aa r X i v : . [ m a t h . N T ] D ec Generalised Weber Functions
Andreas EngeINRIA, LFANTCNRS, IMB, UMR 5251Univ. Bordeaux, IMB33400 [email protected] Fran¸cois MorainINRIA Saclay–ˆIle-de-France& LIX (CNRS/UMR 7161)´Ecole polytechnique91128 Palaiseau [email protected] December 2013
Abstract
A generalised Weber function is given by w N ( z ) = η ( z/N ) /η ( z ),where η ( z ) is the Dedekind function and N is any integer; the originalfunction corresponds to N = 2. We classify the cases where somepower w eN evaluated at some quadratic integer generates the ringclass field associated to an order of an imaginary quadratic field. Wecompare the heights of our invariants by giving a general formulafor the degree of the modular equation relating w N ( z ) and j ( z ). Ourultimate goal is the use of these invariants in constructing reductionsof elliptic curves over finite fields suitable for cryptographic use. Let K be an imaginary quadratic field of discriminant ∆ <
0. We areinterested in orders O of K having discriminant D = c ∆. The principalorder of discriminant ∆ is O K , which is generated by ω = √ ∆2 if ∆ ≡ ω = √ ∆2 if ∆ ≡ O of discriminant D , let K D denote the ring class field that is associated to it. It is well-known that if j denotes the modular invariant, then K D = K ( j ( cω )); so K D /K ≃ K [ X ] / ( H D ( X )), where the class polynomial H D is the minimalpolynomial of j ( cω ). It can be used to obtain elliptic curves over finite fieldswith a number of points known in advance, with applications to cryptology, : 11G15, 14K22 Key words : complex multiplication, class invariants, eta quotients A. Enge and F. Morain in particular based on the Weil or Tate pairing (cf. [13]), and primalityproving [1].Since the class polynomial has a rather large height, it is desirable tofind smaller defining polynomials to speed up the computations. There is along history of such studies, going back to at least Weber [27]; see, e.g., [2,26, 20] for connections with the class number 1 problem. Generally modularfunctions f and special arguments α ∈ O are considered such that the singular value f ( α ) lies in K D , in which case f ( α ) is called a class invariant .Our ultimate goal is to build elliptic curves having CM, and this is doneusing a so-called modular equation (with integer coefficients) relating a mod-ular function f to j . For this to be efficient, we need f ( α ) to have a smallheight and the corresponding modular equation to be of small genus (witha predilection for genus 0).Part of the literature has concentrated on the functions introduced byWeber, quotients of two η -functions with a transformation of level 2 appliedto one of them, see [23, 14, 15, 22] besides the already cited sources. This isa perfect case for us, since the genus of the associated modular curve is 0.Results on more general η -quotients are given in [18, 17, 19, 14, 8, 12].All of them are obtained using the modern tool for determining the Galoisaction of the class group of O on singular values of modular functions,namely Shimura’s reciprocity law [25]. The present article is no exceptionto this rule. For the sake of self-containedness and the reader’s ease, webriefly summarise in § generalised Weber functions w N , defined and stud-ied in §
3, which are quotients of two η -functions with a transformation oflevel N applied to one of them. These appear in [22, Table 1] and as a spe-cial case of [19]. While there is some overlapping between this article and[19], we follow a different approach: The authors of [19] use an ideal in theclass group to transform the η -function, and the norm of the ideal implic-itly determines the level; they then proceed to prove which root of unity isneeded for twisting the function so that a minimal power of it yields a classinvariant. On the other hand, we start with a fixed level and thus a fixedgeneralised Weber function and determine the minimal power yielding classinvariants without using additional roots of unity.A first result on the “canonical” power w sN is readily obtained in § § w eN with e | s yield class invariants in § N (or, equivalently, anideal in the class group) such that the corresponding generalised Weber func-tion yields a class invariant, fixing the level first as we do it in this studyimplies control over the height of the class invariants. Indeed, this height, animportant measure for the complexity of computing a class polynomial, is eneralised Weber Functions j -invariant. Thus, the generalisedWeber functions can be ordered totally with respect to their computationalefficiency, see §
7, and the invariants can be compared directly to other in-variants in the literature, cf. [7, 10].Unlike [19], we explicitly consider levels N that are not coprime to 6, aconsiderable source of complication, which is justified since the correspond-ing functions tend to yield class invariants of lower height, see the formulæin § N alone. This makes it easier to constructthe associated elliptic curves with complex multiplication; in particular, [21]shows how w can be used to directly write down the correct twist of theelliptic curve with the desired number of points over a finite field.Existing results in the literature often only state when a singular valueis a class invariant; to obtain the class polynomial, however, one needs anexplicit description of its algebraic conjugates. These can be worked outusing Shimura reciprocity again; following the approach of N -systems intro-duced in [22], we obtain synthetic and simple descriptions of the conjugates,and moreover determine when the class invariant has a minimal polynomialwith rational coefficients, that is, it defines the real subfield of the class fieldover Q . In the following, we denote by f ◦ M the action of matrices M = (cid:18) a bc d (cid:19) ∈ Γ = Sl ( Z ) / {± } on modular functions given by( f ◦ M )( z ) = f ( M z ) = f (cid:18) az + bcz + d (cid:19) . For n ∈ N , let Γ( n ) = (cid:26)(cid:18) a bc d (cid:19) ≡ (cid:18) (cid:19) (mod n ) (cid:27) be the principalcongruence subgroup of level n ; for a congruence subgroup Γ ′ such thatΓ( n ) ⊆ Γ ′ ⊆ Γ, denote by C Γ ′ the field of modular functions for Γ ′ .One of the most important congruence subgroups is given by Γ ( n ) = (cid:26)(cid:18) a bc d (cid:19) ≡ (cid:18) ∗ ∗ ∗ (cid:19) (mod n ) (cid:27) . Definition 2.1.
The set F n of modular functions of level n rational overthe n -th cyclotomic field Q ( ζ n ) is given by all functions f such that1. f is modular for Γ( n ) and2. the q -expansion of f has coefficients in Q ( ζ n ) , that is, f ∈ Q ( ζ n ) (cid:0)(cid:0) q /n (cid:1)(cid:1) , A. Enge and F. Morain where q /n = e πiz/n . The function field extension F n / Q ( j ) has Galois group isomorphic toGl ( Z /n Z ) / {± } , where the isomorphism is defined by the following actionof matrices on functions: • ( f ◦ M )( z ) = f ( M z ) as above for M ∈ Γ; this implies in particularthat also the q -expansion of f ◦ M has coefficients in Q ( ζ n ); • f ◦ (cid:18) d (cid:19) for gcd( d, n ) = 1 is obtained by applying to the q -expansionof f the automorphism ζ n ζ dn ; • any other matrix M that is invertible modulo n may be decomposedas M ≡ M (cid:18) d (cid:19) M (mod n ) with gcd( d, n ) = 1 and M , M ∈ Γ,and ( f ◦ M )( z ) = (cid:18)(cid:18) ( f ◦ M ) ◦ (cid:18) d (cid:19)(cid:19) ◦ M (cid:19) ( z ) . Shimura reciprocity makes a link between the Galois group of the func-tion field F n and the Galois groups of class fields generated over an imagi-nary-quadratic field by singular values of modular functions. Theorem 2.2 (Shimura’s reciprocity law, Th. 5 of [22], Th. 5.1.2 of [24]) . Let f be a function in F n , ∆ < a fundamental discriminant and O theorder of K = Q ( √ ∆) of conductor c . In the following, all Z -bases of idealsare written as column vectors. Let a = (cid:18) α α (cid:19) Z with basis quotient α = α α ∈ H be a proper ideal of O , m an ideal of O K of norm m prime to cn , m its conjugate ideal and M ∈ Gl ( Z ) a matrix of determinant m such that M (cid:18) α α (cid:19) is a basis of a ( m ∩ O ) . If f does not have a pole in α , then • f ( α ) lies in the ray class field modulo cn over K and • the Frobenius map σ ( m ) acts as f ( α ) σ ( m ) = ( f ◦ mM − )( M α ) . In the following, we are particularly interested in class invariants , that is,values f ( α ) that lie not only in a ray class field, but even in a ring class field.Using Shimura’s reciprocity law, [22, Th. 4] gives a very general criterionfor class invariants, which is the basis for our further investigations. Theorem 2.3.
