X, X ) = 0 ,let us henceforth assume that s > ξ = j ( ̺ ) is a zero of J s ( X, X ) , or ̺ is a zero of the function J s ( j ( ω ) , j ( ω )) = ψ ( s ) Y ν =1 ( j ( S ν ( ω )) − j ( ω ) ) , if and only if there exists a primitive matrix S of determinant s with S ( ̺ ) = ̺. This results in a quadratic equation for ̺ with rational coefficients which are not all 0 since S = E ;thus ̺ is an imaginary quadratic (algebraic) number: every zero of J s ( X, X ) is a singular classinvariant.The multiplicity of a zero ξ = j ( ̺ ) of J s ( X, X ) is equal to the number t of solutions S (1) , . . . , S ( t ) of the equation S ( ̺ ) = ̺ which are inequivalent modulo M . That will be proved if it is shown that J s ( X, X ) ( X − ξ ) − t is regular but = 0 at X = ξ , or that J s ( j ( ω ) , j ( ω )) ( j ( ω ) − j ( ̺ )) − t is regular but = 0 at ω = ̺ . Let us write this function in the form J s ( j ( ω ) , j ( ω )) ( j ( ω ) − j ( ̺ )) − t = t Y µ =1 j ( S ( µ ) ( ω )) − j ( ω ) j ( ω ) − j ( ̺ ) Y S ν inequiv. to all S ( µ ) (cid:0) j ( S ν ( ω )) − j ( ω ) (cid:1) , from which we see that it suffices to prove: j ( ω ) − j ( ̺ ) and j ( S ( µ ) ( ω )) − j ( ω )have the same order at the zero ω = ̺ . Now one has j ( ω ) − j ( ̺ ) = ∞ X n =1 j ( n ) ( ̺ ) n ! ( ω − ̺ ) n ,j ( S ( µ ) ( ω )) − j ( ω ) = ∞ X n =1 j ( n ) ( ω ) n ! ( S ( µ ) ( ω ) − ω ) n = ∞ X n =1 j ( n ) ( ̺ ) + j ( n +1) ( ̺ )( ω − ̺ ) + · · · n ! ( S ( µ ) ( ω ) − ω ) n , ω = ̺ differ by the order of thezero ̺ of S ( µ ) ( ω ) − ω as a factor. But the latter is equal to 1 , because S ( µ ) ( ω ) − ω has the two distinct zeros ̺ and ̺ .Now let us ask conversely: how many times is a given class invariant j ( k R ) a zero of J s ( X, X ) ?Again let α , α be a basis of an ideal from k R ; thus j ( k R ) = j (cid:18) α α (cid:19) . The desired multiplicity isthe number of inequivalent primitive S of determinant s with S (cid:18) α α (cid:19) = α α , or S α α ! = α α ! µ for a certain element µ . Obviously µ lies in R , and S is equal to the representation matrix D µ determined by the basis α , α . Let S ′ = D µ ′ be a second primitive matrix of determinant s with S ′ (cid:18) α α (cid:19) = α α . Then one has S − S ′ = D − µ D µ ′ = D µ − µ ′ , whence S and S ′ are equivalent, i.e. D µ − µ ′ is unimodular if and only if µ − µ ′ is a unit in R :The multiplicity of the zero j ( k R ) of J s ( X, X ) is equal to the number of primitive elements ofnorm s in R which are non-associated in R .Now let R be given. We want to determine: for which s does there exist only one single classassociated to a primitive µ in R with N µ = s ?Now s = 1 always has this property, for then µ must be a unit in R . Henceforth let usassume s > µ , µ is also primitive of norm s ; thus the desired µ must satisfy µ = ε µ for a unit ε in R . As µ cannot be rational (or else µ would have the rational part µ = s > R ), one has ε = 1 . Now let R be the order R = R f ( √− d ) of conductor f in thefield Σ = P ( √− d ) . A basis of R is1 , f √− d for − d , f √− d − d ≡ . Let us leave aside the two cases R = R ( √−
1) and R = R ( √−
3) for now. Then one has µ = − µ , thus µ = b f √− d with a rational b . In the case of − d b must be an integer, and since µ is primitive,one has b = ± µ = ± f √− d, N µ = d f . In the case of − d ≡ µ = − bf + 2 b · f √− d b and bf must be integers, and since µ is primitive in R , it follows that µ = ± f √− d, N µ = d f for odd f ,µ = ± f √− d, N µ = d f for even f . In the case of R = R ( √−
1) with the basis 1 , √− µ = − µ , which gives µ = ±√− N µ = 1 , or µ = ±√− µ ,µ = ± ± √− , N µ = 2 . The four elements µ with N µ = 2 are associated in R .In the case of R = R ( √−
3) with the basis 1 , ζ = ( − √−
3) , one has either µ = − µ , thus µ = ±√− , N µ = 3 , or µ = ± ζ ν µ , ν = 1 , µ = ± ζ ν , which is the excluded case of N µ = 1 , or µ = ± ζ ν √− , N µ = 3 . The six elements with norm 3 are associated in R .Summarizing, we can say:To each R , there exists exactly one value s = s R > R which is associated in R to primitive elements µ of norm N µ = s R , namely s R = d f for R = R f ( √− d ) , − d , excluding R = R ( √− ,s R = 2 for R = R ( √− ,s R = d f for R = R f ( √− d ) , − d ≡ , f odd ,s R = d f for R = R f ( √− d ) , − d ≡ , f even . What possibilities are there that for two distinct orders R and R ′ , the elements s R and s R ′ are equal? 29hen the two orders R = R f ( √− d ) and R ′ = R f ′ ( √− d ) belong to the same field P ( √− d ) , with f < f ′ , the above table shows:One always has s R < s R ′ excluding the case − d ≡ f odd and f ′ = 2 f , in which case s R = s R ′ = d f .As s R differs only by a square factor from the discriminant of the field to which the order R belongs, with the one exception R = R ( √−
1) , s R = 2 , there exists only one possibility for twoorders R , R ′ in different fields to have s R = s R ′ , namely R = R ( √−
1) and R ′ = R ( √−
2) with s R = s R ′ = 2 . Hence there does not at all exist three distinct orders with the same s R .Next let R be an order for which there does not exist an R ′ different from it with s R = s R ′ .Then the simple zeros of J s R ( X, X ) are exactly the class invariants j ( k R ) of R , and so H R ( X ) isthe product of those factors of J s R ( X, X ) which are irreducible over P and which divide J s R ( X, X )only to the first power; thus H R has rational coefficients.Now let us consider two distinct orders R , R ′ with s R = s R ′ . The simple zeros of J s R ( X, X ) arethen exactly the class invariants j ( k R ) of R and those j ( k R ′ ) of R ′ , and it follows as above that H R ( X ) H R ′ ( X ) is a polynomial with rational coefficients.First case: R = R f ( √− d ) , R ′ = R f ( √− d ) , − d ≡ R = R ( √−
3) . Then thereexists a primitive element λ in R with norm d +14 f , namely λ = f √− d R ′ ,because the norm of a primitive element λ ′ = a + b · f √− d a ′ + b f √− d of R ′ , when b = 0 and thus λ ′ = ± N λ ′ = 1 < d +14 f , and when b = 0 , satisfies N λ ′ = a ′ + b f d > d f > d +14 f . It follows that J d +14 f ( X, X ) is divisible by H R ( X ) , but relatively prime to H R ′ ( X ) , whence H R ( X ) = (cid:0) H R ( X ) H R ′ ( X ) , J d +14 f ( X, X ) (cid:1) has rational coefficients, and then the same holds for the second factor H R ′ ( X ) as well.Second case: R = R ( √−
3) , R ′ = R ( √−
3) . Here the first factor H R ( X ) can be easilydetermined: R has only one class, and its invariant is calculated from the basis 1 , ζ to be j ( ζ ) = 0 , so H R ( X ) = X. R = R ( √−
1) , R ′ = R ( √−
2) . Here R ( √−
1) has only one class, whose invariantis calculated via the basis 1 , i to be j ( i ) = 2 · , so H R ( X ) = X − · . We are getting to the proof of the congruence (33), but before that, let us insert an investigationon
13. The singular values of the functions ϕ S ( ω ) . First part. The singular values ϕ S ( α ) ,with α imaginary quadratic, are algebraic integers. This follows from the equationΦ s ( ϕ S ( α ) , j ( α ) ) = 0 , s = | S | , because j ( α ) as a singlar class invariant is an (algebraic) integer, and the coefficients of Φ s ( X, j ( ω ))are rational integers.From (17) it follows that the product ψ ( s ) Y ν =1 ϕ S ν ( α )extended over all classes of level s is a rational integer, which increases as a power of s ; for a primenumber s = p , one has by (18) more precisely(36) p +1 Y ν =1 ϕ P ν ( α ) = p . The construction of the functions ϕ S ( ω ) allows us to determine the divisor generated by ϕ S ( ω ) ;but we only do this for a few special cases here.The determinant of S = P is a prime number p , which firstly does not remain prime in thequadratic field Σ = P ( α ) generated by α , thus becoming the product of two equal or distinctdegree-one prime divisors p and p , and secondly does not divide the conductor of the order R ofΣ for which α is the ideal basis quotient. Let α = α α , where α , α is a basis of an ideal a R of R , and let P p be a matrix of deter-minant p — which is uniquely determined up to equivalence — for which P p α α ! is a basis of a R p R . Then one has (37) ϕ P p ( α ) ≈ p . c R be an integral characteristic ideal of R prime to p , so that c R p R becomes a principal ideal γ R , and let C P p α α ! be a basis of c R p R a R = γ a R . Then | C | = c = N c and N γ = p c . We assume c R is without anyrational part, which is obviously allowed; so C is primitive and we have ϕ CP p ( α ) = c p ∆ C P p α α !! ∆ α α ! = c p ∆ α α ! γ ! ∆ α α ! = c p γ − = γ ≈ c p . On the other hand, it follows directly from the definition of ϕ S ( ω ) that one has the formula ϕ CP p ( ω ) = ϕ C ( P p ( ω )) ϕ P p ( ω ) , whence ϕ CP p ( α ) = ϕ C ( P p ( α )) ϕ P p ( α ) , and since ϕ C ( P p ( α )) as a divisor of a power of c is prime to p , it follows that ϕ P p ( α ) ≈ p . Naturally, corresponding to a matrix P p for which P p α α ! is a basis of a R p R , one has(38) ϕ P p ( α ) ≈ p . Let us now assume that p and p are distinct. Then it follows from (36) that(39) ϕ P ( α ) is a unit for all P of determinant p which is neither equivalent to P p nor to P p . By means of this divisor representation, we prove the following theorem: Let p , p , p and α have the same meaning as above, but suppose p = p . Let f P ( ω ) be anentire integral function from P M P , and moreover let the coefficients of the q -expansion of f (cid:16) p
00 1 (cid:17) ( ω ) be divisible by p . Then the (complex) number f P p ( α ) is algebraic and divisible by p . According to § one has f P ( ω ) Φ ′ p ( ϕ P ( ω ) , j ( ω )) = a ( j ( ω )) + a ( j ( ω )) ϕ P ( ω ) + · · · + a p ( j ( ω )) ϕ P ( ω ) p , Hasse , loc. cit.
