aa r X i v : . [ m a t h . N T ] J a n EVALUATING IGUSA FUNCTIONS
Reinier Br¨oker, Kristin Lauter
Abstract.
The moduli space of principally polarized abelian surfaces is parametrizedby three Igusa functions. In this article we investigate a new way to evaluate thesefunctions by using Siegel Eisenstein series. We explain how to compute the Fouriercoefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluatethe Igusa functions to a prescribed precision.
1. Introduction
The classical theory of complex multiplication gives an explicit description of theHilbert class field of an imaginary quadratic field: for a fundamental discriminant
D <
0, the Hilbert class field of K = Q ( √ D ) is obtained by adjoining the value j (( D + √ D ) /
2) to K . Here, j : H → C is the classical modular function withFourier expansion j ( z ) = 1 /q + 744 + 196884 q + . . . in q = exp(2 πiz ). There arevarious ways to compute the minimal polynomial of j (( D + √ D ) / j -function to highprecision.The j -function is invariant under the action of SL ( Z ) on the upper half plane H .To evaluate j ( τ ), we may assume that τ is in the ‘standard’ fundamental domainfor SL ( Z ) \ H as described in e.g. [21, Sec. VII.1.1]. The naive approach to evaluate j ( τ ) is to simply compute enough Fourier coefficients using for instance the recursiveformulas given in [19]. Alternatively, one can use the relation j ( z ) = 1728 g ( z ) g ( z ) − g ( z ) (1 . j -function in terms of the normalized Eisenstein series g , g ofweight 4 and 6. Better results can be obtained [1] by using the Dedekind η -functiondefined by η ( z ) = q / Q ∞ n =1 (1 − q n ) , and which satisfies j ( z ) = (cid:18) ( η ( z/ /η ( z )) + 16( η ( z/ /η ( z )) (cid:19) . Typeset by
AMS -TEX version 20100506
REINIER BR ¨OKER, KRISTIN LAUTER
The sparsity of the q -expansion of the η -function makes it very efficient for explicitcomputations.The j -function is intrinsically linked to the theory of elliptic curves, and thesituation outlined above can be viewed as the ‘1-dimensional’ case of complex mul-tiplication theory. In dimension 2, suitably chosen invariants of principally polarizedabelian surfaces generate abelian extensions of degree 4 CM-fields, see [22, Sec. 15]for a precise statement. A popular choice of invariants are the three Igusa functions j , j , j defined below. Just as evaluating the elliptic j -function has applicationsto elliptic curve cryptography, evaluating Igusa functions is an important step inconstruction genus 2 curves suitable for use in cryptography, see e.g. [25].The explicit evaluation of Igusa functions is less developed than its dimension-1counterpart. Most people use θ -functions to evaluate Igusa functions. The (ratherunwieldy) formulas expressing Igusa functions in terms of θ -functions are given ine.g. [25, pp. 441–442]. There is also a direct analogue of formula (1.1) which ex-presses the Igusa functions as rational functions in the Siegel Eisenstein series E w .Indeed, Igusa [11, p. 195] defines the normalized cusp forms χ = − · · · ·
53 ( E E − E )and χ = 131 · · · · ·
337 (3 · E + 2 · E − E ) . With that, we have the three
Igusa functions j = 2 · χ χ , j = 2 − E χ χ , j = 2 − · E χ χ + 2 − · E χ χ (1 . . Igusa shows the equivalence with the definition of these functions in terms of thetafunctions in [10, p. 848]. The analogue of the denominator ∆ = g − g appearingin (1.1) is the form χ . The form ∆ is a classical cusp form of weight 12 and χ is a Siegel cusp form of weight 10.A mathematically natural question is whether we can use formula (1.2) directlyto evaluate the Igusa functions, thereby bypassing the θ -functions. The main focusof this paper is to give an explicit algorithm to evaluate the Siegel modular formsoccuring in (1.2) to some prescribed accuracy. Our result gives a relatively easy wayto analyze the precision necessary for the computation to succeed, and we give a rigorous complexity analysis for our method, something which has not been donefor other approaches.Although the asymptotic convergence of our algorithm is slower than the algo-rithm using theta functions, our approach has the advantage that there are fewerhigh precision multiplications required in the