Let f ∈ C Γ ( n ) for some n ∈ N be such that f itself and f ◦ S have rational q -expansions. Denote by α ∈ H a root of the primitiveform [ A, B, C ] of discriminant D with gcd( A, n ) = 1 and n | C . If α is nota pole of f , then f ( α ) ∈ K D . eneralised Weber Functions f ( α ) are then derived generically in a form that is wellsuited for computations in [22, Prop. 3 and Th. 7], [24, Th. 5.2.4]. Theorem 2.4. An n -system for the discriminant D is a complete systemof equivalence classes of primitive quadratic forms [ A i , B i , C i ] = A i X + B i X + C i , i = 1 , . . . , h ( D ) , of discriminant D = B i − A i C i , such that gcd( A i , n ) = 1 and B i ≡ B (mod 2 n ) . Such a system exists for any n . Tothese quadratic forms, we associate in the following the quadratic numbers α i = − B i + √ D A i .Let f ∈ F n be such that f ◦ S with S = (cid:18) −
11 0 (cid:19) has a rational q -expansion. If f ( α ) ∈ K D , then a complete system of conjugates of f ( α ) under the Galois group of K D is given by the f ( α i ) , and the characteristicpolynomial of f ( α ) over K is H D [ f ] = h ( D ) Y i =1 ( X − f ( α i )) . w N In this section we examine the general properties of the function w N , withthe aim in mind of applying Theorem 2.3 to its powers.Let z be any complex number and put q = e iπz . Dedekind’s η -functionis defined by [5] η ( z ) = q / Y m > (1 − q m ) . The Weber functions are [27, §
34, p. 114] f ( z ) = ζ − η (( z + 1) / η ( z ) , f ( z ) = η ( z/ η ( z ) , f ( z ) = √ η (2 z ) η ( z ) . The modular invariant j is recovered via [27, §
54, p. 179]: j ( z ) = ( f − f = ( f + 16) f = ( f + 16) f . The functions − f , f and f are the three roots of the modular polynomialΦ c ( F, j ) = F + 48 F + F (768 − j ) + 4096 , that describes the curve X (2).For an integer N >
1, let the generalised Weber function be defined by w N = η ( z/N ) η ( z ) . A. Enge and F. Morain
As shown in the following, there is a canonical exponent t such that w tN ismodular for Γ ( N ). Its minimal polynomial Φ cN ( F, j ) over C ( j ) is a modelfor X ( N ). The other roots of this polynomial can be expressed in terms of η , too, a topic to which we come back in § w N under unimodular transforma-tions, which can be broken down to the transformation behaviour of η ( z/K )for K = 1 or N . This has been worked out in [9, Th. 3]. Theorem 3.1.
Let M = (cid:18) a bc d (cid:19) ∈ Γ be normalised such that c > , and d > if c = 0 . Write c = c λ ( c ) with c odd; by convention, c = λ ( c ) = 1 if c = 0 . Define ε ( M ) = (cid:18) ac (cid:19) ζ ab + c ( d (1 − a ) − a )+3 c ( a − λ ( c )( a − . For K ∈ N write ua + vKc = δ = gcd( a, Kc ) = gcd( a, K ) . Then η (cid:16) zK (cid:17) ◦ M = ε (cid:18) aδ − v Kcδ u (cid:19) p δ ( cz + d ) η δz + ( ub + vKd ) Kδ ! , where the square root is chosen with positive real part. Theorem 3.2.
The function w N has a rational q -expansion. Denote by S = (cid:18) −
11 0 (cid:19) the matrix belonging to the inversion z
7→ − z . If N is asquare, then w N ◦ S has a rational q -expansion. Otherwise, w N ◦ S has arational q -expansion.Let the subscript 1 and the function λ have the same meaning for apositive integer n as in Theorem 3.1, that is, n = n λ ( n ) with n odd. If M = (cid:18) a N b c d (cid:19) ∈ Γ ( N ) , then w N ◦ M = ε w N with (3.1) ε = (cid:18) aN (cid:19) ζ ( N − − b a + c ( d (1 − a ) − a ))24 ζ c N − a − ( − λ ( N )( a − . Let t = N − , measure how far N − is from being divisible by ,and let e and s be such that t | s | and e | s . If N is a square or e iseven, then w eN is modular for Γ (cid:0) se (cid:1) ∩ Γ (cid:0) se N (cid:1) . Otherwise, w eN is modularfor Γ (cid:0) se N (cid:1) ∩ Γ (cid:0) se N (cid:1) . In both cases, w eN ∈ F se N ⊆ F N .Proof. The q -expansion of w N is rational since that of η is. Let M = (cid:18) a bc d (cid:19) ∈ Γ. By Theorem 3.1 applied to K = 1 and N , we have(3.2) w N ◦ M = ε (cid:18) aδ − v Ncδ u (cid:19) ε (cid:18) a bc d (cid:19) − √ δ η (cid:16) δz +( ub + vNd ) Nδ (cid:17) η ( z ) eneralised Weber Functions δ = gcd( a, N ) = ua + vN c .In the special case M = S we obtain δ = N , v = 1, u = 0 and w N ◦ S = √ N η ( N z ) η ( z ) , which proves the assertion on the q -expansion of w N ◦ S .Assume now that M ∈ Γ ( N ). Letting b = N b , we have δ = 1, u = d and v = − b since ad − bc = 1. Thus, (3.2) specialises as w N ◦ M = ε (cid:18) a b N c d (cid:19) ε (cid:18) a bc d (cid:19) − η ( z/N ) η ( z ) = ε w N ( z )with ε = (cid:18) ac N (cid:19)(cid:18) ac (cid:19) − ζ ( b − b ) a + c ( N − d (1 − a ) − a )+3 c ( N − a − ( λ ( Nc ) − λ ( c ))( a − , which proves (3.1).We need to examine under which conditions ε e = 1. The Legendre symbolvanishes when N is a square, e is even or a ≡ N ). The exponentof ζ becomes divisible by s ( N −
1) and thus by 24 whenever se divides b and c .In the case of odd N , we have λ ( N ) = 0 and N = N , and the conditionon a implies that the exponent of ζ is divisible by 4.In the case of even N , the coefficient a is odd since det M = 1, and ε e = ( − e (cid:16) c N − a − + λ ( N ) a − (cid:17) . For even e , there is nothing to show. If e is odd, then 8 | t | s implies that a ≡ w N To be able to apply Theorem 2.3 directly to powers of w N , we are interestedin the minimal exponent s such that w sN is invariant under Γ ( N ) and w sN ◦ S has a rational q -expansion. From Theorem 3.2, we recover the integer t = 24 / (gcd( N − , s = 2 t if t is odd and N is not asquare, and s = t otherwise. We begin with the following purely arithmetical lemma.
Lemma 4.1.