123 and 125. a ν ( j ( ω )) of j ( ω ) whose coefficients are rational integers; the coefficients of a ( j ( ω )) are in fact divisible by p . Let us put ω = α here, from which it follows that f P ( α )is algebraic, and if we take P = P p , then by (38) it follows that f P p ( α ) Φ ′ p ( ϕ P p ( α ) , j ( α )) ≡ p , but because Φ ′ p ( ϕ P p ( α ) , j ( α )) = Y P P p (cid:0) ϕ P p ( α ) − ϕ P ( α ) (cid:1) ≡ ( − p Y P P p ϕ P p ( α ) mod p is relatively prime to p by (38) and (39), the assertion follows.
14. Proof of the fundamental congruence (33) It is a matter of showing the congruence j ( k R p − R ) − j ( k R ) p ≡ p for degree-one prime divisors p of norm p not dividing the conductor of R . In the case of p = p ,this follows from (30), because in this case the two possibilities undetermined in (30) obviously saythe same thing. But in the case of p = p , we can apply the theorem just proven, because the ideal a R with the basis α , α can be taken from k R , thus a R p R is an ideal from k R p − R , and with f P ( ω ) = j ( P ( ω )) − j ( ω ) p one now has j ( k R p − R ) − j ( k R ) p = f P p ( α ) . The conjugate functions f P ( ω ) obviously have integer q -coefficients, and they are divisible by p ,as seen from f (cid:16) p
00 1 (cid:17) ( ω ) = j ( pω ) − j ( ω ) p and because j ( pω ) ≡ j ( ω ) p mod p.
15. The isomorphism σ of R R with G Ω R Σ . Let p be a degree-one prime divisor of Σ notdividing the conductor of R , and which is relatively prime to all the differences j ( k R ) − j ( k ′ R ) , k R = k ′ R , or, what amounts to the same thing, to the discriminant D ( H R ) of the class polynomial H R ( X ) . Let P be a prime factor of p in Ω R and let F P be the Frobenius automorphism of P over Σ ; then one has(40) j ( k R ) N p ≡ j ( k R ) F P mod P . Hence by (33) one has j ( k R ) F P ≡ j ( k R p − R ) mod P , and because j ( k R ) F P is one of j ( k ′ R ) — H R ( x ) having rational coefficients —, one has(41) j ( k R p − R ) = j ( k R ) F P . h R be a second class of the order R and let p (1) R p (2) R · · · p ( m ) R be an integral ideal from h R prime to the conductor of R and to D ( H R ) . As degree-two prime divisors p determine principalideals p R of R , the p ( ν ) can be assumed to be of degree-one; then (41) yields j ( k R h − R ) = j ( k R ) F P (1) F P (2) · · · F P ( m ) , that is to say: there exists an automorphism σ ( h R ) of Ω R / Σ with(42) j ( k R ) σ ( h R ) = j ( k R h − R )for all classes k R of R . Also, σ ( h R ) is uniquely determined by (42). On one hand, it followsfrom (42) that σ ( h (1) R ) σ ( h (1) R ) = σ ( h (1) R h (2) R ) , while on the other hand, σ ( h R ) = 1 can hold only for h R = e R : thus σ maps R R isomorphically into G Ω R Σ .In fact one has σ ( R R ) = G Ω R Σ , thus σ is an isomorphism of the class group of R with theGalois group of Ω R / Σ . For that, we first show that for a single invariant j ( h R ) , one already hasΣ( j ( h R )) = Ω R , which also justifies the definition of Ω R in § deviating from § . But that is to say: any automor-phism λ of Ω R / Σ( j ( h R )) is the identity, or, for any class k R , one has(43) j ( k R ) λ = j ( k R ) . If (43) holds for a particular class k R , then it also holds for k R p − R , where p is understood to bea degree-one prime divisor of Σ prime to D ( H R ) and to the conductor of R : one has j ( k R p − R ) ≡ j ( k R ) N p mod p , and so since p λ = p , one has j ( k R p − R ) λ ≡ (cid:0) j ( k R ) λ (cid:1) N p = j ( k R ) N p mod p ,j ( k R p − R ) λ ≡ j ( k R p − R ) mod p . But the (algebraic) number j ( k R p − R ) λ is an invariant j ( k ′ R ) , because H R ( x ) has rational coeffi-cients, so it must be equal to j ( k R p − R ) . From the obviously valid equation j ( h R ) λ = j ( h R ) , wecan get to j ( k R ) λ = j ( k R ) for an arbitrary k R by multiplication with a product of degree-one primedivisors.Since j ( h R ) as a zero of the rational integer polynomial H R ( X ) has degree at most h R over Σ ,it follows that G Ω R Σ = G Σ( j ( h R ))Σ has order at most h R , and from that it follows that σ ( R R ) = G Ω R Σ .At the same time it is proven that the class polynomial H R ( X ) is irreducible over Σ .
6. The Galois group G Ω R P . Obviously Ω R is also normal over P . Let us compute the Galoisgroup G Ω R P . Since one has [ P ( j ( h R )) : P ] = h R , it follows that [ Ω R : P ( j ( h R )) ] = 2 , so G Ω R P ( j ( h R )) is of order 2. Let λ ( h R ) be the automorphism of Ω R / P ( j ( h R )) distinct from the identity. For thatone has(44) j ( h R k R ) λ ( h R ) = j ( h R k − R ) for every k R . For k R = e R , the equation is valid. Hence it suffices to prove that, assuming it is valid for a particularclass k R , it is then so for k R p − R , where again p is of degree-one and is prime to D ( H R ) and to theconductor of R . Now λ ( h R ) not the identity on Σ , whence one has p λ ( h R ) = p , and from j ( h R k R p − R ) ≡ j ( h R k R ) N p mod p and (44), it follows that j ( h R k R p − R ) λ ( h R ) = j ( h R k − R ) N p mod p ;but on the other hand one also has j ( h R k − R ) N p ≡ j ( h R k − R p R ) mod p , whence j ( h R k R p − R ) λ ( h R ) = j ( h R ( k R p − R ) − ) . The Galois group G Ω R P consists of all σ ( k R ) , k R ∈ R R , and all λ ( h R ) σ ( k R ) , k R ∈ R R for fixed h R .The structure of G Ω R P is known if we known the order of λ ( h R ) and understand to which σ ( k ′ R ) a σ ( k R ) is transformed by λ ( h R ) . It follows directly from (44) that(45) λ ( h R ) = 1 , and furthermore one has j ( h R ) λ ( h R ) − σ ( k R ) λ ( h R ) = j ( h R k − R ) λ ( h R ) = j ( h R k R ) = j ( h R ) σ ( k − R )and because λ ( h R ) − σ ( k R ) λ ( h R ) lies in G Ω R Σ , it is one of σ ( k ′ R ) , but on the other hand it is uniquelydetermined by its effect on the primitive element j ( h R ) of Ω R / Σ , so one has(46) λ ( h R ) − σ ( k R ) λ ( h R ) = σ ( k − R ) . The automorphism λ ( h R ) σ ( h R ) − is independent of h R because j ( k R ) · λ ( h R ) σ ( h R ) − = j ( k − R ) ;consequently one has λ ( h ′ R ) = λ ( h R ) σ ( h ′ R h − R ) , in particular λ ( h R ) = λ ( e R ) σ ( h R )and λ ( h R ) λ ( h ′ R ) = σ ( h − R h ′ R ) . The automorphism λ ( e R ) is the transition to the complex conjugate, because one has α · λ ( e R ) = α for α ∈ Σ and j ( k R ) · λ ( e R ) = j ( k − R ) , and j ( k R ) and j ( k − R ) are complex conjugates because, taking α , α to be a basis of an ideal a R from k R , one has α , − α as a basis of the ideal a R from k − R ,and j ( k R ) = j (cid:18) α α (cid:19) and j ( k − R ) = j (cid:18) α − α (cid:19) are thus values of the function j ( ω ) at points whichare mirror-images in the imaginary axis, on which j ( ω ) takes real values. P ( j ( e R )) is the maximal real subfield of Ω R .35
7. The reciprocity law for the ring class field Ω R / Σ . As Ω R is abelian over Σ , the Frobenius automorphisms F P of all the Ω R -prime factors P of a prime divisor p of Σ coincideand henceforth will be denoted by F p . Let us now prove that for the prime divisors p of Σunramified in Ω R , the decomposition assertion of the reciprocity law holds:(47) F p = σ ( e R p R );the proof here accomplishes this for almost all p .If p is of degree-one and is relatively prime to the conductor of R and to D ( H R ) , this followsdirectly from the definition of σ ( k R ) . For the p of degree-two unramified in Ω R , we prove itas follows: since one has e R p R = e R in this case, it must be shown that the decomposition group Z Ω R Σ ( P ) of the Ω R -prime factor P of p over Σ has order 1. As Σ does not lie in the decompositionfield of P over P , it follows on one hand that[ Z Ω R P ( P ) : Z Ω R Σ ( P ) ] = 2 , and on the other hand that a generating element of the cyclic group Z Ω R P ( P ) cannot lie in G Ω R Σ ,and so must be one of λ ( h R ) . But that means[ Z Ω R P ( P ) : 1 ] = 2; [ Z Ω R Σ ( P ) : 1 ] = 1 .