evaluation, and thus less precision lossand fewer rounding errors occur. Furthermore, we give a detailed analysis of theEisenstein series and cusp forms, including an algorithm for computing them usingclassical modular forms of half-integral weight and explicit bounds on the size of VALUATING IGUSA FUNCTIONS 3 the coefficients in their Fourier expansions. Indeed, one of the main contributionsof the paper is the detailed analysis of various aspects of the computation of Siegelmodular forms. Finally, our approach may lend itself to improvement in variousways and is a new direction in this area which could produce further progress.Any Siegel modular form f admits a Fourier expansion f ( τ ) = X T a ( T ) exp(2 πi Tr(
T τ )) (1 . T ranges over certain 2 × Z . We proposeto evaluate the functions occuring in (1.2) by truncating the sum in (1.3) to onlyinclude matrices with trace below some bound. The Eisenstein series are Siegel mod-ular forms with a considerable amount of extra structure. We show that computingthe Fourier coefficients of the Eisenstein series ultimately boils down to computingFourier coefficients of classical modular forms of half-integral weight. One of themain results of this paper is the following theorem, proved in Section 4. Theorem 1.1.
For
A, C ∈ Z ≥ and B ∈ Z with B ≤ AC , the Fourier coefficientsof the Siegel Eisenstein series E w for all matrices (cid:16) ab/ b/ c (cid:17) satisfying ≤ a ≤ A , ≤ c ≤ C , | b | ≤ B can be computed in time O (( ABC ) ε ) for every ε > . Theconstant in the O -symbol depends on the weight w . By examining the size of the Fourier coefficients more closely, we derive the followingresult in Section 6.
Theorem 1.2.
Let τ ∈ H be given, and let δ = δ ( τ ) be the supremum of all δ ′ ∈ R such that Im( τ ) − δ ′ is positive semi-definite. Assume that δ ( τ ) ≥ .Assume χ ( τ ) is non-zero and choose n ∈ Z such that | χ ( τ ) | ≥ − n holds.For a positive integer k , let B ∈ Z > be such that Z ∞ B − t exp( − πtδ ( τ )) d t ≤ − k − max { , n } (1 . holds.Then the following holds: if we approximate the modular forms E , E , χ , χ using their truncated Fourier expansions consisting of all the matrices of trace atmost B , then the values j ( τ ) , j ( τ ) , j ( τ ) computed via the formulas in (1.2) areaccurate to precision − k . The condition δ ( τ ) ≥ B in case δ ( τ ) <
1. We assume inTheorem 1.2 that we can bound | χ ( τ ) | from below. This lower bound will allowus to bound the precision loss that occurs when we divide by χ ( τ ). Using theexplicit bounds on the Fourier coefficients of χ , proved in Section 5, we give asimple method to find a value of n in Section 6. This method works in general and REINIER BR ¨OKER, KRISTIN LAUTER does not depend on the value of δ ( τ ). Hence, Theorem 1.2 gives an effective methodto evaluate the three Igusa functions up to some prescribed precision.Just as the elliptic j -function is invariant under SL ( Z ), the Igusa functions j , j , j are invariant under the symplectic group Sp ( Z ). Hence, we may translatethe argument τ by a matrix M ∈ Sp ( Z ) to obtain an Sp ( Z )-equivalent τ ′ ∈ H .The value δ ( τ ′ ) can be significantly different from δ ( τ ), see e.g. Example 7.1. Beforeapplying Theorem 1.2, we therefore move, using e.g. the method from [23], τ to the‘standard’ fundamental domain for Sp ( Z ) \ H described in [8].The outline of the article is as follows. In Section 2 we recall basic facts aboutSiegel modular forms and their Fourier expansions. Section 3 introduces Jacobiforms and their relation to Eisenstein series. The approach we follow in this section is‘classical’ and most likely well-known to experts working with Siegel modular forms.In Section 4 we go one step further, and relate Jacobi forms to classical modularforms of half-integral weight. This gives a very efficient method of computing theFourier coefficients of the 2-dimensional Eisenstein series. The functions χ and χ are Siegel cusp forms, and we explain in Section 5 how to compute the Fouriercoefficients of these forms. We investigate the convergence of the Fourier expansionsof E , E , χ and χ in Section 6. This leads to the proof of Theorem 1.2. A finalSection 7 contains two detailed examples.