Let N be an integer. For a prime p , denote by v p the p -adicvaluation. Let D = c ∆ be a discriminant with fundamental part ∆ . Then D admits a square root B modulo N if and only if for each prime p dividing N , one of the following holds. A. Enge and F. Morain (cid:0) ∆ p (cid:1) = +1 ;2. (cid:0) ∆ p (cid:1) = − and v p ( N ) v p ( c ) ;3. (cid:0) ∆ p (cid:1) = 0 and v p ( N ) v p ( c ) + 1 .Proof. The Chinese remainder theorem allows to argue modulo the differentprime powers dividing N . The argumentation is slightly different for p oddand even, and we give some hints only for p = 2.When ∆ ≡ ≡ c ∆ is a square modulo4 N if and only if v ( c ) + 3 > v (4 N ).When ∆ ≡ v ( c ) + 2 > v (4 N ) is needed in that case.In the following, arithmetical conditions on a prime p to be representableby the principal form of discriminant D will be needed. We take the follow-ing form of Dirichlet’s theorem from [3, Ch. 4] (alternatively, see [4, Chap18, G]). For an integer p , let χ ( p ) = (cid:0) − p (cid:1) and χ ( p ) = (cid:0) p (cid:1) . The genericcharacters of D = c ∆ are defined as follows:(a) (cid:0) pq (cid:1) for all odd primes q dividing D ;(b) if D is even:(i) χ ( p ) if D/ ≡ , , χ ( p ) if D/ ≡ χ ( p ) · χ ( p ) if D/ ≡ χ ( p ) and χ ( p ) if D/ ≡ D is fundamental (i.e., c = 1), then case (iv) cannot occur andin case (i), we may have D/ ≡ , Theorem 4.2.
An integer p such that gcd( p, cD ) = 1 is representable bysome class of forms in the principal genus of discriminant D if and only ifall generic characters χ ( p ) have value +1 . In particular, this condition isnecessary for representability by the principal class. Theorem 4.3.
Let N be an integer and t = N − , . If t is odd and N is not a square, let s = 2 t , otherwise, let s = t . Suppose D satisfies Lemma4.1. Consider an N -system of forms [ A i , B i , C i ] with roots α i = − B i + √ D A i such that B i ≡ B (mod 2 N ) , as introduced in 2.4. Then the singular values w sN ( α i ) lie in the ring class field K D , and they form a complete set of Galoisconjugates. eneralised Weber Functions Proof.
Once the existence of B is verified, the form [1 , B, C ] with C = B − D is of discriminant D and satisfies N | C . The assertion of the theorem isthen a direct consequence of Theorems 2.3 and 3.2.Sometimes, the characteristic polynomial of w sN is real, so that its coeffi-cients lie in Z instead of the ring of integers of Q ( √ D ). It is then interestingto determine the pairs of quadratic forms that lead to complex conjugates. Theorem 4.4.
Under the assumptions of Theorem 4.3, let B ≡ N ) ,which is possible whenever N is odd and N | D , or N is even and N | D .Then the characteristic polynomial of w sN is real. More precisely, if α i and α j are roots of inverse forms of the N -system, then w sN ( α j ) = w sN ( α i ) .Proof. Notice that B ≡ N ) and B i ≡ B (mod 2 N ) imply − B i ≡ B (mod 2 N ), so that [ A i , − B i , C i ], the inverse form of [ A i , B i , C i ], satisfies the N -system constraint; thus w sN ( α j ) = w sN (cid:16) B i + √ D A i (cid:17) = w sN ( − α i ) . On theother hand, q ( − α i ) = q ( α i ), which implies w N ( − α i ) = w N ( α i ) since w N has a rational q -expansion.These first results, direct consequences of the Shimura reciprocity law,are meant to set the stage for the detailed and much more involved analysisof lower powers in the following chapters. For gcd( N,
6) = 1, [19, Theo-rem 20] determines a 48-th root of unity ζ and an exponent e | s suchthat ζ w eN yields a class invariant. With a bit of work, it can be shown that ζ s/e = 1 in our context, which provides an alternative proof of Theorem 4.3without giving the algebraic conjugates of the singular value. Throughout the remainder of this section, we assume that N is a squareor e is even, so that f = w eN and f ◦ S have rational q -expansions byTheorem 3.2. Let α be a root of the primitive quadratic form [ A, B, C ] ofdiscriminant D with gcd( A, N ) = 1. By Theorems 3.2 and 2.2, the singularvalue f ( α ) lies in the ray class field modulo c te N over K , and the Galoisaction of ideals in O K can be computed explicitly. We eventually need toshow that the action of principal prime ideals generated by elements in O is trivial, which implies that the singular value lies in the ring class field K D . Then Theorems 3.2 and 2.4 show that the conjugates are given by thesingular values in a te N -system.We are only interested in the situation that N | C . Notice that undergcd( A, N ) = 1 this is equivalent to 4 N | AC = B − D , or B ≡ D (mod 4 N ). The remainder of this section is devoted to computing in thiscase the Galois action of principal prime ideals ( π ) with π ∈ O coprime to6 cN on the singular values according to the arithmetic properties of N and D . § A. Enge and F. Morain
To apply Shimura reciprocity in the formulation of Theorem 2.2, weneed to explicitly write down adapted bases for the different ideals. So let a = (cid:18) AαA (cid:19) Z be an ideal of O = (cid:18) Aα (cid:19) Z with basis quotient α . Without lossof generality, we may assume that p = N( π ) | C by suitably modifying α :Indeed, notice that the quadratic form associated to α ′ = α − kN for some k ∈ Z is given by [ A, B ′ , C ′ ] = [ A, B +2 A (24 kN ) , A (24 kN ) + B (24 kN )+ C ].This form still satisfies N | C ′ , and furthermore f ( α ′ ) = f ( α ) since f isinvariant under translations by 24 N according to Theorem 3.2. Since p splitsin O and is prime to c , the equation AX + BX + C has a root x modulo p .Choosing k ∈ Z such that k ≡ x (24 N ) − (mod p ), which is possible since p ∤ N , we obtain p | C ′ .Let π = u + vAα with u , v ∈ Z . From(5.1) p = N( π ) = u ( u − vB ) + v AC and p | C we deduce that p divides u or u ′ = u − vB . Using Aα = − Aα − B and N ( Aα ) = AC , we compute pa = π (cid:18) AαA (cid:19) = (cid:18) uAα + vACuA − vA α − vAB (cid:19) = (cid:18) u vC − vA u − vB (cid:19) (cid:18) AαA (cid:19)
