Table Of Content.
CLASS POLYNOMIALS FOR NONHOLOMORPHIC
MODULAR FUNCTIONS
JAN HENDRIK BRUINIER, KEN ONO, AND ANDREW V. SUTHERLAND
3 Abstract. We give algorithms for computing the singular moduli of suitable nonholomor-
1 phic modular functions F(z). By combining the theory of isogeny volcanoes with a beauti-
0 ∗
ful observation of Masser concerning the nonholomorphic Eisenstein series E (z), we obtain
2 2
CRT-based algorithms that compute the class polynomials H (F;x), whose roots are the
D
n discriminant D singular moduli for F(z). By applying these results to a specific weak Maass
a
form F (z), we obtain a CRT-based algorithm for computing partition class polynomials, a
J p
sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the
3
2 expected running time of this algorithm is O(n5/2+o(1)). Key to these results is a fast CRT-
basedalgorithmforcomputingthe classicalmodularpolynomialΦm(X,Y)thatweobtainby
] extending the isogeny volcano approach previously developed for prime values of m.
T
N
.
h
t
a 1. Introduction and Statement of results
m
[ As usual, we let
1 3
1+240 ∞ d3qn
v
n=1 d|n
2 (1.1) j(z) := (cid:16) (cid:17) = q−1 +744+196884q+21493760q2+
7 q ∞P(1 Pqn)24 ···
n=1 −
6
Q
5 be Klein’s classical elliptic modular function on SL (Z) (q := e2πiz throughout). The values
2
.
1 of j(z) at imaginary quadratic arguments in the upper-half of the complex plane are known
0 as singular moduli. Two examples are the Galois conjugates
3
1 1+√ 15 191025 85995√5 1+√ 15 191025+85995√5
: j − = − − and j − = − .
v
(cid:18) 2 (cid:19) 2 (cid:18) 4 (cid:19) 2
i
X
These numbers play an important role in algebraic number theory. Indeed, they are algebraic
r
a integers that generate ring class field extensions of imaginary quadratic fields, and they are
the j-invariants of elliptic curves with complex multiplication [3, 9, 10, 11].
The problem of computing singular moduli has a long history that dates back to the works
of Kronecker, and is highlighted by famous calculations by Berwick [2] and Weber [35]. His-
torically, these numbers have been difficult to compute. More recently, Gross and Zagier [21]
determined the prime factorization of the absolute norm of suitable differences of singular
moduli (further work in this direction has been carried out by Dorman [12, 13]), and Zagier
[36] identified the algebraic traces of singular moduli as coefficients of half-integral weight
modular forms.
ThefirstauthorwassupportedbyDFGgrantBR-2163/2-2. ThesecondauthorthanksthesupportofNSF
grant DMS -1157289 and the Asa Griggs Candler Fund. The third author received support from NSF grant
DMS-1115455.
1
2 JAN HENDRIKBRUINIER,KEN ONO,ANDANDREWV. SUTHERLAND
Here we consider the problem of computing the minimal polynomials of singular moduli,
the so-called Hilbert class polynomials. This problem has been the subject of much recent
study. For example, Belding, Bro¨ker, Enge, and Lauter [1], and the third author [33], have
provided efficient methods of computation that are based on the theory of elliptic curves with
complex multiplication (CM). The basic approach in that work is simple. One uses theoretical
facts about elliptic curves with CM to quickly compute the reductions of these polynomials
modulo a set of suitable primes p, and one then compiles these reductions via the Chinese
Remainder Theorem (CRT) to obtain the exact polynomials. The primes p are chosen in a
way that facilitates the computation, and in particular, they split completely in the ring class
field K of the imaginary quadratic order associated to the singular moduli whose minimal
O
O
polynomial one wishes to compute. Under the Generalized Riemann Hypothesis (GRH), this
algorithm computes the discriminant D Hilbert class polynomial with an expected running
time of O( D ).1 In [6], the third author, together with Bro¨ker and Lauter, further developed
| |
these ideas to compute modular polynomials Φ using the theory of isogeny volcanoes.