18. The genus field of the order R . The genus field A R of R can be defined as the maximalsubfield of Ω R which is abelian over P . It is the fixed field of the commutator group of G Ω R P ;since the commutator of two σ ( k R ) equals 1, the commutator of λ ( h R ) σ ( k R ) and σ ( k ′ R ) equals σ ( k ′ R − ) and the commutator of λ ( h R ) σ ( k R ) and λ ( h ′ R ) σ ( k ′ R ) equals σ (( k − R k ′ R ) ) , it follows thatthis commutator group is equal to σ ( R R ) , where R R consists of all the square classes k R andstands for the principal genus of R . The kernel of the homomorphism k R → k R from R R onto R R is the group R (0) R of ambiguous classes of R . The index [ R R : R (0) R ] is consequently equal to theorder of R R , and hence the index [ R R : R R ] , which is the number of genus of R (residue classesof R R modulo the principal genus), is equal to the number of ambiguous classes (order of R (0) R ).The genus group R R / R R is (like R (0) R ) of the type (2 , , . . . , a ) . The factor group G Ω R P /σ ( R R )is then of the type (2 , , . . . , a , a +1 ) , which proves that the genus field A R is the composite of a +1 independent quadratic fields , of which one can be taken to be equal to Σ .By (44) one has λ ( k ′ R ) = λ ( k R ) if and only if k R = k ′ R ; thus one has P ( j ( k R )) = P ( j ( k ′ R )) ifand only if the class k R k ′ R − is ambiguous.
19. The correspondence theorem for the ring class field.
Let R and R ′ be orders of animaginary quadratic field Σ , and suppose R ⊇ R ′ . Then one has Ω R ⊆ Ω R ′ , and an automorphism σ ( h R ′ ) of Ω R ′ / Σ induces on Ω R the automorphism σ ( h R ) when the class h R ′ from R ′ belongs tothe class h R from R ; the Galois group G Ω R ′ Ω R thus consists of all σ ( h R ′ ) for the h R ′ belonging tothe principal class of R . The conductor f ′ of R ′ is divisible by the conductor f of R ; let us say f ′ /f = s .Let a class k R of R be given, let k R ′ be a class of R ′ contained in k R . If j ( k R ′ ) = j ( α ) , then by § ,exactly one zero j ( S ( α )) of J S ( X, j ( k R ′ )) = Y S ( X − j ( S ( α ))) is a class invariant of R , and in factequal to j ( k R ) . Thus X − j ( k R ) is the greatest common divisor of J S ( X, j ( k R ′ )) and H R ( X ) ,which proves that j ( k R ) lies in Ω R ′ , and thus one has Ω R ⊆ Ω R ′ . Then σ ( h R ′ ) sends X − j ( k R )to the greatest common divisor of J S ( X, j ( k R ′ ) σ ( h R ′ )) = J S ( X, j ( k R ′ h − R ′ )) and H ( X ) , that is to sayto X − j ( k R h − R ′ ) : j ( k R ) σ ( h R ′ ) = j ( k R h − R ′ ) = j ( k R h − R ) = j ( k R ) σ ( h R ) , q.e.d.
20. The singular values of functions from P M S . The fact that a function f S ( ω ) from P M S is a rational function of j ( ω ) and j ( S ( ω )) with rational coefficients does not allow one to concludethat the function value f S ( α ) for a (complex) number α lies in the field P ( j ( α ) , j ( S ( α ))) . For anentire function f S ( ω ) , the product f S ( ω ) J ′ S ( j ( S ( ω )) , j ( ω )) is a rational integer polynomial of j ( ω )and j ( S ( ω )) , and from that it follows that f S ( α ) ∈ P ( j ( α ) , j ( S ( α ))) if j ( S ( α )) is a simple zero of J S ( X, j ( α )) . But for imaginary quadratic α that is in general not the case.Let us assume that among the conjugates j ( S ν ( ω )) , ν = 1 , , . . . , ψ ( s ) , of j ( S ( ω )) over P M S , exactly r of them, say j ( S ( ω )) , . . . , j ( S r ( ω )) , take on the value j ( S ( α ))for ω = α , so that S is equivalent to one of the matrices S , . . . , S r . As we are going to show, f S ( α ) is then algebraic over P ( j ( α ) , j ( S ( α ))) and the conjugates of f S ( α ) over this field are among f S ( α ) , . . . , f S r ( α ) . For the proof, let us denote by G the Galois group of the normal closure K = P ( j ( ω ) , j ( S ( ω )) , . . . , j ( S ψ ( s ) ( ω ))) of P M S over P M , and by H the subgroup of G of theautomorphisms which permute the functions j ( S ( ω )) , . . . , j ( S r ( ω ))among one another, and by K the fixed field of H in K . The polynomial F ( X, ω ) = r Y ν =1 ( X − j ( S ν ( ω )))is sent to itself by every σ ∈ H ; in contrast, for σ ∈ G , σ / ∈ H , not only is F ( X, ω ) σ = F ( X, ω ) ,but in fact F ( X, α ) σ = F ( X, α ) , whence for suitable rational a , ℓ ( ω ) = F ( a, ω )is a primitive element of K / P M with the following property: writing L ( X, j ( ω )) for the irreduciblepolynomial over P M with L ( ℓ ( ω ) , j ( ω )) = 0 , one has ℓ ( α ) as a simple zero of L ( X, j ( α )) . For anentire function g ( ω ) from K , the product g ( ω ) L ′ ( ℓ ( ω ) , j ( ω )) is a rational integer polynomial of Hasse , Monatsh. f. Math. u. Phys. , p. 325 (1931). ( ω ) and ℓ ( ω ) , whence g ( α ) is an element of P ( j ( α ) , ℓ ( α )) , and thus of P ( j ( α ) , j ( S ( α ))) because ℓ ( α ) = ( a − j ( S ( α ))) r .For an entire function f S ( ω ) from P M S , the polynomial r Y ν =1 ( X − f S ν ( ω )) has entire functionsfrom K as coefficients; thus r Y ν =1 ( X − f S ν ( α )) has coefficients in P ( j ( α ) , j ( S ( α ))) , from which infact it follows that f S ( α ) is algebraic over P ( j ( α ) , j ( S ( α ))) and that its conjugates over this fieldare among f S ( α ) , . . . , f S r ( α ) .In a special case, it can also be deduced that f S ( α ) ∈ P ( j ( α ) , j ( S ( α ))) when r > Let α , α be a basis of a characteristic ideal a R of the order R of Σ and let S α α ! belikewise a basis of a characteristic ideal a R b R of R . Set α α = α . The entire function f S ( ω ) from P M S has the property that a suitable power f S ( ω ) n , n = 1 , , . . . of it is represented in the form f S ( ω ) n = H S ω ω ! ! H ω ω ! with two modular forms H , H of the same weight t distinct from 0, and furthermore H ( α ) = 0 .Thus f S ( α ) lies in the field Ω R = Σ( j ( α )) . Proof. From j ( S ν ( α )) = j ( S ( α )) , ν = 1 , , . . . , r , it follows that there exist matrices M ν ∈ M with S ( α ) = M S ( α ) = · · · = M r S r ( α ) . More precisely, one has S α α ! ξ ν = M ν S ν α α ! , ν = 1 , , . . . , r with elements ξ ν = 0 from Σ . These ξ ν are congruent to rational integers modulo the conductor f of R . To see that, let us take an integral characteristic ideal c R of R so that a R c R becomes aprincipal ideal a R c R = γ R . Let ̺, f ̺, R . Hence one has ̺ f γγ ! = C α α ! with a rational integer matrix C , and it follows that ̺ f ξ ν ξ ν ! = C S − M ν S ν C − ̺ f ! . Hasse , J. f. Math. , 82 to 83 (1931). There, a special function f S ( α ) is investigated and R is the principalorder. | S | = N b and | C | = N c are prime to f , the entries of the matrix C S − M ν S ν C − are(rational) numbers with denominators prime to f , so that the second row of the last equation readsas ξ ν = A f ̺ + B with f -integral rationals A, B ; consequently one has ξ ν ≡ B mod f. The matrices S , . . . , S r equivalent to S are thereby characterized by ξ ν being a unit. Since S − M ν S ν α α ! = α α ! ξ ν ,S − M ν S ν is the representing matrix of ξ ν with respect to the basis α , α , and hence it followsthat one has S − M ν S ν ∈ M , i.e. S is equivalent to S ν if and only if ξ ν is a unit in R ; but when ξ ν is a unit, it certainly lies in R , because it is congruent to a rational number modulo f .Let σ be an automorphism of the normal field of f S ( α ) over P ( j ( α ) , j ( S ( α ))) = Ω R ;thus f S ( α ) σ = f S ν ( α ) for a suitable ν = 1 , , . . . , r . One has f S ( α ) n σ = f S ν ( α ) n = H S ν α α ! ! H α α ! = H M − ν S α α ! ξ ν ! H α α ! = ξ tν f S ( α ) n . If σ has order N , then one has ξ tNν = 1 , and as t = 0 , it follows that ξ ν is a root of unity, whence S ν is equivalent to S , so f S ( α ) σ = f S ν ( α ) = f S ( α ) , but that means that f S ( α ) lies in Ω R .