So if p | u , the matrix M of Theorem 2.2 is given by M = (cid:18) u vC − vA u − vB (cid:19) = (cid:18) p
00 1 (cid:19) M with M = (cid:18) up v Cp − vA u ′ (cid:19) ∈ Γ ( N )since N | C and p ∤ N .If f is invariant under M − , the rationality of its q -expansion impliesthat f ◦ mM − = f ◦ M − ◦ (cid:18) p (cid:19) = f, so that f ( α ) σ ( p ) = f ( M α ) = f (cid:18) uα + vC − vAα + u − vB (cid:19) = f (cid:18) παπ (cid:19) = f ( α ) . For p | u ′ , we decompose in a similar manner M = M (cid:18) p (cid:19) = M S (cid:18) p
00 1 (cid:19) S with M = u v Cp − vA u ′ p ! ∈ Γ ( N ) , and the rationality of the q -expansion of f ◦ S allows to conclude if f isinvariant under M − .So we need the transformation of f under M − = (cid:18) u ′ − v Cp vA up (cid:19) . eneralised Weber Functions f ◦ M − = ζ eθ f with θ =( N − v (cid:18) u ′ CN p + A (cid:18) up (1 − u ′ ) − u ′ (cid:19)(cid:19) + 3 v A ( N − u ′ −
1) + 3 λ ( N )( u ′ − . (5.2)We obtain invariance provided eθ ≡ M − iscompletely analogous and omitted.) In the following, we classify the valuesof D and B for which θ is 0 modulo some divisor of 24. It is natural to studyseparately θ mod 3 and θ mod 2 ξ for 1 ξ N . We will give code names to the following propositions for future use. θ modulo To be able to use some exponent e not divisible by 3, we need to impose3 | θ . From the reduction of (5.2) modulo 3, namely θ = ( N − v (cid:18) u ′ CN p + A (cid:18) up (1 − u ′ ) − u ′ (cid:19)(cid:19) mod 3 , we immediately see that 3 | θ for N ≡ ∤ s in this case.For N B ≡ D (mod 4 N ) to obtain divisibilityof C by N (see the discussion above), and define r ∈ { , , } such that(5.3) A CN = B − D N ≡ r (mod 3) . Notice that r = 1 implies A ≡ CN (mod 3), while r = 2 implies A ≡ − CN (mod 3). N ≡ Proposition 5.1 (PROP30) . Let N ≡ , B ≡ D (mod 4 N ) and r as in (5.3) . Then | θ if(a) | D and r = 1 ;(b) D ≡ and r = 2 .In these cases, B satisfies the following congruences modulo :(a) | B ;(b) ∤ B .Proof. Since 3 | N | C and 3 ∤ p , u ≡ u ′ ≡ θ ≡ ± v (cid:18) CN p − A (cid:19) mod 3 . A. Enge and F. Morain (a) If 3 | B , or equivalently 3 | D , then p ≡ u ≡ ∤ B , which is equivalent with D ≡ ∤ v needs to be examined. Then u u ′ (mod 3) and p ≡ N ≡ Proposition 5.2 (PROP32) . Let N ≡ , B ≡ D (mod 4 N ) and r ∈ { , } as in (5.3) . If D ≡ r (mod 3) , then | θ and | B .Proof. Notice that D ≡ r (mod 3) is equivalent with 3 | B by (5.3). Then u ′ ≡ u (mod 3) and θ ≡ uv (cid:18) CN p + Ap (1 − u ) − A (cid:19) (mod 3) . If 3 divides u or v , we are done.Otherwise, u ≡ v ≡ θ ≡ ± (cid:18) CN p − A (cid:19) (mod 3) . Writing p ≡ AC ≡ − r (mod 3), we see that this case is possible only for r = 2 and p ≡ A ≡ − CN (mod 3) and 3 | CNp − A .Note that the proposition does not hold for r = 0, since then 3 | D ,3 | B , 3 | AC , and exactly one of A and C is divisible by 3 (if both were,then [ A, B, C ] would not be primitive), causing θ u or v is divisible by 3. θ modulo powers of N odd Since N = N and λ ( N ) = 0, (5.2) becomes θ ≡ ( N − ρ (mod 8)for ρ = v (cid:18) u ′ CN p + A (cid:18) up (1 − u ′ ) − u ′ (cid:19)(cid:19) + 3 v A ( u ′ − . So θ is divisible by 8 if N ≡ N is a square. Otherwise, e is supposed to be even, so eθ is divisible by 4;if N ≡ eθ is even divisible by 8. So the only remaining case ofinterest is N ≡ e ≡ | eθ is equivalent with ρ even. We have ρ ≡ v (cid:0) u ′ C + A ( u (1 + u ′ ) + u ′ ) (cid:1) + u ′ + 1 mod 2 . eneralised Weber Functions Proposition 5.3 (PROP21) . Let N be odd. If D is odd, then θ ≡ ( N − ρ (mod 8) with ρ even.Proof. Since B is odd, u ′ ≡ u + v (mod 2).If one of v , A and C is even, then u and u ′ are odd by (5.1) (so that infact v is even), and ρ is even.Otherwise, v , A and C are odd, u ′ = u + 1 (mod 2) and ρ is even aswell. N even Let N = 2 λ ( N ) N with N odd and λ ( N ) >
1. We study divisibility of θ by2 ξ for increasing values of ξ . The value ξ = 3 is of interest only when e isodd, in which case N and thus N are squares. We start with an elementaryremark. Lemma 5.4. If | N | C , then(a) u and u ′ are odd and (5.4) θ ≡ ( N − vu ′ (cid:18) CN p − A (cid:19) (mod 4); (b) moreover, if | C , then | vB .Proof. (a) u and u ′ are odd by (5.1), so that u ′ ≡ N is odd, almost all terms disappear from (5.2).(b) We have p = u + v ( − uB + vAC ) ≡ u ( u − vB ) mod 4. Since u is oddby (a), we deduce that vB must be even.As discussed above, N | C is equivalent with B ≡ D (mod 4 N ). Then A CN = B − D N ; by gradually imposing more restrictions modulo powers of 2times 4 N , we fix A CN modulo powers of 2. Proposition 5.5 (PROP20) . When N is even, θ is even in the followingcases:(a) B ≡ D + 4 N (mod 8 N ) ;(b) B ≡ D (mod 8 N ) and D ≡ .Proof. (a) The conditions imply that A ( C/N ) is odd, and Lemma 5.4(a)allows to conclude since p is odd.(b) In that case A ( C/N ) is even. Since A is prime to N , it is odd andtherefore C/N is even, which implies in turn 4 | C . By Lemma 5.4(b),we get 2 | vB . Since D is odd, B is odd and v is even, and (5.4) finishesthe proof.4 A. Enge and F. Morain
Divisibility of θ by B ≡ D + r (4 N ) mod (16 N )to have a solution. Lemma 5.6.