ℓ
e
We extend these results to the setting of nonholomorphic modular functions such as
E (z) 7j(z) 6912
(1.2) γ(z) := 4 E∗(z) − ,
6E (z)j(z) · 2 − 6j(z)(j(z) 1728)
6
−
where
∞
3
(1.3) E∗(z) := 1 24 dqn
2 − πIm(z) −
Xn=1Xd|n
is the weight 2 nonholomorphic modular Eisenstein series, and where
∞ ∞
E (z) := 1+240 d3qn and E (z) := 1 504 d5qn
4 6
−
Xn=1Xd|n Xn=1Xd|n
are the usual weight 4 and weight 6 modular Eisenstein series.
Remark. The function γ(z) plays an important role in this paper. We shall make use of an
observation of Masser [27] which gives a description of its singular moduli in terms of the
coefficients of certain representations of the classical modular polynomials.
We first recall the setting of Heegner points on modular curves (see [20]). Let N > 1,
and let D < 0 be a quadratic discriminant coprime to N. The group Γ (N) acts on the
0
discriminant D positive definite integral binary quadratic forms
Q(X,Y) = [a,b,c] := aX2 +bXY +cY2
with N a. This action preserves b (mod 2N). Therefore, if β2 D (mod 4N), then it
| ≡
is natural to consider , the set of those discriminant D forms Q = [a,b,c] for which
N,D,β
Q
0 < a 0 (mod N) and b β (mod 2N), and we may also consider the subset prim
≡ ≡ QN,D,β
obtained by restricting to primitive forms. The number of Γ (N) equivalence classes in
0 N,D,β
Q
is the Hurwitz-Kronecker class number H(D), and the natural map defines a bijection
/Γ (N) /SL (Z),
N,D,β 0 D 2
Q −→ Q
1We use the “soft” asymptotic notation O(n) to denote bounds of the form O(nlogcn).
e
CLASS POLYNOMIALS FOR NONHOLOMORPHIC MODULAR FUNCTIONS 3
where isthe set ofdiscriminant D positive definite integral binary quadratic forms(see the
D
Q
proposition on p. 505 of [20]). This bijection also holds when restricting to primitive forms,
in which case the number of Γ (N) equivalence classes in prim , and the number of SL (Z)
0 QN,D,β 2
equivalence classes in prim, is given by the class number h(D).
QD
For modular functions F(z) on SL (Z), our goal is to calculate the class polynomial
2
(1.4) H (F;x) := x F(α ) ,
D Q
−
Q∈QpDrYim/SL2(Z)(cid:0) (cid:1)
whereα HisarootofQ(x,1) = 0. FormodularfunctionsF(z)onΓ (N)anddiscriminants
Q 0
∈
D < 0 coprime to N, our goal is to calculate the class polynomial
(1.5) H (F;x) := x F(α ) .
D,β Q
−
Q∈QpNr,iYDm,β/Γ0(N)(cid:0) (cid:1)
We first consider the special case of the SL (Z) nonholomorphic modular function γ(z),
2
defined by (1.2), and we compute the Q-rational polynomials H (γ;x).
D
Theorem 1.1. For discriminants D < 4 that are not of the form D = 3d2, Algorithm 1
− −
(see 3.1) computes H (γ;x). Under the GRH, its expected running time is O( D 7/2) and it
D
§ | |
uses O( D 2) space.
| | e
e
Remark. The discussion on p. 118 of [27], makes it clear how to modify Algorithm 1 to handle
discriminants of the form D = 3d2. We expect that the bound on its expected running time
−
can be improved to O( D 5/2) using tighter bounds on the size of the coefficients of H (γ;x).
D
| |
e
A key building block of Algorithm 1 is a new algorithm to compute the classical modular
polynomial Φ (X,Y), which parameterizes pairs of elliptic curves related by a cyclic isogeny
m
of degree m. Here we extend the iosgeny volcano approach that was introduced in [6] to
compute Φ for prime m so that we can now efficiently handle all values of m. The result is
m
Algorithm 1.1 (see 3.1), which, under the GRH, computes Φ in O˜(m3) time. For suitable
m
§
primes p it can compute Φ modulo p in O˜(m2) time (and space), which is crucial to the
m
efficient implementation of Algorithm 1.