21. The principal ideal theorem for imaginary quadratic number fields.
We will need thefunctions ϕ S ( ω ) = | S | ∆ S ω ω ! ! ∆ ω ω ! for arbitrary rational matrices with | S | > S is uniquelyrepresentable as S = r S with primitive S and a positive rational number r , and one has ϕ S ( ω ) = r ϕ S ( ω ) . Accordingly, we set p ϕ S ( ω ) = √ r p ϕ S ( ω ) with √ r > . α , α be a basis of a characteristic ideal a R of the order R of Σ and let S α α ! be abasis of a characteristic ideal a R s R of R . As in § it will be proven that the divisor of ϕ S ( α ) isequal to s .From the theorem proven in § , it follows that ϕ S ( α ) belongs to the field Ω R , and that bymeans of § , the equation for p ϕ S ( α ) holds if additionally it is assumed that | S | is the square ofa rational number prime to 6. From that we conclude that Let b be a given divisor of Σ prime to the conductor f of R and to 6. Let α , α be a basisof a characteristic ideal a R of R and let the rational-entry matrix B transform α , α to a basisof a R b − R . Set α = α α ; then p ϕ B ( α ) is an element of the field Ω R with the divisor b . If the principal order of Σ is taken for R ( f = 1 ), this is an explicit version of the principalideal theorem for the field Σ , for then Ω R is the absolute class field of Σ .
22. The singular values of the functions ϕ S ( ω ) . Second part. The divisors of the singularvalues ϕ S ( α ) can be completely determined. First a preliminary remark. Let α , α be a basisof a characteristic ideal a R f of the order R f with conductor f ; let S and S ′ be two (primitive)matrices of determinant s such that S α α ! and S ′ α α ! are bases of characteristic ideals b R fs and b ′ R fs of the order R fs with conductor f s . Then the two (algebraic) numbers ϕ S ( α ) and ϕ S ′ ( α ) are associated (where α = α α ). Proof. In the class of the ideal b ′ R fs b − R fs , let c R fs be an integral ideal which is prime to f s andto a given prime number p ; then c R fs is canonically associated to a divisor c prime to f sp . Onehas(48) b ′ R fs γ = b R fs c R fs . By § it follows by multiplication with R f that a R f γ = a R f c R f ,γ R f = c R f , and by multiplication with R that γ R = c R ;the divisor of γ is thus c . From (48) it further follows that S ′ α α ! γ = C S α α ! cf. Fricke , p. 361–362, also
Hasse , Monatsh. f. Math. u. Phys. , 315–322 (1931). C whose determinant | C | = [ b R fs : b R fs c R fs ]is equal to the norm N c = N γ . Hence one has ϕ S ′ ( α ) ϕ S ( α ) = ∆ S ′ α α ! ! ∆ S α α ! ! = ∆ C S α α ! γ − ! ∆ S α α ! ! = γ ∆ C S α α ! ! ∆ S α α ! ! = γ N c − ϕ C ( S ( α )) = γ − ϕ C ( S ( α )) , and as ϕ C ( S ( α )) is a factor of a power of N c = N γ by (17), it follows that ϕ S ′ ( α ) ϕ S ( α ) is prime tothe arbitrarily specified prime number p , whence a unit.Since a primitive matrix S of determinant | S | can be put in the form S = P P · · · · · P r , where each | P ν | is a prime number, by the formula ϕ AB ( α ) = ϕ A ( B ( α )) ϕ B ( α ) , in order to computethe divisor of a singular value ϕ S ( α ) , it suffices to know the divisors of the ϕ P ( α ) where | P | is aprime number. Now let p be a prime number, let R f be the order with conductor f , and let p t be the powerof p contained in f . Let P denote the primitive matrix of determinant p . Let α , α be a basisof a characteristic ideal a R f of R f . Then the divisors of the singular value ϕ P ( α ) are given asfollows:
1. If p is the product of two distinct prime divisors p and p in Σ :1.1. ϕ P ( α ) is a unit when P α α ! is a basis of a characteristic ideal of R fp ,1.2. ϕ P ( α ) ≈ p when P α α ! is a basis of a characteristic ideal of R fp − ,1.3. ϕ P p ( α ) ≈ p and ϕ P p ( α ) ≈ p when p ∤ f ( t = 0 ) and P p α α ! is a basis of a R f p R f and P p α α ! is a basis of a R f p R f .2. If p is the square of a prime divisor p in Σ :2.1. ϕ P ( α ) ≈ p p t +1 when P α α ! is a basis of a characteristic ideal of R fp . Essentially in
Hasse , Monatsh. f. Math. , p. 331 (1931). .2. ϕ P ( α ) ≈ p − p t when P α α ! is a basis of a characteristic ideal of R fp − .2.3. ϕ P p ( α ) ≈ p when P p α α ! is a basis of a R f p R f .3. If p is prime in Σ :3.1. ϕ P ( α ) ≈ p p t ( p +1) when P α α ! is a basis of a characteristic ideal of R fp .3.2. ϕ P ( α ) ≈ p h − p t − ( p +1) i when P α α ! is a basis of a characteristic ideal of R fp − . We prove this by induction on t . For t = 0 the assertions of 1.3 and 2.3 are proven in § ;for 3.1 the assertion follows immediately from the preliminary remark, for then all α α ! arebases of characteristic ideals in R fp , thus letting P , . . . , P p +1 be a system of representatives of theequivalence classes of matrices of determinant p , one has ϕ P ν ( α ) p +1 ≈ p +1 Y ν =1 ϕ P ν ( α ) = p , and we conclude just as in the cases of 1.1 and 2.1.Now let t > § there exists exactly one of the representatives P ν , say P , for which P α α ! is a basis of a characteristic ideal of R fp − . Since P ν α α ! for the remaining P ν arebases of characteristic ideals of R fp , it follows that one has p = p +1 Y ν =1 ϕ P ν ( α ) ≈ ϕ P ( α ) ϕ P ν ( α ) p , ν > , so it suffices to determine the divisor of ϕ P ( α ) . Now one has ϕ p P − ( P ( α )) ϕ P ( α ) = ϕ p E ( α ) = p , and P α α ! is a basis of a characteristic ideal of R fp − , which is transformed by the matrix p P − of determinant p to the basis α , α of the characteristic ideal a R f of R f , and hence by theinduction hypothesis one has ϕ p P − ( α ) ≈ p p t − ( p +1) and ϕ P ( α ) ≈ p h − p t − ( p +1) i , q.e.d.