Let r ∈ { , , , } and N be even. Given D , suppose theequation B ≡ D + 4 rN (mod 16 N ) admits a solution in B . Then either D ≡ which implies B is odd, or D is even and D satisfies one ofthe conditions of the following table depending on rN mod 8 , which in turngives properties of B . rN mod 8 condition on D ⇒ D/ B/
20 4 mod 32 1 odd | D even odd | D even
20 mod 32 5 odd || D oddProof. Since B ≡ D mod 8, the only possible value for odd D is D ≡ B odd. If D is even, then (cid:18) B (cid:19) ≡ D rN mod 8and since N is even, the above table makes sense.Remembering that the only squares modulo 8 are { , , } , the table iseasily constructed and left as an exercise to the reader.Now, we are ready to extend the result of Proposition 5.5 by considering B ≡ D + r (4 N ) (mod 16 N ) with r ∈ { , } , which yields A CN ≡ r (mod 4).Note that case (b) cannot be extended and we leave the proof of this to thereader. Proposition 5.7 (PROP44) . Let N be even, and suppose B ≡ D + 4 N (mod 16 N ) has a solution. Then θ is divisible by if one of the followingconditions is met:(a) D ≡ ;(b) | D ;(c) || N and || D . eneralised Weber Functions Proof. If D is odd, the condition follows from Lemma 5.6. Then u ′ = u − vB leads to 2 | v and 4 | θ .Assuming D even, Theorem 4.2 implies that χ ( p ) = 1 (or, equivalently, p ≡ D/ ∈ { , , , } , which immediately settlescase (b). When D/ | N whencomparing with the table of Lemma 5.6, and this gives us (c).In the other cases, when p ≡ v odd since AC ≡ θ ≡ Proposition 5.8 (PROP412) . Let N be even, and suppose B ≡ D + 12 N (mod 16 N ) . Then θ is divisible by if one of the following conditions ismet:(a) D ≡ ;(b) || D and || N ;(c) || D and | N .In the cases of D even, B satisfies the following congruences modulo :(b) | B ;(c) || B .Proof. The proof for D odd as well as the case distinctions for D even arethe same as in Proposition 5.7. However, we now have A CN ≡ − χ ( p ) = 1 (i.e., D/ ∈ { , , , } ), we get p ≡ CNp − A ≡ v should be even, θ may or may not be divisible by 4.So we have to turn our attention to the four other cases, i.e., D/ ∈{ , , , } , with Lemma 5.6 in mind. If 4 | B , 8 || D and 2 || N , then 2 || C ,and either v is even or p ≡ | θ . If 2 || B and 4 || D , suppose that furthermore 4 | N . Then 4 | AC ,and again v is even or p ≡ Divisibility of θ by As discussed at the beginning of § θ mod 8 when N is a square, that is, λ ( N ) is even and N is asquare; in particular, N ≡ Lemma 5.9. If N is an even square dividing C , then θ ≡ ( N − vu ′ (cid:18) CN p − A (cid:19) (mod 8) . A. Enge and F. Morain
From the results obtained for B ≡ D + 4 rN (mod 16 N ) for r ∈ { , } ,it is natural to look at B ≡ D + 4 rN (mod 32 N ) for r ∈ { , , , } . Then A CN ≡ r (mod 8). Proposition 5.10 (PROP8) . Let N be an even square, and suppose B ≡ D + 4 rN (mod 32 N ) . Then θ is divisible by if one of the following con-ditions holds:(a) r = 3 or r = 7 , and D ≡ ;(b) r = 1 , and | D ;(c) r = 5 , and || D .In the cases of D even, B satisfies the following congruences modulo :(b1) || B if || N ;(b2) | B if | N .(c1) || B if | N ;(c2) | B if || N .Proof. Since 4 | N | C , we have p ≡ u ( u − vB ) (mod 4) by (5.1).For D odd, B is odd and v is even as seen in Proposition 5.7. If v isdivisible by 4, then θ is divisible by 8 by Lemma 5.9. If 2 || v , then p ≡ r ≡ | CNp − A , and 8 | θ byLemma 5.9.In the remaining cases of the proposition, 16 | D , 4 | B , r ≡ p ≡ v is even, Lemma 5.9 implies that 8 | θ . From nowon, we assume that v is odd. Then p = u − uvB + AC (mod 8), and weneed to verify that 8 | CNp − A .The results now follow from close inspection of AC ≡ rN (mod 8) and (cid:18) B (cid:19) ≡ D
16 + r N . Consider first the case r = 1 and 32 | D . By Theorem 4.2, we have χ ( p ) = χ ( p ) = 1, which yields p ≡ CNp − A by 8.Consider now r = 5; it is sufficient to show that p ≡ || D and 16 | N | C , then B ≡ p ≡ || D and4 || N , then AC ≡ | D + 4 rN , whence 8 | B and p ≡ eneralised Weber Functions w N The aim of this section is to determine conditions under which singularvalues of lower powers of w N than those given in Theorem 4.3 yield classinvariants. When N is not a square, only even powers are possible by The-orems 3.2 and 2.3. So we specialise the propositions of § N (mod 12). When N is a square, odd powers may yield class in-variants, and we need to distinguish more finely modulo 24. Note that then N ∈ { , , , , , } (mod 24).Throughout this section, we use the notation of Theorem 4.3. The num-ber α is a root of the quadratic form [ A, B, C ] of discriminant D and N isan integer such that A is prime to N and B is a square root of D mod-ulo 4 N according to Lemma 4.1, so that N | C . The canonical power s such that w sN ( α ) is a class invariant, that is, w sN ( α ) ∈ K D , is defined as inTheorem 4.3, and we wish to determine the minimal exponent e such that w eN ( α ) is still a class invariant. The general procedure is as follows: Giventhe value of N , we assemble the propositions of § B as well as the periodof D for which class invariants are obtained. In general, we can combine acondition on B related to θ mod 3 and another one related to θ mod 2 ξ . TheChinese remainder theorem is then used to find compatible values. Whenno particular condition modulo 3 or powers of 2 is imposed, that is, e and s have the same 3-adic or 2-adic valuation, then Theorem 4.3 already leadsto the desired conclusion.Once a power w eN ( α ) is identified as a class invariant, its conjugates maybe obtained by an M -system for M = se N containing [ A, B, C ] as shownthrough Theorems 2.4 and 3.2. In more detail, one may proceed as follows:1. Determine a form [
A, B, C ] with root α satisfying gcd( A, M ) = 1 andthe constraint on B so that w eN ( α ) is a class invariant; in general, onemay choose A = 1.2. Enumerate all reduced forms [ a i , b i , c i ], i = 1 , . . . , h ( D ) of discriminant D , numbered in such a way that [ a , b , c ] ≡ [ A, B, C ].3. Let [ A , B , C ] = [ A, B, C ]. For i >
2, find a form [ A i , B i , C i ] ≡ [ a i , b i , c i ] such that gcd( A i , M ) = 1 and B i ≡ B (mod 2 M ), using, forinstance, the algorithm of [22, Prop. 3], [24, Th. 3.1.10].Then a floating point approximation of the class polynomial can be com-puted as h D Y i =1 (cid:0) X − w eN ( α i ) (cid:1) with α i = − B i + √ D A i . Using the algorithms of [11], one obtains a quasi-linearcomplexity in the total size of the class polynomial.8 A. Enge and F. Morain