Example. We have used Algorithm 1 to compute H (γ,x) for D > 20000. Some small
D
−
examples are listed below.
4 JAN HENDRIKBRUINIER,KEN ONO,ANDANDREWV. SUTHERLAND
D H (γ,x)
D
3 x 23
− − 211·33
4 x
−
7 x 181
− − 36·53·7
8 x+ 61
− 26·53·72
11 x 289
− − 214·72·11
12 x+ 67
− 23·33·53·112
15 x2 + 313 x 1045769
− 34·5·113 · − 38·53·74·115
16 x+ 179
− 36·72·113
19 x 275
− − 214·36·19
20 x2 43925 x 2307859
− − 26·113·192 · − 218·53·115·192
23 x3 + 8123835989 x2 + 6062055706222 x 346923509992369
− 53·72·113·173·192·23 · 56·74·114·173·192·23 · − 56·74·114·173·192·23
We extend Algorithm 1 to compute class polynomials for a large class of nonholomorphic
modular functions. This class includes, for example, the SL (Z)-function
2
E∗(z)E (z)E (z)+3E (z)3 +2E (z)
K(z) := 288 2 4 6 4 6
· E (z)3 E (z)2
4 6
−
considered by Zagier (see 9 of [36]) in his famous paper on traces of singular moduli. More
§
generally, it includes suitable modular functions F(z) of the form
F(z) := ∂ ∂ ∂ (F),
−2 −4 2−2k
◦ ···
where F(z) is a weight 2 2k weakly holomorphic modular form on Γ (N) whose Fourier
0
−
expansions at cusps are algebraic. Here the differential operator ∂ , which maps weight h
h
modular forms to weight h+2 modular forms, is defined by
1 ∂ h
(1.6) ∂ := .
h
2πi · ∂z − 4πIm(z)
We consider a specific class of such modular functions. Let be the imaginary quadratic
O
order with discriminant D, ring class field K , and fraction field K = Q(√D). Let c and c
O 1 2
denote fixed positive integers. Let F(z) be a modular function (i.e. weight 0) for Γ (N) that
0
CLASS POLYNOMIALS FOR NONHOLOMORPHIC MODULAR FUNCTIONS 5
can be written in the form
F(z) = A (z)γ(z)n,
n
X
where each A Q(j) is a rational function of j(z). The function F(z) is said to be good for
n
∈
a discriminant D < 0 coprime to N if it satisfies the following:
(1) Each A (α ) lies in K for all Q prim/SL (Z).
n Q O ∈ QD 2
(2) The polynomial c D c2hH (F;x) has integer coefficients.
1 D
| |
Remark. The class of good modular functions includes many nonholomorphic modular func-
tions, such as γ(z), and it includes meromorphic modular functions which may have poles in
the upper half plane (poles at CM points are excluded by condition (1)).
For good modular functions, we obtain the following general result.
Theorem 1.2. For discriminants D < 4 not of the form D = 3d2, Algorithm 2 (see 3.2)
− − §
computes class polynomials for good modular functions F(z). Under the GRH, its expected
running time is O( D 5/2), and it uses O( D 2) space.
| | | |
In fact, Algoritehm 2 can be readily aedapted to treat modular functions of the form F(z) =
A (z)γ(z)n where the coefficient functions A (z) do not necessarily lie in Q(j), using the
n n
tPechniques developed by Enge and the third author in [16]. As an example, we apply Algo-
rithm 2 to obtain a CRT-based algorithm for computing the “partition polynomials” defined
by the first two authors in [8]. These are essentially the class polynomials of the Γ (6) non-
0
holomorphic modular function
1 5 29
(1.7) F (z) := ∂ (P(z)) = 1 q−1 + + 29+ q +...,
p −2
− (cid:18) − 2πIm(z)(cid:19) πIm(z) (cid:18) 2πIm(z)(cid:19)
where P(z) is the weight 2 weakly holomorphic modular form
−
1 E (z) 2E (2z) 3E (3z)+6E (6z)
P(z) := 2 − 2 − 2 2 = q−1 10 29q+....