2. Siegel modular forms
Let H = { τ ∈ Mat ( C ) | τ = τ T , Im( τ ) > } be the Siegel upper half plane.With J = (cid:16) − (cid:17) , the symplectic group Sp ( R ) is defined as Sp ( R ) = { M ∈ GL ( R ) | M J M T = J } . The group Sp ( R ) naturally acts on the Siegel upper halfplane via (cid:16) ac bd (cid:17) τ = aτ + bcτ + d , where dividing by cτ + d means multiplying on the right with the multiplicativeinverse of the 2 × cτ + d . The matrix − acts trivially, and it is wellknown that the automorphism group of H equals PSp ( R ) = Sp ( R ) / {± } .A holomorphic function f : H → C is called a Siegel modular form of weight w ≥ f ( (cid:16) ac bd (cid:17) τ ) = det( cτ + d ) w f ( τ )for all τ and all matrices in the subgroup Sp ( Z ) ⊂ Sp ( R ). The integer w is calledthe weight of the form f . Whereas we have to demand that f is ‘holomorphic atinfinity’ for classical modular forms H → C , this is not necessary for Siegel modularforms. Indeed, the Koecher principle implies that f is bounded on sets of the form { τ ∈ H | Im( τ ) > α } for α >
0, see [14].The matrix (cid:16)
10 11 (cid:17) is contained in Sp ( Z ), and a Siegel modular function f isinvariant under the transformation τ τ + 1. In particular, a Siegel modular VALUATING IGUSA FUNCTIONS 5 function f admits a Fourier expansion f ( τ ) = X T a ( T ) exp(2 πi Tr(
T τ )) . Here, the sum ranges over all symmetric matrices T ∈ Mat( Z ) with integer diag-onal entries. The coefficients a ( T ) are called the Fourier coefficients of f . By theKoecher principle, they are zero in case T is negative definite.We embed the group GL ( Z ) in Sp ( Z ) via M (cid:16) M M T ) − (cid:17) . As M T hasdeterminant ±
1, we see that a Siegel modular function f is invariant under thetransformation τ M τ M T for M ∈ GL ( Z ). This invariance is the key ingredientin the proof of the following well known lemma. Lemma 2.1.
The Fourier coefficients a ( T ) of a Siegel modular form f satisfy a ( M T T M ) = a ( T ) for every M ∈ GL ( Z ) . Proof.
Writing τ = x + iy with x, y ∈ Mat ( R ), the Fourier coefficient a ( T ) isgiven by a ( T ) = Z f ( τ ) e − πi Tr(
T τ ) d x. Here, d x means the Euclidean volume of the space of x -coordinates and the integralranges over the ‘box’ − / ≤ x ij ≤ /
2. Using the invariance of f we compute a ( M T T M ) = Z f ( M τ M T ) e − πi Tr(
T MτM T ) d x, and the lemma follows. (cid:3) In the remainder of this section we investigate how many different values a ( T )attains for a fixed value of n = det( T ) > f .To a matrix T = (cid:16) ab/ b/ c (cid:17) with a, b, c ∈ Z we associate the binary quadraticform f T = aX + bX Y + cY of discriminant b − ac = − n . An explicit checkshows that for M = (cid:16) αγ βδ (cid:17) the quadratic forms associated to M T T M equals f M T T M = f T ( αX + βY, γX + δY ) , which means that the GL ( Z )-action on H is compatible with the GL ( Z )-actionon quadratic forms of discriminant − n . In fact, the GL ( Z )-action on quadraticforms originally considered by Lagrange is not used that much as it leads to a‘wrong’ kind of equivalence. For quadratic forms, the ‘correct’ action is the actionof the subgroup SL ( Z ) ⊂ GL ( Z ) studied by Gauß. The difference between thesetwo actions is implicit in the following lemma. REINIER BR ¨OKER, KRISTIN LAUTER
Lemma 2.2.