Note that the conditions on B of § B ≡ D +4 rN (mod 4 RN ), where r is defined modulo R and the only primes dividing R are 2 and 3. For the sake of brevity, we denote such a condition by r : R .So if no particular condition beyond B ≡ D (mod 4 N ) is required, this isdenoted by 0:1.We will give more details for the first non-trivial cases and be brieferin the sequel, since the results rapidly become unweildy. We add numericalexamples for these cases. N odd N N ≡ ∤ D , then PROP32 applies; moreover, theresulting condition 3 | B is automatically satisfied, and we gain a factor of3 in the exponent. Similarly if D is odd, then PROP21 applies without anyrestriction on B , and we gain a factor of 2 in the exponent. N mod 12 s B D e proposition(s)5 6 1:3 D ≡ D ≡ ∤ D ∤ D D ≡ D ≡ D ≡ D ≡ D = c ∆, we put ω = p ∆ / | ∆ and ω = (1+ √ ∆) / N f − D H D [ f ]5 w X − ω − w X − − ω w X − ω + 111 w X + (27 ω − X + (1656 ω − X + (7947 ω − X − ω + 100069311 w X + 7 + 6 ω w X + 8 ω − w X − ω + 111 w X − ω + 3 N ≡ N, = 1. For N ≡ s = 12, and N cannot be a square. Therefore we need eneralised Weber Functions e . Since already the full power w N can only be usedwhen D is a square modulo 4 N , we only have to consider D ∈ { , , , } (mod 12). Then PROP30 applies; moreover, PROP21 applies whenever D is odd, resulting in the following table. N mod 12 s B D mod 12 e propositions(s)3 12 0:1 1 , , , D ≡ B .The entry D ≡ D ≡ B , the lower exponent is available for preciselythe same quadratic forms. In the following, we will present only tables thathave been reduced accordingly.However, the previous table does not yet contain the full truth. A linein the table means that if there is a solution to B ≡ D + 4 rN (mod 4 RN )with D in the given residue class D modulo 12, then w eN yields a classinvariant. Examining this equation modulo the part of 4 RN that containsonly 2 and 3 yields further restrictions. Write N = N N ′ such that theonly primes dividing N are 2 and 3 and gcd( N ′ ,
6) = 1. Then we need toensure that D + 4 rN ≡ D is a square modulo N ′ ; this is guaranteed byLemma 4.1, since otherwise we would not even consider the full power w sN .We furthermore need to examine under which conditions D + 4 N rN ′ is a square modulo 4 RN and D ≡ D (mod 12) . Concerning the second to last line, for instance, the condition becomes D + 12 N D ≡ . Thus, D + 12 N ≡ N mod 3, only one valueof D (mod 36) remains.For N = 3, for instance, or more generally N ≡ B D mod 36 e ,
12 120:1 9 ,
21 61:3 24 42:3 4 , ,
28 41:3 33 22:3 1 , ,
25 20
A. Enge and F. Morain
To illustrate this, we give the following table of examples:
N f − D H D [ f ]3 w X − X + 7293 w X − ω − X − w X − w X − − ω w X − ω − w X − ω N ≡ s = 3 for squares in that family (for instance, N = 3 n ) and maythen reach w N . Otherwise, s = 6, and the only possible smaller exponent is2. N s B D e propositions(s)9 mod 12, = (cid:3) = (cid:3) (cid:3) (cid:3) N = 21, the second for N = 9. For theformer, we find B D mod 252 e N f − D H D [ f ]21 w X + (108 + 102 ω ) X − ω w X + ω + 421 w X + ( − ω ) X − − ω For N = 9, we get: B D mod 108 e N f − D H D [ f ]9 w X − X + 279 w X − ω − w X − − ω eneralised Weber Functions N even A look at § N iseven. We distinguish the cases λ = 1 (in which N cannot be a square) and λ > N a square or not. λ = 1Three values are concerned, namely N mod 12 ∈ { , , } . We have s = 24for N mod 12 ∈ { , } , whereas s = 8 for N ≡
10 (mod 12). N mod 12 s B D e proposition(s)2 24 1:2 — 12 PROP20a2 24 0:2 1 mod 8 12 PROP20b2 24 1:3 1 mod 3 8 PROP322 24 2:3 2 mod 3 8 PROP322 24 1:4 1, 4 mod 8; 0 mod 16 6 PROP442 24 3:4 1 mod 8; 8 mod 16 6 PROP412ab2 24 1:2 ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ The case N = 2 corresponds to Weber’s classical functions. We presentthe case N = 6 in more detail, illustrating the complexity of the process.2 A. Enge and F. Morain
B D mod 288 e ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ N f − D H D [ f ]6 w X + 1866246 w X − ω X + 12597126 w X + (432 ω − X + 207366 w X + (112 + 64 ω ) X − − ω w X − X − w X + (3 ω − X + (486 ω + 108) X + ( − ω + 9072) X + 6561 ω − w X + (324 + 60 ω ) X + 14688 X + (69984 − ω ) X + 466566 w X + (144 − ω ) X + 2196 X + (5184 + 432 ω ) X + 12966 w X + ( −
32 + 4 ω ) X + ( − − ω ) X + (1152 − ω ) X −
752 + 256 ω w X − ω X − w X + ( − ω − X + 3 ω − w X + ( − − ω ) X + (132 + 6 ω ) X − X − X − X + (4752 − ω ) X + ( − ω ) X + 12966 w X + 2 + ω w X − X + ( − ω + 15) X + ω − w X + (2 − ω ) X − − ω λ > N mod 12, namely, 0, 4 and 8, for which s = 24, 8, and 24, respectively. The cases N ≡ eneralised Weber Functions N ≡ N s B D e proposition(s)4 mod 12 8 1:2 — 4 PROP20a4 mod 12 8 1:2 1 mod 8 4 PROP20b4 mod 12 8 1:4 1 mod 8 2 PROP44a4 mod 12 8 1:4 0 mod 16 2 PROP44b4 mod 12 8 3:4 1 mod 8 2 PROP412a4 mod 12 8 3:4 4 mod 8 2 PROP412c4 mod 12, = (cid:3) (cid:3) (cid:3) (cid:3) N ≡ N mod 12 s B D e proposition(s)8 24 1:2 — 12 PROP20a8 24 1:2 1 mod 8 12 PROP20b8 24 1:4 1 mod 8 6 PROP44a8 24 1:4 0 mod 16 6 PROP44b8 24 3:4 1 mod 8 6 PROP412a8 24 3:4 4 mod 8 6 PROP412c8 24 1:3 1 mod 3 8 PROP328 24 2:3 2 mod 3 8 PROP328 24 1:2 ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ N ≡ A. Enge and F. Morain
N s B D e proposition(s)12 24 1:2 — 12 PROP20a12 24 1:2 1 mod 8 12 PROP20b12 24 1:4 1 mod 8 6 PROP44a12 24 1:4 0 mod 16 6 PROP44b12 24 3:4 1 mod 8 6 PROP412a12 24 3:4 4 mod 8 6 PROP412c12 24 1:3 0 mod 3 8 PROP30a12 24 2:3 1 mod 3 8 PROP30b12 24 1:2 ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ N = 4, these results translate as follows: B D mod 128 e ≡ ≡
20 (mod 32) 21:4 64 23:8 ≡ ≡
48 (mod 64) 1 eneralised Weber Functions N f − D H D [ f ]4 w X − ω + 324 w X − ω X − w X − ω w X + ( − − ω ) X + 4 ω w X − ω w X + ( − − ω ) X + 6 ω X + (8 − ω ) X − w X − − ω The precise results for N = 16 are the following: B D mod 512 e ≡
16 (mod 128) 81:2 64, 128, 320, 384 43:4 ≡ ≡ ≡
80 (mod 128) 1
N f − D H D [ f ]16 w X + (12288 ω − X − ω − w X + (128 + 192 ω ) X + 6656 ω X + ( − ω ) X − w X + 2 ω − w X + (16 − ω ) X + ( −
288 + 288 ω ) X + (768 − ω ) X − ω w X − ω − w X − X + 416 w X + 4 X + 4 The argumentation of the proof of Theorem 4.4 carries over to the lowerpowers of w N and shows that the class polynomial is real whenever forsome form [ A, B, C ] in the se N -system the inverse form [ A, − B, C ] satisfiesthe congruence constraints of the system as well. This is precisely the casewhen B is divisible by se N . In particular, this implies that N | D , andinspection of the previous results proves the following theorem. Theorem 6.1.
Under the general assumptions of §
6, the characteristic poly-nomial of w eN ( α ) is real whenever N | D and se N | B . For e < s , this ispossible only in the following cases: A. Enge and F. Morain (a) N odd: N s B D e : : : : : , = (cid:3) : , = (cid:3) : (b) || N and | D (b1) se is even and || D (b2) se = 3 (c) | N and | D Proof.
We again start from B ≡ D +4 rN (mod 4 RN ), where in fact R = se is a non-trivial divisor of 24. Then the hypotheses of the theorem translateas B = N RB ′ and D = N D ′ , so that(6.1) N R B ′ ≡ D ′ + 4 r (mod 4 R ) . This immediately implies D ′ ≡ − r (mod 3) if 3 | R (6.2) 4 | D ′ if 2 | R (6.3)(a) The assertions are a direct consequence of (6.2) and (6.3), togetherwith the tables in § N is even, from N | D we immediately have 4 | D .If R is even, then moreover (6.3) yields that 8 | D . Going throughthe table in § r is odd, and (6.1) implies that D ′ ≡ − r ≡ || D .(c) If 4 | N , then (6.1) shows that 4 | D ′ , whence 16 | D .We end this section with related results concerning the functions √ D w eN .Since √ D ∈ O , a singular value √ D w eN ( α ) is a class invariant if and onlyif w eN ( α ) is, and integrality of the class polynomial coefficients carries over.In some cases, however, the additional factor √ D may lead to rational classpolynomials. eneralised Weber Functions Lemma 6.2.