2 · η(z)2η(2z)2η(3z)2η(6z)2 − −
These polynomials are defined as
(1.8) Hpart(x) := (x P(α )).
n − Q
Y
Q∈Q6,1−24n,1
In contrast with (1.4) and (1.5), we stress that the roots of these polynomials include singular
moduli for imprimitive forms (if any). The interest in these polynomials arises from the fact
that (see Theorem 1.1 of [8])
(1.9) Hpart(x) = xh(1−24n) (24n 1)p(n)xh(1−24n)−1 +...,
n − −
where p(n) is the usual partition function. Since the roots P(α ) are algebraic numbers that
Q
lie in the usual discriminant 1 24n ring class field, we have the following finite algebraic
−
formula
1
p(n) = P(α ).
Q
24n 1
− Q∈QX6,1−24n,1
Remark. By the work of the first two authors [8], combined with recent results by Larson and
Rolen [26], it is known that each (24n 1)P(α ) is an algebraic integer.
Q
−
6 JAN HENDRIKBRUINIER,KEN ONO,ANDANDREWV. SUTHERLAND
As a consequence of Theorem 1.2, we obtain the following result.
Theorem 1.3. For all positive integers n, Algorithm 3 (see 3.3) computes Hpart(x). Under
§ n
the GRH, its expected running time is O(n5/2) and it uses O(n2) space.
Remark. For the simpler task of compueting individual valuees of p(n), an efficient implemen-
tation of Rademacher’s formula such as the one given in [23] is both asymptotically and
practically faster than Algorithm 3.
Example. We have used Algorithm 3 to compute Hpart(x) for n 750. Some small examples
n ≤
are listed below.
n (24n 1)p(n) Hpart(x)
− n
1 23 x3 23x2 + 3592x 419
− 23 −
2 94 x5 94x4 + 169659x3 65838x2 + 1092873176x+ 1454023
− 47 − 472 47
3 213 x7 213x6 + 1312544x5 723721x4+ 44648582886x3
− 71 − 712
+9188934683x2 + 166629520876208x+ 2791651635293
71 713 712
4 475 x8 475x7 + 9032603x6 9455070x5+ 3949512899743x4
− 95 − 952
97215753021x3 + 9776785708507683x2
− 19 953
53144327916296x 134884469547631
− 192 − 54·19
This paper is organized as follows. In 2 we recall essential facts about elliptic curves with
§
complex multiplication, and singular moduli for modular forms and certain nonholomorphic
modular functions. In 3 we use these results to derive our algorithms. In 4 we conclude
§ §
with a detailed example of the execution of Algorithm 3 for n = 1 and n = 24.
2. nuts and bolts
Webeginwithsomepreliminariesonellipticcurveswithcomplexmultiplicationandsingular
moduli for suitable modular functions.
2.1. Elliptic curves with complex multiplication. We recall some standard facts from
the theoryof complex multiplication, referring to [9, 25, 30] forproofsandfurther background.
Let beanimaginaryquadraticorder, identified byitsdiscriminant D. Thej-invariantofthe
O
lattice is an algebraic integer whose minimal polynomial is the Hilbert class polynomial H .
D
O
If a is an invertible -ideal (including a = ), then the torus C/a corresponds to an elliptic
O O
curve E/C with complex multiplication (CM) by , meaning that its endomorphism ring
O
End(E) is isomorphic to , and every such curve arises in this fashion. Equivalent ideals
O
CLASS POLYNOMIALS FOR NONHOLOMORPHIC MODULAR FUNCTIONS 7
yield isomorphic elliptic curves, and this gives a bijection between the ideal class group cl( )
O
and the set
(2.1) Ell (C) := j(E/C): End(E) = ,
O ∼
{ O}
the j-invariants of the elliptic curves defined over C with CM by . We then have
O
(2.2) H (x) := (x j ) = H (j;x).
D i D
−
Y
ji∈EllO(C)
The splitting fieldof H over K = Q(√D) isthe ring class field K . It isanabelianextension
D O
whose Galois group is isomorphic to cl( ), via the Artin map.