Fix a Siegel modular form f and n ∈ Z > . Suppose that − n is afundamental discriminant and let O be the maximal order of Q ( √− n ) . Then theset { a ( T ) | det( T ) = n } has size at most ( O ) + { a ∈ Pic( O ) | a = 0 } ) . Proof. If − n is fundamental, then any integer binary quadratic form aX + bX Y + cY of discriminant − n is primitive . The set of PSL ( Z )-equivalence classes ofprimitive quadratic forms of discriminant − n is in bijection with the class groupPic( O ) via aX + bX Y + cY a Z + − b + √− n Z by [2, Th. 5.2.8].It remains to investige when a GL ( Z )-equivalence class decomposes as 2 disjunctSL ( Z )-equivalence classes. If a fractional O -ideal a is GL ( Z )-equivalent but notSL ( Z )-equivalent to b , then b equals the inverse a − and we have 2 a = 0. Thelemma follows. (cid:3) For the general case of not necessarily fundamental discriminants, we note that anybinary quadratic form aX + bX Y + cY of discriminant − n determines a primitive quadratic form ( aX + bX Y + cY ) / gcd( a, b, c ) of discriminant − n/ gcd( a, b, c ) .Arguing as in the proof of Lemma 2.2, we see that the set { a ( T ) | det( T ) = n ∈ Z > } has at most 12 X O O ) + { a ∈ Pic( O ) | a = 0 } elements. Here, the sum ranges over all imaginary quadratic orders O that containthe order of discriminant − n . Corollary 2.3.
Let m be the index of the order of discriminant − n in the maximalorder of the quadratic field Q ( √− n ) and let ϕ denote the Euler ϕ -function. For afixed Siegel modular form f , the set { a ( T ) | det( T ) = n ∈ Z ≥ } then has as most √ n log(4 n )( m/ϕ ( m )) elements. Proof.
The class number for the imaginary order of discriminant D is bounded by | D | / log | D | by [18, Sec. 2]. The result now follows from the class number formula,see e.g. [17, Sec. 1.6]. (cid:3)
3. Eisenstein series
For w ≥
0, the space M w of Siegel modular forms of weight w has a natural structureof a C -vector space. For even w ≥
4, the primordial example of a degree w Siegelmodular form is the Eisenstein series E w defined by E w ( τ ) = X c,d ( cτ + d ) − w . (3 . c d ) of elements of Sp ( Z )with respect to left-multiplication by SL(2 , Z ). The restriction w ≥ w = 2.The direct product M = ` ∞ w =0 M w has a natural structure of a graded C -algebra. By restricting the product to even w , we get a graded subalgebra M e . Thefollowing lemma gives the structure of these two algebras. VALUATING IGUSA FUNCTIONS 7
Lemma 3.1.
The Eisenstein series E , E , E and E are algebraically inde-pendent and generate M e . There exists a polynomial P in variables such that M is isomorphic to M e [ X ] / ( X − P ( E , E , E , E )) . The element X corresponds toa Siegel modular form of weight . Proof.