23. Congruences for the singular values of functions from P M S . The theorem provenin § on the divisibility of the singular value of functions from P M P , | P | = p prime, by a Σ -primefactor p of p , is for the case p = p p , p = p . Let us now establish a similar theorem for the casesof p = p and p = p put aside. 42et p be a prime number, and let P , . . . , P p +1 be a system of representatives of the equivalenceclasses of matrices of determinant p ; let P stand for an arbitrary matrix of determinant p . Let f P ( ω ) be an entire integral function from P M P . By § the principal polynomial F ( X, j ( ω )) = p +1 Y ν =1 ( X − f P ν ( ω )) = p X µ =0 Q µ ( j ( ω )) X µ of f P ( ω ) over P M has rational integer coefficients as a polynomial in X and j ( ω ) ; thus f P ( α ) isan integer for imaginary quadratic α . The entire integral function f P ( ω ) from P M P has the property that the q -coefficients of theconjugate f (cid:16) p
00 1 (cid:17) ( ω ) are divisible by p . Let α be imaginary quadratic, P ( α ) = Σ , and let α = α α where α , α is a basis of a characteristic ideal a R f of the order R f with conductor f in Σ . Let p be prime or ramified in Σ ; thus p = p or p = p where p is a prime divisor of Σ . Let P be aprime factor of p in the number field Λ P = P ( j ( α ) , f P ( α )) . Then in the following cases, f P ( α ) isdivisible by P :1. p = p , f p , P arbitrary.2. p = p , f p , P α α ! is a basis of a R f p R f .3. f ≡ p , P α α ! is a basis of a characteristic ideal of R fp − .But we also make these two assumptions on p : p > and p is unramified in Ω R f where f = f p t with f p . Proof. By § one has the congruence in the q -coefficients F ( X, j ( ω )) ≡ (cid:0) X − f (cid:16) p
00 1 (cid:17) ( ω ) (cid:1) (cid:0) X p − f (cid:16) p (cid:17) ( ω ) p (cid:1) mod (1 − ζ p );thus by the assumption on f (cid:16) p
00 1 (cid:17) ( ω ) , one even has F ( X, j ( ω )) ≡ X (cid:0) X p − f (cid:16) p (cid:17) ( ω ) p (cid:1) mod (1 − ζ ) , and from that, as in § , it follows that one has the congruence F ( X, j ( ω )) ≡ X ( X p − Q ( j ( ω ))) mod p for F as a polynomial in X and j ( ω ) , and thus also(49) F ( X, j ( α )) ≡ X ( X p − Q ( j ( α ))) mod p. One has P ( j ( α )) = Ω R f , Λ P = Ω R f ( f P ( α )) and Λ ∗ = Ω R f ( f P ( α ) , . . . , f P p +1 ( α )) is an extensionfield of Λ normal over Ω R f . Let P ∗ be a prime factor of P in Λ ∗ . From (49) it follows that at43east one of the p +1 elements f P ν ( α ) is divisible by P ∗ . Let us say f P ( α ) ≡ P ∗ ; thenone has f P ( α ) p ≡ Q ( j ( α )) mod P ∗ for ν > f P ( α ) is a multiple zero of F ( X, j ( α )) , then one has Q ( j ( α )) ≡ P ∗ , f P ν ( α ) ≡ P ∗ for all ν and in particular f P ( α ) ≡ P .Thus let us assume that f P ( α ) is a simple zero of F ( X, j ( α )) . Then f P ( ω ) is all the morea simple zero of F ( X, j ( ω )) and consequently f P ( ω ) is a generating element of P M P over P M .From this we conclude as in § that F ′ (cid:0) f P ( ω ) , j ( ω ) (cid:1) ϕ P ( ω ) = p X µ =0 b µ ( j ( ω )) f P ( ω ) µ with polynomials b µ , whose coefficients are rational numbers (even rational integers). As one has F ′ ( f P ( α ) , j ( α )) = 0 , the (algebraic) number ϕ P ( α ) lies in the field Λ P .Let us first consider the case p = p , f p . Then one has ϕ P ( α ) ≈ p p +1 and because p is unramified in Ω R f and p >
12 , it follows that P is ramified over Ω R f . The inertia group of P ∗ over Ω R f is consequently not contained in the Galois group of Λ ∗ / Λ P , and so there exists aninertia automorphism λ of P ∗ / Ω R f with f P ( α ) λ = f P ( α ) .Thus f P ( α ) λ = f P ν ( α ) for a suitable ν > P ∗ , Q ( j ( α )) ≡ P ∗ , f P ν ( α ) ≡ P ∗ for all ν and in particular f P ( α ) ≡ P .In the two other cases, p = p but f ≡ p and p = p , the matrix P to be examined isequivalent to a certain P ν . If P ∼ P , then the assertion holds. If P is not equivalent to P , then P α α ! is a basis of a characteristic ideal of R fp and hence ϕ P ( α ) ≈ p p t ( p +1) in the case p = p and ϕ P ( α ) ≈ p p t +1 in the case p = p . But the ramification index of p in Ω R f is a factor of thedegree [ Ω R f : Ω R f ] = h R f / h R f = e R f e R f ( p t − ( p +1) , p = p p t , p = p (because p is unramified in Ω R f ). In the course of this we have used the correspondence theoremfrom § . We have to assume the formula for h R f /h R f as well-known. Since p >
12 , it followsthat P is ramified over Ω R f and as above we conclude that f P ν ( α ) ≡ P ∗ for every ν , inparticular f P ( α ) ≡ P . 44 . The second main theorem
24. The singular elliptic functions.
Let K ( w ) denote the field of elliptic functions with theperiod lattice w . The field K ( w ) is called singular if w = a R is an ideal in the order R of animaginary quadratic number field Σ .Every unit ε of R generates an automorphism ̺ ε of the singular field K ( a R ) : For a function f ( z ) from K ( a R ) one sets f ( z ) ̺ ε = f ( ε z ) . The map ε → ̺ ε is an isomorphism from the unitgroup of R onto an automorphism group of order e (number of units of R ) of K ( a R ) , becauseone has ̺ ε · ̺ ε = ̺ ε ε and since ℘ ( z, a R ) ℘ ′ ( z, a R ) − · ̺ ε = ε ℘ ( z, a R ) ℘ ′ ( z, a R ) − one has ̺ ε = 1 only for ε = 1 .The singular Weber function τ R ( z, a R ) is invariant under the ̺ ε , thus also under the transfor-mations z → ε z + α, ε unit in R , α in a R . In the group of these transformations, the group of the translations z → z + α, α in a R has index e , and since τ R ( z, a R ) as an elliptic function has order e , one has τ R ( z ′ , a R ) = τ R ( z, a R ) if and only if one has z ′ ≡ ε z mod a R , where ε is a unit in R . The field K ( a R ) has degree e over the field of rational functions in τ R ( z, a R ) , which we will denote by K ( a R ) , and hence K ( a R ) is the fixed field of the ̺ ε . In the power series ℘ ( z, w ) = ∞ X ν = − a ν ( w ) z ν the coefficient a ν ( w ) is obviously an entire modular form of weight − ν , and as a ν ( ω ) is knownto be a polynomial in g and g with rational coefficients, the Fourier coefficients of a ν ( w ) arerational. It follows that τ R ( z, w ) = ∞ X ν = − e b ν ( w ) z ν , where b ν ( w ) is an entire modular form of weight − ν with rational Fourier coefficients. The singular function τ R ( z, a R ) depends only on z e , i.e. one has τ R ( z, a R ) = ∞ X ν = − b eν ( a R ) z eν . We introduce u = z e g ( e ) ( a R ) − as a new variable: τ R ( z, a R ) = ∞ X ν = − b eν ( a R ) g ( e ) ( a R ) ν u ν . b eν ( w ) g ( e ) ( w ) ν is an entire modular function with rational Fourier coefficients, thus a polyno-mial in j ( w ) with rational coefficients, so b eν ( a R ) g ( e ) ( a R ) ν lies in Ω R . We thus have (50) τ R ( z, a R ) = ∞ X ν = − η ν u ν , η ν in Ω R , u = z e g ( e ) ( a R ) − .
25. The singular values of the Weber function. Ray class invariants.
In the following, letΣ be a fixed imaginary quadratic number field. By a singular value of the Weber function τ R ( z, w ) associated to an order R of Σ , we mean a function value τ R ( γ, a R ) , where γ is an element of Σand a R is a characteristic ideal of R . For this, we assume that γ a R , so that one has τ R ( γ, a R ) = ∞ . The singular values τ R ( γ, a R ) are algebraic numbers, because taking α , α tobe a basis of a R , one has γ = N − ( x α + x α ) with rational integers N, x , x , ( N, x , x ) = 1 ,and hence τ R ( γ, a R ) is a zero of the division polynomial T N ( X, j ( α )) , whose coefficients lie in Ω R .The denominator (-divisor) of τ R (1 , a R ) contains at most prime factors of N and τ R (1 , a R ) is evenintegral when N is not a prime power.The main result on the singular values of τ R ( z, w ) is: the number field Ω R ( τ R ( γ, a R )) is theclass field of Σ for a divisor class group determined in a simple way by a R and γ . We will provethis only for the case when R is the principal order of Σ (second main theorem); the general casecan be treated quite similarly, but it requires some complicated considerations from the ideal theoryof non-principal orders, which unnecessarily encumbers the simple train of thought.Thus let R from now on be the principal order of Σ . Aside from that, only the orders R p withprime conductor p play a role.Let m be an integral divisor of Σ distinct from 1 . The singular value τ R ( γ, a R ) is called an m -th division value of a if one has m γ ≡ a , i.e. if γ ≡ a m − . The value τ R ( γ, a R )is called a characteristic m -th division value of a or a division value of order m of a if m isthe smallest (in the sense of divisibility) integral divisor with m γ ≡ a ; in other words: if( γ ) = a r m − with integral r prime to m , or also: if m is the denominator of the divisors γ a − .The number N in the representation x = N − ( x α + x α )with relatively prime rational integers x , x , N , N > m of τ R ( γ, a R ) ; since N m | N and N | N m , it follows that the denominator of τ R ( γ, a R ) is at most divisible by the prime factors of N m , and τ R ( γ, a R ) is even integral when N m is not a prime power.The value τ R ( γ, a R ) depends only on γ modulo a R , and one has τ R ( γ, a R ) = τ R ( γ ′ , a R )if and only if one has γ ′ ≡ γ ε mod a R with a unit ε of R .46f τ R ( γ, a R ) is the m -th division value of a R , then one has ( γ ) = a m − r with integral r , and r − lies in the (absolute) class of the divisor a m − . Then one has τ R ( γ, a R ) = τ R (1 , m R r − R ) . Conversely, if r is an integral divisor, with r − in the class of a m − , then one has τ R (1 , m R r − R ) = τ R ( γ, a R )with ( γ ) = a m − r , so τ R (1 , m R r − R ) is an m -th division value of a R . We will henceforth regard τ R (1 , m R r − R ) also as an m -th division value of the divisor class k of a , i.e. of m r − . The value τ R (1 , m R r − R ) is a characteristic m -th division value of the class k of m r − (i.e. of m r − ) if and only if r is prime to m . The value τ R (1 , m R r − R ) depends only on the ray class k m to which r − belongs. (The rayclasses of Σ modulo m will in general be denoted by k m , k ′ m , k ( ν ) m , h m , . . . .) This is because if r and r ′ are equivalent modulo m , then r ′ = λ r with λ ≡ ∗ m , and since the denominator of λ divides r , it follows that λ ≡ m r − , thus τ R (1 , m R r − R ) = τ R ( λ, m R r − R ) = τ R (1 , ( m R r − R ) λ − ) = τ R (1 , m R r ′ R − ) . Henceforth, when r − lies in the ray class k m modulo m , we will call τ R (1 , m R r − R ) the invariantof the ray class k m and denote it by τ R ( k m ) , analogous to the class invariant j ( k R ) .But unlike the j ( k R ) , the invariant τ R ( k m ) does not necessarily determine the ray class k m ;rather, one only has the much weaker statement: τ R ( k m ) = τ R ( k ′ m ) implies k m = k ′ m if k m and k ′ m belong to the same absolute class. For then one has τ R ( k m ) = τ R (1 , m R r − R ) and τ R ( k ′ m ) = τ R (1 , m R r ′ R − ) with integral r , r ′ primeto m and r ′ = λ r with λ ∈ Σ ; thus τ R ( k ′ m ) = τ R (1 , m R r − R λ − ) = τ ( λ, m R r − R ) , consequently one has λ ≡ ε mod m r − with a unit ε of R , and since r is prime to m , it followsthat λ ≡ ε mod ∗ m , thus r ′ and r are equivalent modulo m , which is to say k m = k ′ m .