O
This isomorphism can be made explicit via isogenies. Let E/C be an elliptic curve with CM
by and let a be an invertible -ideal. There is a uniquely determined separable isogeny
O O
whose kernel is the subgroup of points annihilated by every endomorphism in a ֒
⊂ O →
End(E). The image of this isogeny is an elliptic curve that also has CM by , and this
O
defines an action of the ideal group of on the set Ell (C). Principal ideals act trivially, and
O
O
the induced action of the class group is regular. Thus Ell (C) is a principal homogeneous
O
space, a torsor, for the finite abelian group cl( ).
O
If p is a (rational) prime that splits completely in K , equivalently, for p > 3, a prime
O
satisfying the norm equation
(2.3) 4p = t2 v2D
−
for some nonzero integers t and v, then H splits completely in F [x] and its roots form the
D p
set
(2.4) Ell (F ) := j(E/F ): End(E) = ,
O p p ∼
{ O}
Conversely, every ordinary (not supersingular) elliptic curve E/F has CM by some imaginary
p
quadratic order in which the Frobenius endomorphism corresponds to an element of norm p
O
and trace t.
2.2. Modular polynomials via isogeny volcanoes. For each positive integer m, the clas-
sical modular polynomial Φ is the minimal polynomial of the function j(mz) over the field
m
C(j). As a polynomial in two variables, Φ Z[X,Y] is symmetric in X and Y. If E/k is
m
∈
an elliptic curve and N is prime to the characteristic of k, then the roots of Φ (j(E),Y) are
m
precisely the j-invariants of the elliptic curves that are related to E by a cyclic m-isogeny;
see [25] for these and other properties of Φ .
m
For distinct primes ℓ and p, we define the graph of ℓ-isogenies G (F ), with vertex set F
ℓ p p
and edges (j ,j ) present if and only if Φ (j ,j ) = 0. Ignoring the connected components
1 2 ℓ 1 2
of 0 and 1728, the ordinary components of G (F ) are ℓ-volcanoes [17, 24], a term we take
ℓ p
to include cycles as a special case; see [34] for further details on isogeny volcanoes. In this
paper we focus on ℓ-volcanoes of a special form, for which we can compute Φ mod p in a
ℓ
particularly efficient way, using [6, Alg. 2.1].
Let be an order in an imaginary quadratic field K with maximal order , and let ℓ be
K
O O
an odd prime not dividing [ : ]. Assume D = disc( ) < 4. Suppose p is a prime of the
K
O O O −
form 4p = t2 ℓ2v2D with p 1 mod ℓ and ℓ v; equivalently, p splits completely in the ray
− ≡ 6 |
class field of conductor ℓ for and does not split completely in the ring class field of the order
O
8 JAN HENDRIKBRUINIER,KEN ONO,ANDANDREWV. SUTHERLAND
with index ℓ2 in . Then the components of G (F ) that intersect Ell (F ) are isomorphic
ℓ p O p
O
ℓ-volcanoes with two levels: the surface, whose vertices lie in Ell (F ), and the floor, whose
O p
vertices lie in EllO′(Fp), where ′ is the order of index ℓ in . Each vertex on the surface is
O O
connected to 1+ D = 0,1 or 2 siblings on the surface, and ℓ D children on the floor. An
ℓ − ℓ
example with ℓ =(cid:0)7(cid:1)is shown below: (cid:0) (cid:1)
Provided h( ) ℓ+2, this set of ℓ-volcanoes contains enough information to completely
O ≥
determine Φ mod p. This is the basis of the algorithm in [6, Alg. 2.1] to compute Φ mod p,
ℓ ℓ
which we make use of here. Selecting a sufficiently large set of such primes p allows one to
compute Φ over Z (via the CRT), or modulo an arbitrary integer M (via the explicit CRT).
ℓ
Our requirements for the order and the primes p are summarized in the definition below.
O
Definition 2.1. Let ℓ > 2 be prime, and let c > 1 be an absolute constant independent of ℓ.
An imaginary quadratic order is said to be suitable for ℓ if ℓ [ : ] and ℓ+2 h( )
K
O 6 | O O ≤ O ≤
cℓ. A prime p is then said to be suitable for ℓ and if p 1 mod ℓ and 4p = t2 ℓ2v2disc( ),
O ≡ − O
for some t,v Z with ℓ v.