Let N , α = − B + √ D and e be such that se is even, s e N | B and se N ∤ B . Then w N ( α ) e ∈ i R .Proof. Write w N = f f , where f = q − N − N and f is a power series in q /N .Notice that if N | B , then q /N ( α ) = e πiα/N ∈ R . So w eN ( α ) is real up tothe factor f ( α ) e , which itself is real up to the factor e πi · s ( N − · eBsN . This isan odd power of i under the hypotheses of the lemma; N s ( N − is odd. Lemma 6.3.
Let f be a modular function and α ∈ O such that f ( α ) is aclass invariant and a real number. Then H D [ f ] ∈ Q [ X ] .Proof. This is a trivial application of Galois theory. The complex conjugate f ( α ) is a root of H D [ f ]. Since f ( α ) = f ( α ), this implies that H D [ f ] is amultiple of the minimal polynomial H D [ f ] of f ( α ), so both are the same,and H D [ f ] has coefficients in K ∩ R = Q .Combining the lemmata yields the following result. Theorem 6.4.
Under the general assumptions of §
6, the characteristic poly-nomial of √ D w eN ( α ) is real whenever N , N | D , se is even, s e N | B and se N ∤ B . For instance, we may apply this theorem to the cases N ∈ { , , , } , inwhich Propositions 5.3 or 5.5 hold: N D B e ± ±
12 63 9 mod 12 ± ± ± H − [ √− w ] = X + 720 X + 576 ,H − [ w ]( X ) = X + 6 √− X − ,H − [ √− w ]( X ) = X − X + 1377 . Let f be a modular function yielding class invariants and Φ[ f ]( F, J ) theassociated modular polynomial such that Φ[ f ]( f, j ) = 0. It is shown in [7]8 A. Enge and F. Morain that asymptotically for | D | → ∞ , the height of the class invariant f ( α ) is c ( f ) times the height of j ( α ), where(7.1) c ( f ) = deg J (Φ[ f ])deg F (Φ[ f ])depends only on f . It is then clear that c ( f r ) = rc ( f ) for rational r . Soto obtain c ( w eN ), it is sufficient to determine the degrees of the modularpolynomials of the full power w sN , where s is as defined in Theorem 4.3. w sN Since w sN is modular for Γ ( N ) by Theorem 3.2, we haveΦ c N := Φ[ w sN ] = Y M ∈ Γ ( N ) \ Γ ( F − w sN ◦ M ) . So deg F Φ c N = ψ ( N ) = N Q p prime, p | N (cid:16) p (cid:17) . The degree in J is obtainedby examining the q -developments of the conjugates w sN ◦ M of w sN . Proposition 7.1 (Oesterl´e) . The cosets of Γ ( N ) \ Γ can be split into thefollowing three families: T ν = (cid:18) ν (cid:19) , ν < N,S = (cid:18) −
11 0 (cid:19) ,M k,k ′ = (cid:18) k kk ′ − k ′ (cid:19) with < k < N , gcd( k, N ) > and k ′ < µ ( k ) where µ ( k ) is thesmallest integer for which gcd( µ ( k ) k − , N ) = 1 . Using (3.2), we find
Proposition 7.2. ( w sN ◦ T )( z ) = w N ( z + ν ) s , ν < N, ( w sN ◦ S )( z ) = (cid:18) √ N η ( N z ) η ( z ) (cid:19) s , ( w sN ◦ M k,k ′ )( z ) = ζ k,k ′ p δ k η (cid:16) δ k z + c k,k ′ N/δ k (cid:17) η ( z ) s , where δ k = gcd( k, N ) , ζ k,k ′ is a 24-th root of unity and c k,k ′ is a rationalinteger. eneralised Weber Functions w sN have inte-gral and that w sN and w sN ◦ S have rational q -expansions. The q -expansionprinciple now implies that Φ c N ∈ Z [ F, J ], cf. [6, § Theorem 7.3. deg J Φ cN = s
24 ( N − S ( N )) where (7.2) S ( N ) = X k :1 Consider Φ c N as a polynomial in F with coefficients in Z [ J ]. Followingthe same reasoning as in [9], we see that the coefficient of highest degree in J is obtained when all conjugates are multiplied together whose q -expansionshave strictly negative order; since the q -expansion of j starts with q − , thedegree in J is then the opposite of this order. The w N ( z + ν ) s have negativeorder − s ( N − N and contribute a total of − s ( N − . The function w sN ◦ S haspositive order. The conjugates coming from M k,k ′ have order s (cid:16) δ k N − (cid:17) ,which is negative whenever δ k < √ N .Let us note a list of useful corollaries. Proposition 7.4. When N = ℓ n for a prime ℓ and n > , then S ( N ) = (cid:26) ( ℓ m − if n = 2 m + 1 , ( ℓ m − ℓ m +1 − if n = 2 m + 2 . Proof. The k occurring in (7.2) are the ( k + ℓk ) ℓ r with 1 k < ℓ ,1 r m and 0 k < ℓ n − r − (so that k < N ); they yield δ k = ℓ r and µ ( k ) = 1. Hence, S ( N ) = m X r =1 ( ℓ − ℓ n − r − (cid:0) − ℓ r − n (cid:1) = (cid:0) ℓ n − m − − (cid:1) ( ℓ m − . Corollary 7.5. When N is prime or the square of a prime, then deg J Φ c N = s ( N − . Proposition 7.6. When N = p p for two primes p > p , then S ( N ) = p − p .Proof. The case p = p is already proven. So it remains to consider p < √ N < p , and the integers k contributing to S ( N ) are the ˜ kp with 1 ˜ k < p . Among these, only one is such that gcd( k − , N ) = 1, namely the k with ˜ k ≡ /p (mod p ); for this one, µ ( k ) = 2. Therefore S ( N ) = (cid:0) ( p − · · (cid:1) (cid:18) − p N (cid:19) = p − p . A. Enge and F. Morain With some more effort, the constant coefficient Φ c N (0 , J ) could be ob-tained as the product of all conjugates, but it is not needed in the following. Knowing the degrees of the modular polynomials, we can compare classinvariants obtained from w eN among themselves and with others using (7.1).Of special interest is the infinite family of invariants obtained in [8] fromthe double η -quotients w σp ,p ( z ) = η (cid:16) zp (cid:17) η (cid:16) zp (cid:17) η (cid:16) zp p (cid:17) η ( z ) σ , where p , p are (not necessarily distinct) primes and σ = , ( p − p − .These functions yield class invariants whenever (cid:0) Dp (cid:1) = (cid:0) Dp (cid:1) = 1, and insome cases when (cid:0) Dp (cid:1) = 0 or (cid:0) Dp (cid:1) = 0, see [8, Cor. 3.1]. The degrees of theirmodular polynomials have been worked out in [9, Th. 9], and we summarisethe results in the following table, in which ℓ and p = p are supposed to beprime numbers. f c ( f ) deg J Φ c N w eℓ e ( ℓ − ℓ +1) s ( ℓ − w eℓ e ( ℓ − ℓ ℓ − if ℓ > w ep p e ( p − p +1) s ( p − p +1)24 w eN e ( N − S ( N ))24 ψ ( N ) s ( N − S ( N ))24 w eℓ,ℓ e ( ℓ − ℓ ( ℓ +1) σ ( ℓ − w ep ,p e ( p − p − p +1)( p +1) σ ( p − p − Notice that asymptotically for ℓ or p , p → ∞ , the factors c ( f ) tendto e/ for w eℓ (here, e is necessarily even), e for the double η quotientsand e for w eℓ . For any discriminant D , there are suitable choices of primesin arithmetic progressions modulo D such that e/ e = 1 arereachable, and c ( f ) may become arbitrarily close to resp. . However,at the same time, the degrees of