The first statement can be found in [11, pp. 194–195]. The second statementis proven in [10] with an explicit polynomial P at page 849. (cid:3) The remainder of this section is devoted to deriving a ‘formula’ for the Fouriercoefficient a ( T ) of the Eisenstein series E w . The approach we follow is intrinsicallyrelated to the theory of Jacobi forms , see [7] for a good introduction. Let f : H → C be a Siegel modular form of weight w . We write τ ∈ H as τ = (cid:16) τ ε ετ (cid:17) . Because f is periodic with respect to τ , it admits a Fourier expansion f ( τ ) = ∞ X m =0 ϕ m ( τ , ε ) e πimτ where ϕ m is a function from H × C to C . The functions ϕ m have the followingproperties: ⋄ ϕ m ( aτ + bcτ + d , εcτ + d ) = ( cτ + d ) ω e πimcε/ ( cτ + d ) ϕ m ( τ , ε ) , (cid:16) ac bd (cid:17) ∈ SL ( Z ) ⋄ ϕ m ( τ, ε + λτ + µ ) = e − πim ( λ τ +2 λε ) ϕ m ( τ , ε ) , ( λ, µ ) ∈ Z ⋄ ϕ m admits a Fourier expansion of the form ∞ X n =0 X r ∈ Z r ≤ nm c ( n, r ) e πi ( nτ + rε ) . The first two properties follow from the transformation law of Siegel modular formsunder the symplectic matrices a b
00 1 0 0 c d
00 0 0 1 and µλ µ
00 0 1 − λ and the third property follows from the Koecher principle.A holomorphic function g : H × C → C satisfying the three properties above forsome w and m is called a Jacobi form of weight w and index m . Jacobi forms canbe seen as an ‘intermediate’ between Siegel modular forms and classical modularforms. Indeed, the ‘Fourier coefficients’ of a Siegel modular form of weight w areJacobi forms of weight w and for a Jacobi form g , the function g ( τ,
0) is a classicalmodular form of weight w .The space of all Jacobi forms of weight w and index m is denoted by J w,m , andwe have maps M w ֒ −→ Y m ≥ J w,m pr − ։ J w, , REINIER BR ¨OKER, KRISTIN LAUTER where pr denotes the projection onto the first factor. For this article, the key prop-erty of Jacobi forms is that we can also construct a map J w, → M w which willallow us to identify certain Siegel modular forms with its ‘first’ Jacobi form. Aswe have J w, = 0 for odd w by [7, Th. 2.2], we restrict to even weight w for theremainder of this section.For m ≥
0, we define the ‘Hecke operator’ V m : J w, → J w,m as follows. For g ∈ J w, with Fourier expansion P n,r c ( n, r ) e πi ( nτ + rε ) , we put V m ( g ) = X n,r X a | gcd( n,r,m ) a w − c (cid:16) nma , ra (cid:17) e πi ( nτ + rε ) for m >
0. This is the natural generalization of the Hecke operators for classicalmodular forms, see e.g. [21, Prop. VII.12]. For m = 0, we put V ( g ) = − B w c (0 , w − wB w X n ≥ σ w − ( n ) e πinτ with σ n ( x ) the sum of the n th powers of the divisors of x and B w the w th Bernoullinumber defined by t/ ( e t −
1) = P ∞ n =0 B n t n /n !. In particular, the function V ( g ) isa multiple of the classical Eisenstein series of weight w . It is not hard to show thatthe function Ψ( g ) = X m ≥ V m ( g )( τ , ε ) e πimτ defines a Siegel modular form of weight w , see [7, Th. 6.2]. Lemma 3.2.
The map
Ψ : J w, → M w is injective. Proof.
This follows directly from the fact that the composition J w, −→ M w ֒ −→ Y m ≥ J w,m pr − ։ J w, is the identity. (cid:3) We stress that the map Ψ is in general not surjective. The image Ψ( J k, ) is knownas the Maaß Spezialschar . However, the Eisenstein series E w ∈ M w do occur at theimage of a Jacobi form. They are the images of the Jacobi Eisenstein series E Jw defined by the (rather awkward looking) formula E Jw ( τ, z ) =12 X c,d ∈ Z gcd( c,d )=1 X λ ∈ Z ( cτ + d ) − w exp (cid:18) πi (cid:18) λ aτ + bcτ + d + 2 λ zcτ + d − cz cτ + d (cid:19)(cid:19) for w ≥
4. Here, a and b are integers such that (cid:16) ac bd (cid:17) is contained in SL ( Z ). VALUATING IGUSA FUNCTIONS 9
Lemma 3.3.