∈ 6 |
The definition of suitability above is weaker than that used in [6], but this only impacts
logarithmic factors in the running time that are hidden by our soft asymptotic notation.
2.3. Selecting primes with the GRH. In order to apply the isogeny volcano method to
compute Φ (or the polynomials H (F;x) we wish to compute), we need a sufficiently large
ℓ D
set S of suitable primes p. We deem S to be sufficiently largewhenever logp B+log2,
p∈S ≥
where B is an upper boundon the logarithmic height of the integer coeffiPcients that we wish to
computewiththeCRT.2ForΦ (X,Y) = a XiYj,wemayboundht(Φ ) := logmax a
ℓ i,j ij ℓ i,j| ij|
using P
(2.5) ht(Φ ) 6ℓlogℓ+18ℓ,
ℓ
≤
as proved in [7].
Heuristically (and in practice), it is easy to construct the set S. Given an order of
O
discriminant D suitable for ℓ, we fix v = 2 if D 1 mod 8 and v = 1 otherwise, and for
≡
increasing t 2 mod ℓ of correct parity we test whether p = (t2 v2ℓ2D)/4 is prime. We add
≡ −
each such prime to S, and stop when S is sufficiently large.
Unfortunately, we cannot prove thatthis method will find any primes, even under theGRH.
Instead, we use Algorithm 6.2 in [6], which picks an upper bound x and generates random
integers t and v in suitable intervals to obtain candidate primes p = (t2 v2ℓ2D)/4 x that
− ≤
are then tested for primality. The algorithm periodically increases x, so its expected running
time is O(B1+ǫ), even without the GRH.
Under the GRH, there are effective constants c ,c > 0 such that x c ℓ6log4ℓ guarantees
1 2 1
≥
at least c ℓ3log3ℓ suitable primes less than x, by [6, Thm. 4.4]. Asymptotically, this is far
2
more than the O(ℓ) primes we need to compute Φ . We note that S contains O(B/logB)
ℓ
primes (unconditionally), and under the GRH we have logp = O(logB +logℓ) for all p S.
∈
2When the coefficients are rational numbers that are not integers, we first clear denominators.
CLASS POLYNOMIALS FOR NONHOLOMORPHIC MODULAR FUNCTIONS 9
2.4. Modular singular moduli. The results from the previous section can be cast in terms
of the CM values of the j-function. Indeed, we have the following classical theorem (for
example, see [3, 9]) which summarizes some of the most important properties of singular
moduli for Klein’s j-function.
Theorem 2.2. Suppose that Q = ax2 + bxy + cy2 is a primitive positive definite binary
quadratic form with discriminant D = b2 4ac < 0, and let α H be the point for which
Q
− ∈
Q(α ,1) = 0. Then the following are true:
Q
(1) The singular modulus j(α ) is an algebraic integer whose minimal polynomial has
Q
degree equal to the class number h(D).
(2) The Galois orbit of j(α ) consists of the j(z)-singular moduli associated to the h(D)
Q
equivalence classes in prim/SL (Z).
QD 2
(3) If K = Q(√D), then the discriminant D singular moduli are conjugate over K. More-
over, K(j(α )) is the ring class field of the quadratic order of discriminant D; in the
Q
case that D is a fundamental discriminant, K(j(α )) is the Hilbert class field of K.
Q
Theorem 2.2 and the properties of the weight 2 nonholomorphic Eisenstein series E∗(z)
2
at CM points shall play a central role in the construction of the algorithms described in the
next section. To this end, we make use of the special nonholomorphic function γ(z) defined
in (1.2).
Masser nicely observed thatthesingular moduli forγ(z) canbecomputed using thesingular
moduli for j(z) and certain expressions for modular polynomials. Here we make this precise
for discriminants D < 4 that are not of the form D = 3d2.
− −
Remark. Masser explains how to handle discriminants D = 3d2; see p. 118 of [27].