Φ c N in F and J tend to infinity, which maybe undesirable in complex multiplication applications where the modularpolynomial needs to be factored over a finite field.In Table 7.2, we list in decreasing order of attractiveness the functions f together with the factors 1 /c ( f ) they allow to gain in height comparedto j and with the degree of the modular polynomial in J , thus completingthe tables of [7] and [10, p. 21]. We limit ourselves to functions gaining afactor of at least 13 and with degree in J at most 20. The function w is in eneralised Weber Functions f , and leads to the same height as the other twoWeber functions f and f . Notice that, as indicated by the explicit formulæ,transformation levels divisible by 2 or 3 (or, in general, small primes) tendto yield smaller class invariants.Table 1: Comparison of class invariants: height factor and degree in J w , > w , > w , , > w , / , > w , = w , > w , > w , > w , , = w , > w , > w / , > w , > w / , > w , / , = w / , > w , / , > w , , = w , = w , = w , = w , > w , , = w , , > w / , > w , , > w , / , > w , , = w , , > w , / , > w , / , > w , , = w , = w , = w , = w , = w , > w , / , > w , , = w , = w , = w , = w , = w , = w , = w , = w , = w , > w / , > w , / , > w / , = w / , = w / , = w / , = w / , = w / , > w , = w , = w , > w / , > w , / , = w / , = w / , = w / , > w , / , = w · / , = w / , = w / , > w / , The presented results concern singular values of powers of w N as class in-variants. It is possible to obtain smaller invariants by authorising 24-th rootsof unity to enter the game. This was already done by Weber for N = 2 (theclassical f -functions) and by Gee in [16] for N = 3. For instance, ζ w is aninvariant for D = − 40, leading to the minimal polynomial X + ( − ω ) X + 3 − ω. Similarly, when N is not a square and e is odd, then w eN ◦ S has a q -expansion that is rational up to a factor √ N , so that Theorems 2.3 and 2.42 A. Enge and F. Morain are not applicable any more. Nevertheless, w eN may yield class invariants;this is well-known for Weber’s original functions in certain cases. Acknowledgements. The second author wants to thank the Universityof Waterloo for its hospitality during his sabbatical leave; he also wants toacknowledge the warm and studious atmosphere of the Asahi Judo club ofKitchener where large parts of the results were proven. The first author waspartially funded by ERC Starting Grant ANTICS 278537. References [1] A. O. L. Atkin and F. Morain. Elliptic curves and primality proving. Mathematics of Computation , 61(203):29–68, 1993.[2] B. J. Birch. Weber’s class invariants. Mathematika , 16:283–294, 1969.[3] D. A. Buell. Binary quadratic forms (Classical theory and moderncomputations) . Springer-Verlag, 1989.[4] H. Cohn. A classical invitation to algebraic numbers and class fields .Universitext. Springer-Verlag, 1978.[5] R. Dedekind. Erl¨auterungen zu den vorstehenden Fragmenten. InR. Dedekind and H. Weber, editors, Bernhard Riemann’s gesammeltemathematische Werke und wissenschaftlicher Nachlaß , pages 438–447.Teubner, Leipzig, 1876.[6] M. Deuring. Die Klassenk¨orper der komplexen Multiplikation. In Enzyklop¨adie der mathematischen Wissenschaften mit Einschluß ihrerAnwendungen , volume I 2, Heft 10, Teil II. Teubner, Stuttgart, 1958.[7] A. Enge and F. Morain. Comparing invariants for class fields of imag-inary quadratic fields. In C. Fieker and D. R. Kohel, editors, Algorith-mic Number Theory , volume 2369 of Lecture Notes in Comput. Sci. ,pages 252–266. Springer-Verlag, 2002. 5th International Symposium,ANTS-V, Sydney, Australia, July 2002, Proceedings.[8] A. Enge and R. Schertz. Constructing elliptic curves over finite fieldsusing double eta-quotients. Journal de Th´eorie des Nombres de Bor-deaux , 16:555–568, 2004.[9] A. Enge and R. Schertz. Modular curves of composite level. ActaArith. , 181(2):129–141, 2005.[10] Andreas Enge. Courbes alg´ebriques et cryptologie . Habilitation `a dirigerdes recherches, Universit´e Denis Diderot, Paris 7, 2007. eneralised Weber Functions Mathematics of Computation ,78(266):1089–1107, 2009.[12] Andreas Enge and Reinhard Schertz. Singular values of multiple eta-quotients for ramified primes. Technical Report 768375, HAL-INRIA,2012. To appear in LMS Journal of Computation and Mathematics.[13] David Freemann, Michael Scott, and Edlyn Teske. A taxonomy ofpairing-friendly elliptic curves. Journal of Cryptology , 23(2):224–280,2010.[14] A. Gee. Class invariants by Shimura’s reciprocity law. J. Th´eor. Nom-bres Bordeaux , 11:45–72, 1999.[15] A. Gee and P. Stevenhagen. Generating class fields using Shimurareciprocity. In J. P. Buhler, editor, Algorithmic Number Theory , volume1423 of Lecture Notes in Comput. Sci. , pages 441–453. Springer-Verlag,1998. Third International Symposium, ANTS-III, Portland, Oregon,June 1998, Proceedings.[16] Alice Chia Ping Gee. Class fields by Shimura reciprocity . Proefschrift,Universiteit Leiden, 2001.[17] F. Hajir. Elliptic units of cyclic unramified extensions of complexquadratic fields. Acta Arith. , LXIV:69–85, 1993.[18] F. Hajir. Unramified elliptic units . Phd in mathematics, PrincetonUniversity, June 1993.[19] F. Hajir and F. R. Villegas. Explicit elliptic units, I. Duke Math. J. ,90(3):495–521, 1997.[20] C. Meyer. Bemerkungen zum Satz von Heegner–Stark ¨uber die ima-gin¨ar-quadratischen Zahlk¨orper mit der Klassenzahl Eins. Journal f¨urdie reine und angewandte Mathematik , 242:179–214, 1970.[21] Fran¸cois Morain. Computing the cardinality of CM elliptic curvesusing torsion points. Journal de Th´eorie des Nombres de Bordeaux ,19(3):663–681, 2007.[22] R. Schertz. Weber’s class invariants revisited. J. Th´eor. Nombres Bor-deaux , 14(1):325–343, 2002.[23] Reinhard Schertz. Die singul¨aren Werte der Weberschen Funktionen f , f , f , γ , γ . Journal f¨ur die reine und angewandte Mathematik ,286/287:46–74, 1976.[24] Reinhard Schertz. Complex Multiplikation . Cambridge UniversityPress, Cambridge, 2009.4 A. Enge and F. Morain [25] Goro Shimura. Introduction to the Arithmetic Theory of AutomorphicFunctions . Iwanami Shoten and Princeton University Press, 1971.[26] H. M. Stark. On the ”gap” in a theorem of Heegner. J. Number Theory ,1:16–27, 1969.[27] H. Weber.