26. The second main theorem.
The field Ω R ( τ R ( k m )) is the ray class field of Σ modulo m . Again we first give a proof by means of the general theory of abelian number fields and then aproof which, except for the elementary parts of number theory, relies only on the theory of modularfunctions and elliptic functions.The first proof, quite similar to that in § III.2 , is based on a congruence for the ray class invariants,namely: α ≡ β mod ∗ m means multiplicative congruence. Hasse , loc. cit. p. 134. et p be a degree-one prime divisor of Σ not dividing N m , which is distinct from its conjugate p , and let p be the prime number divisible by p . Then one has (51) τ R ( k m p − ) ≡ τ R ( k m ) p mod P for every prime factor P of p in Ω R ( τ R ( k m ) , τ R ( k m p − )) . Proof. We write τ R ( k m ) in the form τ R ( k m ) = τ R (1 , m R r − R ) = τ R (cid:16) N − ( x , x ) α α ! , α α ! (cid:17) , in which α , α is a basis of a R = m R r − R and 1 = N − ( x α + x α ) with relatively prime rationalintegers x , x , N ; here N contains the same prime factors as N m and is hence not divisible by p .The matrix P p of determinant p transforms α , α to a basis P p α α ! of a R p R . Then one has τ R ( k m p − ) = τ R (1 , m R r R p − R ) = τ R ( p, m R r − R p R ) = τ R (cid:16) p N − ( x , x ) , P p α α ! (cid:17) and thus τ R ( k m ) p − τ R ( k m p − ) = δ P p ( ( x , x ) ; α ) , (cid:18) α = α α (cid:19) . Here δ P p ( ( x , x ) ; α ) is a zero of the polynomial S P p ( X, α ) = X ν D ( ν ) P p ( α ) X ν , whose coefficients D ( ν ) P p ( α ) are divisible by p by § I.7 and § III.5 . From this the assertion follows.The proof of the second main theorem now goes quite like the proof of the first main theoremin § III.2 . If p is a degree-one prime divisor of Σ in the ray modulo m which does not divide the discrim-inant of Σ , then by (51), for every ray class k m and the absolute divisor class k R belonging to k m ,one has τ R ( k m ) p ≡ τ R ( k m ) , j ( k R ) p ≡ j ( k R ) mod P for every prime factor P of p in Ω R ( τ R ( k m )) . If now it is also assumed that p is prime to thediscriminant of j ( k R ) and τ R ( k m ) over Σ , it follows that for all p -integral α from Ω R ( τ R ( k m )) ,one has the congruence α p ≡ α mod P . Thus P is of degree-one, and p splits completely in Ω R ( τ R ( k m )) . Conversely we suppose that p splits into prime factors P of degree-one in Ω R ( τ R ( k m )) . Then one has j ( k R ) p ≡ j ( k R ) mod P ; Hasse , loc. cit. p. 137 to 138. j ( k R ) p ≡ j ( k R p − ) mod P , thus j ( k R p − ) ≡ j ( k R ) mod P , and if we assume p is prime to the differences j ( k R ) − j ( k ′ R ) , k R = k ′ R , then j ( k R p R ) = j ( k R ) , so k R p R = k R , and p is a principal divisor. Furthermore one has τ R ( k m ) p ≡ τ R ( k m ) mod P ;since τ R ( k m ) p ≡ τ R ( k m p − ) mod P , thus τ R ( k m p − ) ≡ τ R ( k m ) mod P . As k m and k m p − belong to the same absolute class k R , it follows that if we still assume p to beprime to the (non-0) difference τ R ( k m ) − τ R ( k ′ m ) , k m = k ′ m , k m and k ′ m in the same absolute class , then τ R ( k m p − ) = τ R ( k m ) , so k m p = k m , and p lies in the ray modulo m .
27. Proof of the second main theorem not relying on general class field theory.
Theproof of the second main theorem not relying on general class field theory also runs parallel to thecorresponding proof of the first main theorem. We setΩ m = Ω R ( τ R ( k (1) m ) , . . . , τ R ( k ( h m ) m ) )and prove that Ω m is the ray class field of Σ modulo m , by showing that the Artin reciprocity lawholds for Ω m (as class field of Σ for the ray modulo m ), thereby also gettingΩ m = Ω R ( τ R ( k m )) = Σ( j ( k R ) , τ R ( k m )) . The polynomial with roots τ R ( k m ) , thus S m ( X ) = Y k m ( X − τ R ( k m )) , is called the ray class polynomial of Σ modulo m . If k R is a divisor class of Σ , then S m , k R ( X ) = Y k m in k R ( X − τ R ( k m ))49hall be called the ray class polynomial of Σ modulo m over the class k R ; one has S m ( X ) = Y k R S m , k R ( X ) . The coefficients of S m , k R ( X ) lie in Ω R . This fact plays the same role in the proof of the secondmain theorem as the fact that the class polynomial H R ( X ) has rational coefficients plays in theproof of the first main theorem; in order to not interrupt the train of thoughts, we postpone theproof of this fact to the end of our deliberations.
28. The fundamental congruence.
While we got by for the proof of the first main theoremwith the congruence j ( k R p − ) ≡ j ( k R ) p mod p which is valid for prime divisors p of degree one , the corresponding congruence τ R ( k m p − ) ≡ τ R ( k m ) p mod p is sufficient for the proof of the second main theorem only if we want to make use of the fact that ineach ray class modulo m there exist divisors which are composed of only prime factors of degree one,which is a fact that can only be seen by means of analytic methods or by the use of essential toolsfrom general class field theory. This prompts us to generalize the congruence (51) as follows: Let p be a prime divisor in Σ , in which the prime number p divisible by p is larger than 12.Suppose p is not a factor of N m . Then one has (52) τ R ( k m p − ) ≡ τ R ( k m ) N p mod P for every prime factor P of p in Ω m . Proof. If p = p p , p = p , that is the assertion proven in § (even without the assumption p >
12 ). In the case of p = p , the proof in § can be taken word-for-word, except instead ofthe theorem of § one has to use the theorem of § . There remains the case of p = p . Weagain set τ R ( k m ) = τ R (1 , m R r − R ) by means of a basis α , α of m R r − R and the representation1 = N − ( x α + x α ) , ( x , x , N ) = 1 , p ∤ N in the form τ R ( k m ) = τ R (cid:16) N − ( x , x ) α α ! , α α ! (cid:17) . Let P denote an arbitrary (but chosen fixed) matrix of determinant p . Then τ R ( k m ) p − τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) = τ R (cid:16) N − ( x , x ) α α ! , α α ! (cid:17) p − τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) = δ P (cid:0) ( x , x ) ; α (cid:1) , ( α = α α )50s a zero of S P ( X, α ) = X ν D ( ν ) P ( α ) X ν , whose coefficients D ( ν ) P ( α ) are divisible by p by § I.7 and § III.15 , and hence for every prime factor P ∗ of p in Ω m τ R (cid:16) p N − ( x , x ) α α ! , P α α !(cid:17)! one has the congruence(53) τ R ( k m ) p ≡ τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) mod P ∗ . Now P ′ = p P − is a matrix of determinant p . We set P α α ! = α ′ α ′ ! , so P ′ α ′ α ′ ! = α α ! p ;furthermore ( x , x ) P ′ = ( x ′ , x ′ ) , so p ( x , x ) = ( x ′ , x ′ ) P , and one obviously has ( x ′ , x ′ , N ) = 1 .We compute τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) in terms of ( α ′ , α ′ ) and ( x ′ , x ′ ) as: τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) = τ R (cid:16) N − ( x ′ , x ′ ) α ′ α ′ ! , α ′ α ′ ! (cid:17) . On the other hand, one has τ R ( k m p − ) = τ R ( k m p − ) = τ R ( 1 , m R r − R p − ) = τ R ( p , m R r − R p )= τ R (cid:16) p N − ( x , x ) α α ! , α α ! p (cid:17) = τ R (cid:16) p N − ( x ′ , x ′ ) α ′ α ′ ! , P ′ α ′ α ′ ! (cid:17) , thus τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) p − τ R ( k m p − ) = δ P ′ (cid:0) ( x ′ , x ′ ) ; α ′ (cid:1) , with α ′ = α ′ α ′ . As above this now yields τ R (cid:16) p N − ( x , x ) α α ! , P α α ! (cid:17) p ≡ τ R ( k m p − ) mod P ∗ ;together with (53), one has τ R ( k m ) p ≡ τ R ( k m p − ) mod P , and since N p = p , that is the assertion.It should be noted that the congruence j ( k R p − ) ≡ j ( k R ) N p mod P for all prime factors P of p in Ω R can also be proven in this way for degree-two prime divisors(under the assumption p >
12 ); but of course it follows directly from the Artin reciprocity law forΩ R / Σ proven in § III.9 . 51