−
To state his observation, we let be the imaginary quadratic order of discriminant D, and
O
let Q ,...,Q be a set of representatives for prim/SL (Z) cl( ), where h = h(D) is the
{ 1 h} QD 2 ≃ O
class number. To simplify notation, we use Φ to denote the classical modular polynomial
D
Φ (X,Y). For any Q = Q we may write Φ in the form
|D| i D
µ ν
(2.6) Φ (X,Y) = β X j(α ) Y j(α ) ,
D µ,ν Q Q
− −
0≤Xµ,ν≤n (cid:0) (cid:1) (cid:0) (cid:1)
where n = ψ( D ) is determined by the Dedekind ψ-function
| |
(2.7) ψ(m) := m (1+p−1),
Y
p|m
which satisfies ψ(m) = O(mloglogm); see [32]. The coefficients β = β (α ) are algebraic
µ,ν µ,ν Q
integers that lie in the ring class field K , and we have β = β (by the symmetry of Φ ).
O µ,ν ν,µ D
Masser [27, p. 118] gives the following formula for γ(α ).
Q
Lemma 2.3. Assuming the notation and hypotheses above, we have
2β (α ) β (α )
0,2 Q 1,1 Q
(2.8) γ(α ) = − .
Q
β (α )
0,1 Q
10 JAN HENDRIKBRUINIER,KEN ONO,ANDANDREWV. SUTHERLAND
Two remarks.
1) Masser proves (see [27, Lem. A2]) that β (α ) is nonzero.
0,1 Q
2) From (2.6), one finds that
β (α ) = [Y]Φ j(α ),Y +j(α ) ,
0,1 Q D Q Q
(2.9) β (α ) = [Y]Φ′ (cid:0)j(α ),Y +j(α )(cid:1),
1,1 Q D Q Q
β (α ) = [Y2]Φ(cid:0) j(α ),Y +j(α (cid:1)) ,
0,2 Q D Q Q
(cid:0) (cid:1)
where Φ′ (X,Y) = ∂ Φ (X,Y), and for any polynomial f(Y), the notation [Yk]f(Y) indi-
D ∂X D
cates the coefficient of Yk in f(Y).
3. The Algorithms
Here we apply and extend the results in 2 to derive our algorithms.
§
3.1. Algorithm 1. We now give an algorithm to compute the class polynomial H (γ;x),
D
where γ(z) is the nonholomorphic modular function defined in (1.2) and D is an imaginary
quadratic discriminant. In order to simplify the exposition as above, we shall assume D < 4
−
and that D is not of the form D = 3d2; these special discriminants are in principle no more
−
difficult to handle than the general case, but the details are more involved; see [27, p. 118].3
To make use of Lemma 2.3, we need to compute the singular moduli j(α ). These shall be
Q
obtained as the roots of the Hilbert class polynomial H (x). Thus if we know Φ and H ,
D D D
then we can apply Lemma 2.3 to compute
H (γ;x) = x γ(α ) .
D Q
−
Q∈QpDrYim/SL2(Z)(cid:0) (cid:1)
Using algorithms for fast multipoint polynomial evaluation and fast integer arithmetic (see
[18], for example), this yields an algorithm that computes H (γ;x) in O( D 3) expected time
D
| |
using O( D 3) space, under the GRH. However, this approach is quite memory intensive and
| | e
quickly becomes impractical, even for moderate values of D. As an alternative, we give a
e
CRT-based algorithm that uses O( D 7/2) expected time and O( D 2) space, under the GRH.4
| | | |
It is clear from equations (2.6) and (2.8) that the coefficients of H (γ;x) lie in Q; indeed,
D
e e
the coefficients of Φ are integers, as are the elementary symmetric functions of the j(α ),
D Q
which are the coefficients of the Hilbert class polynomial H (x). If we let
D
(3.1) δ := β (α ),
0,1 Q
Q∈QpDrYim/SL2(Z)
then δ Z is divisible by the denominator of every coefficient of H (γ;x) and δH (γ;x)
D D
∈ ∈
Z[x]. We now present the algorithm.
Algorithm 1
Input: An imaginary quadratic discriminant D that is not special.
3We note that the discriminants D =1 24n needed to compute Hpart(x) are not special.
− n
4As remarkedin the introduction, we expect this running time can be improved, possibly to O(D 5/2), by
| |
obtaining tighter bounds on the coefficients of H (γ;x).
D e