9. The generation of the ray class field by a single ray class invariant.
We now prove,with the reasoning already used in § III.7 , that one has Ω m = Ω R ( τ R ( k m )) . In any case Ω m is normalover Ω R , because the coefficients of the polynomial S m , k R ( X ) lie in Ω R . Let λ be an automorphismof Ω m / Ω R ( τ R ( k m )) (where k m is a fixed ray class). We have to show that for all ray classes h m , onehas τ R ( h m ) λ = τ R ( h m ) . This is true for a certain ray class h m . If p is a prime divisor of Σ not dividing N m , unramifiedin Ω m , one has τ R ( h m p − ) ≡ τ R ( h m ) N p mod p ;and since p λ = p , thus τ R ( h m p − ) λ ≡ ( τ R ( h m ) λ ) N p = τ R ( h m ) N p mod p , and so τ R ( h m p − ) λ ≡ τ R ( h m p − ) mod p . But since Ω R remains elementwise fixed under λ , τ R ( h m p − ) λ is also a zero of S m , h R p − ( X ) justas τ R ( h m p − ) is, where h R denotes the absolute class to which h m belongs. As it is assumed that p is prime to the discriminant of S m , h R p ( X ) , it follows that τ R ( h m p − ) λ = τ R ( h m p − ) . From τ R ( k m ) λ = τ R ( k m ) we can now go step-by-step to τ R ( h m ) λ = τ R ( h m ) for an arbitrary h m .
30. The reciprocity law for the ray class field.
We still do not know whether Ω m is normalover Σ . Let Ω ∗ be an extension field of Ω m normal over Σ . Let p be a prime divisor of Σ notdividing N m , let P ∗ be a prime factor of p in Ω ∗ and let F P ∗ be the Frobenius automorphismof P ∗ / Σ . Then one has j ( k R ) N p ≡ j ( k R ) F P ∗ , τ R ( k m ) N p ≡ τ R ( k m ) F P ∗ mod P ∗ , and since j ( k R ) N p ≡ j ( k R p − R ) , τ R ( k m ) N p ≡ τ R ( k m p − ) mod P ∗ , it follows that j ( k R ) F P ∗ ≡ j ( k R p − R ) , τ R ( h m ) F P ∗ ≡ τ R ( h m p − ) mod P ∗ for all absolute classes k R and all ray classes k m . We now assume that p is prime to the discriminantof H R ( X ) and to all the differences of the (algebraic) numbers which are conjugate over Σ to theray class invariants τ R ( k m ) , as long as they are distinct from 0; that excludes only finitely many p .Then it follows that j ( k R ) F P ∗ = j ( k R p − R ) , τ R ( k m ) F P ∗ = τ R ( k m p − ) . Let h m be a ray class, let h R be the absolute class to which h m belongs, let p p · · · p m be anintegral divisor in h m with none of its prime factors p ν occuring among the excluded prime divisors,and let P ∗ ν be a prime factor of p ν in Ω ∗ . The automorphism σ m ( h m ) = F P ∗ F P ∗ · · · F P ∗ m
52f Ω ∗ / Σ satisfies the equations(54) j ( k R ) σ m ( h m ) = j ( k R h − R ) , τ R ( k m ) σ m ( h m ) = τ R ( k m h − m ) . The automorphism σ ( h m ) is not uniquely determined by these; rather, the most general automor-phism λ of Ω ∗ / Σ with j ( k R ) λ = j ( k R h − R ) , τ R ( k m ) λ = τ R ( k m h m )is of the form λ = ̺ σ m ( h m )with an arbitrary ̺ from the Galois group of Ω ∗ / Ω m . One has ̺ σ m ( h m ) = ̺ ′ σ m ( h ′ m )if and only if one has ̺ = ̺ ′ and h m = h ′ m . This is because from ̺ σ m ( h m ) = ̺ ′ σ m ( h ′ m ) it followsthat j ( k R h − R ) = j ( k R h ′ R − ) , thus h R = h ′ R , so h m and h ′ m lie in the same absolute class, whence τ R ( k m h − m ) = τ R ( k m h ′ m − ) , and as h m and h ′ m lie in the same absolute class, one has h m = h ′ m , and ̺ = ̺ ′ .The ̺ σ m ( h m ) together are thus h m [ Ω ∗ : Ω m ] elements of the Galois group of Ω ∗ / Σ . But thedegree [ Ω ∗ : Σ ] is at most equal to h m [ Ω ∗ : Ω m ] , because from Ω m = Ω R ( τ R ( k m )) , it follows that[ Ω m : Ω R ] is at most equal to the degree h m /h R of S m , k m ( X ) . Consequently the ̺ σ m ( h m ) areelements of the whole Galois group of Ω ∗ / Σ . From this we can conclude that Ω m is normal overΣ . For a conjugate ( ̺ σ ( h m )) − ̺ ( ̺ σ ( h m )) of an automorphism ̺ of Ω ∗ / Ω m , one has namely j ( k R ) ( ̺ σ ( h m )) − ̺ ( ̺ σ ( h m )) = j ( k R ) σ ( h m ) − ̺ − ̺ ̺ σ ( h m )= j ( k R h R ) ̺ − ̺ ̺ σ ( h m ) = j ( k R h R ) σ ( h m ) = j ( k R ) ,τ R ( k R ) ( ̺ σ ( h m )) − ̺ ( ̺ σ ( h m )) = τ R ( k m ) σ ( h m ) ̺ − ̺ ̺ σ ( h m )= τ R ( k m h m ) ̺ − ̺ ̺ σ ( h m ) = τ R ( k m h m ) σ ( h m ) = τ R ( k m ) . It is thus also an automorphism of Ω ∗ / Ω m ; the Galois group of Ω ∗ / Ω m is a normal subgroup of theGalois group of Ω ∗ / Σ , as claimed.We can therefore assume Ω ∗ = Ω m ; then σ m ( h m ) is uniquely determined by the equations (54),from which it follows that σ m ( h (1) m ) σ m ( h (2) m ) = σ m ( h (1) m h (2) m ) , so σ m is an isomorphism from the ray class group of Σ modulo m onto the Galois group of Ω m / Σ ,and the definition of σ m ( h m ) by means of the Frobenius automorphisms shows that σ m is exactly theisomorphism asserted by the Artin reciprocity law. For that, however, finitely many prime divisorsof Σ are excluded.
31. The ray class polynomial.
It still remains to carry out the proof that S m , k R ( X ) has coeffi-cients in Ω R . Hasse , loc. cit. , 78 to 81.
53e introduce the polynomial T m , k R ( X ) = Y γ mod a ,γ ≡ a m − ,γ a ( X − τ R ( γ, a R )) , whose zeros are thus the m -th division values of the ideal a R of R ; in the notation it is expressedthat T m , k R ( X ) depends only on the class k R of the divisor a .The τ R ( γ, a R ) are exactly the division values of a R whose orders t divide m . The τ R ( γ, a R )whose order is a given divisor t of m are exactly the invariants τ R ( k t ) of those ray classes of Σmodulo t which lie in the absolute class k R , and since two invariants τ R ( γ ′ , a R ) , τ R ( γ, a R ) of order t are equal if and only if the prinicipal divisor ( γ ′ γ − ) lies in the ray modulo t , it follows that everyinvariant τ ( k t ) occurs as a zero of T m , k R ( X ) exactly e t times, where e t denotes the number ofresidue classes of R modulo t represented by units. If we set S , k R ( X ) = 1 , one then has T m , k R ( X ) = Y t | m S t , k R ( X ) e t , and this holds also for m = 1 if we set T , k R ( X ) = 1 . The M¨obius inversion formula yields S m , k R ( X ) e t = Y t | m T t , k R ( X ) µ ( mt ) . It thus suffices to prove that the coefficients of T m , k R ( X ) lie in Ω R .The singular elliptic function τ R ( z, a R m − R ) belongs to the field K ( a R ) , because the periodlattice a R of K ( a R ) is contained in a R m − R and τ R ( z, a R m − R ) is invariant under the ̺ ε . Hence τ R ( z, a R m − R ) is a rational function of τ R ( z, a R ) :(55) τ R ( z, a R m − R ) = Z ( τ R ( z, a R )) N ( τ R ( z, a R )) , where Z ( X ) , N ( X ) are relatively prime polynomials, N with highest coefficient 1. We will show that N ( X ) = T m , k R ( X ) ; for that, we compare the poles of both sides of (55). The poles of τ R ( z, a R m − R )are at z ≡ a R m − R ;they are of order e . The right hand side of (55) has order e (deg N − deg Z ) at z ≡ a R ,whence one has deg Z = deg N + 1 and N ( τ R ( z, a R )) must have e -th order zeros at z ≡ a R m R , z a R , but is otherwise = 0 . The zeros of N ( X ) are thus the numbers τ R ( γ, a R ) , γ ≡ a R m − R , γ a R , thus exactly the zeros of T m , k R ( X ) , and it remainsto show that τ R ( γ, a R ) as a zero of N ( X ) has the same multiplicity as it does as a zero of T m , k R ( X ) .If n γ is the multiplicity of τ R ( γ, a R ) as a zero of N ( X ) , then ( τ R ( z, a R ) − τ R ( γ, a R )) n γ has a zeroof order e at z = γ , and thus z = γ is a zero of order e/n γ of τ R ( z, a ) − τ R ( γ, a R ) . On theother hand, the multiplicity n ′ γ of τ R ( γ, a R ) as a zero of T m , k R ( X ) is equal to the number of zerosof τ R ( z, a R ) − τ R ( γ, a R ) which are incongruent modulo a , because from τ R ( γ ′ , a R ) − τ R ( γ, a R ) = 0it follows that γ ′ ≡ ε γ mod a , where ε is a unit of R , so γ ′ ≡ a m − . The zeros z ≡ ε γ mod a of τ R ( z, a R ) − τ R ( γ, a R ) are mapped to one another under the transformations z → ε z + α α from a R , whence they all have the same order e/n γ ; thus the order of the elliptic function τ R ( z, a R ) − τ R ( γ, a R ) which is e is equal to en γ · n ′ γ , i.e. one has n γ = n ′ γ .We thus have τ R ( z, a R m − R ) = Z ( τ R ( z, a R )) T m , k R ( τ R ( z, a R )) . Now if N denotes a natural number divisible by m , then T m , k R ( X ) = Y γ mod a ,γ ≡ a m − ,γ a ( X − τ R ( γ, a R ))is a factor of L ( X ) = Y γ mod a ,γ ≡ a N − ,γ a ( X − τ R ( γ, a R )) , but L ( X ) can be expressed via the division polynomials: L ( X ) = Y t | N , t> T t ( X, j ( k R )) , so its coefficients lie in Ω R . If L ( X ) = T m , t R ( X ) Q ( X ) and Z ( X ) Q ( X ) = M ( X ) , then thecoefficients of T m , t R ( X ) = L ( X ) (cid:14) ( M ( X ) , L ( X ) ) lie in Ω R , if that holds for the coefficients of M ( X ) .Now to see this, we substitute for τ R ( z, a R ) in M ( τ R ( z, a R )) the series expansion τ R ( z, a R ) = ∞ X ν = − η ν u ν , u = z e g ( e ) ( a R ) − ;since the η ν lie in Ω R , it suffices to show that the coefficients of the expansion of M ( τ R ( z, a R )) inpowers of u lie in Ω R , because the coefficients of the polynomial M ( X ) are then obtained as theuniquely determined solutions of a system of linear equations in the field Ω R . But the expansion of M ( τ R ( z, a R )) is obtained when, in the right hand side of the equation M ( τ R ( z, a R )) = L ( τ R ( z, a R )) τ R ( z, a R m − R )we substitute the expansions τ R ( z, a R ) = ∞ X ν = − η ν u ν , u = z e g ( e ) ( a R ) − and τ R ( z, a R m R ) = ∞ X ν = − η ′ ν u ′ ν , u ′ = z e g ( e ) ( a R m − R ) − i.e. τ R ( z, a R m − R ) = ∞ X ν = − η ′ ν (cid:18) g ( e ) ( a R ) g ( e ) ( a R m − R ) (cid:19) ν u ν . L lie in Ω R , and the η ν and the η ′ ν lie in Ω R . But g ( e ) ( a R ) g ( e ) ( a R m − R ) also lie inΩ R , because α , α is a basis of a R m − R , so a basis of a R can be taken in the form S α α ! t witha primitive matrix S and a natural number t , whence g ( e ) ( a R ) g ( e ) ( a R m − R ) = t e g ( e ) S α α ! ! g ( e ) α α ! and g ( e ) S α α ! ! g ( e ) α α ! lies in Ω R by § III.12 . With that the proof is finished.
E. Remarks
We have concisely but completely depicted the theory of the interrelations between elliptic functions,modular functions and algebraic numbers, usually called “complex multiplication” in short; in theprocess we have assumed as known the elementary parts of the theory of algebraic numbers. Itremains for us to add some references to the literature.Our theory emerged largely before the completion of general class field theory; it was an importantreason for the latter’s development. Therefore, the classical results, for which we refer to
Weber , seen from today’s point of view, are incomplete and convoluted in reasoning. Nevertheless,our presentation of the theory of singular moduli (the ring class field), as summarized under theheading “First Main Theorem”, proceeds basically as in
Weber , Fricke or Fueter . However, thefundamental congruence (33) was first given by
Hasse (previously only (30) was known) and theestablishment of the isomorphism σ : R R → G Ω R Σ has been greatly simplified here.The theory of the singular division values of τ R ( z, a R ) (second main theorem) was developedfrom an idea of Hasse . It consists of introducing the division values τ R x ω + x ω N , ω ω !! as higher level modular functions and thus largely avoiding the transformation theory of ellipticfunctions, which only appears in § III.8 . The establishment of the fundamental congruence (52) alsofor degree-two prime divisors p is new, as is the preparatory theorem in § III.15 . Weber and
Fueter after him have established the theory of ray class fields, in particular thecongruence (51) (or the corresponding statement), based more on the transformation theory of ellipticfunctions ; however, instead of the function τ R which is an elliptic function of level one (invariant Weber , see Monographs.
Fueter , see Monographs.
Weber and
Fueter , see Monographs. M ), they were forced to use functions of higher levels ( Jacobi functions), which are easier tohandle. That it is possible to manage with the function τ R of level one alone was first shown by Hasse . However, a method was later given by
Deuring to carry out the approach of
Weber alsowith functions of level one such as τ R .But from another point of view, it is advantageous to use other elliptic functions besides τ R . Fueter succeeds in this way to treat the decomposition behavior in the ray class field Ω m of thefinitely many prime divisors of the field Σ which for us must remain excluded, and to computethe discriminants of the intermediate class fields. For the goal of Fueter , to prove the completenesstheorem asserting that all abelian number fields over Σ are contained in the ray class fields Ω m , thatwas unavoidable. Today one would simply take this result from Takagi’s converse theorem of generalclass field theory (as Takagi himself has already done) , because the arguments which
Fueter usesfor his proof of the completeness theorem are basically the same ones that appear in the generalconsiderations of
Takagi .It should also be noted that it is actually quite inappropriate to study a number field by fixingone (or some) generating element (as we have done in the case of Ω R and Ω m ). The exclusionof finitely many prime ideals has its reason. The purely algebraic theory, which springs from thetheory of algebraic function fields, is largely free from this defect, and will be reported in thisencyclopedia I 2, 26.On specific questions which we did not address here: Weber , Fricke , Fueter , Watson treat the problem of explicit computation of class invariants .On the question of whether Ω m = Σ( j ( k R ) , τ R ( k m )) is generated over Σ by τ R ( k m ) alone, see .On the number fields which can be generated by the singular values of certain modular functionsof higher level, see .On the division values of τ R z , ω ω !! with respect to non-maximal orders, see A new proof of the first main theorem which, essentially, gets by without using ∆( ω ) , is givenby Eichler . Hasse loc. cit. and . Deuring , Abhandl. Akad. Mainz, Nat.-Math. Kl. 1954.
Takagi , J. of the Coll. of Sci. Tokyo , no. 5 (1922). cf. the monographs of Weber , Fricke and
Fueter , furthermore
Watson , J. London Math. Soc. , 65 to 70 and 126to 132 (1930); J. f. Math. , 238 to 251 (1933); Proc. London Math. Soc. II. , 398 to 409 (1936); Mitra , Indianphys.-math. J. , 7 to 10 (1931); Bull. Calcutta Math. Soc. , 135 to 136 (1933). Sugawara , Proc. Phys.-Math. Soc. Japan III, 99 to 107 (1933); J. f. Math. , 189 to 191 (1936).
S¨ohngen , Math. Ann. , 102 to 328 (1935).
Franz , J. f. Math. , 60 to 64 (1935).
Eicher , Math. Zeitschr. , 229 to 242 (1956)., 229 to 242 (1956).