We have E w = Ψ (cid:16) − wB w E Jw (cid:17) . Proof.
It follows from [7, Th. 6.3] that E w is a multiple of Ψ( E Jw ). Both theSiegel Eisenstein series E w and the Jacobi Eisenstein series E Jw are normalized withconstant coefficient 1. The lemma follows. (cid:3) It is now a straightforward matter to compute the Fourier coefficients of the SiegelEisenstein series. The result is the following theorem.
Theorem 3.4.
Let E w be the Siegel Eisenstein series of weight w , and let T = (cid:16) ab/ b/ c (cid:17) ∈ Mat( Z ) be a positive semi-definite matrix with integer entries on thediagonal. Write D = b − ac ≤ and let D be the discriminant of Q ( √ D ) . Thenthe Fourier coefficient a ( T ) equals for a = b = c = 0 and − wB w X d | gcd( a,b,c ) d w − α ( D/d ) otherwise. Here, B k is the k th Bernoulli number and α is defined by α (0) = 1 and α ( D ) = 1 ζ (3 − w ) C ( w − , D ) ( D < where C is Cohen’s function defined by C ( s − , D ) = L D (2 − s ) X d | f µ ( d ) (cid:18) D d (cid:19) d s − σ s − ( f /d ) , D = D f . Here, ζ denotes the Dedekind ζ -function, L D is the quadratic Dirichlet L -series, µ is the M¨obius function, (cid:0) ·· (cid:1) is the Kronecker symbol and σ n ( x ) denotes the sumof the n th powers of the divisors of x . Proof.
By [7, Th. 2.1], the term α ( D ′ ) equals the Fourier coefficient α ( n, r ) of theJacobi Eisenstein series E Jw with D ′ = r − n . By Lemma 3.3, we have to applythe Hecke operators V m to these coefficients. The theorem follows. (cid:3) A formula for a ( T ) is also given in Corollary 2 to [7, Th. 6.3]. In this formula, theBernoulli numbers and the ζ -function from Theorem 3.4 are missing.We see that Theorem 3.4 gives a much better bound than Lemma 2.2 for thecardinality of { a ( T ) | det( T ) = n ∈ Z } for the Eisenstein series. Indeed, for fun-damental discriminants − n , we have only one Fourier coefficient a ( T ). In general,the number of coefficients is bounded by the number of square divisors of − n which in turn is bounded by O ( n ε ) for all ε >
0. These bounds hold in general forfunctions in the Spezialschar Ψ( J w, ) ⊂ M w . Indeed, the Fourier coefficients c ( n, r )of a function g ∈ J w, only depend on the value 4 n − r , cf. [7, Th. 2.2]. Corollary 3.5.
Let n k be the numerator of the k th Bernoulli number B k . Thenthe Fourier coefficient a ( T ) of the Siegel Eisenstein series E w for the matrix T = (cid:16) ab/ b/ c (cid:17) is contained in the set / ( n w n w − ) Z ⊂ Q . Proof.
As we have ζ (3 − w ) = − B w − / (2 w −
2) all we have to do is examinethe denominator of the value L D (2 − w ) occuring in Theorem 3.4. This is mosteasily done using p -adic L -series as in [3, Ch. 11]. The corollary follows from [3, Cor.11.4.3] except in the following case: the discriminant of Q ( √ b − ac ) equals − p foran odd prime p with w − ≡ ( p − / p − L -value could be divisible by p ( w − p then satisfies ( p − | (2 w −
2) and by the Clausen-von Staudt theorem[3, Cor. 9.5.15] the prime p also occurs in the denominator of B w − . Finally, w − ζ (3 − w ). (cid:3) Corollary 3.6.
The Fourier coefficient a ( T ) of the Siegel Eisenstein series E w for the matrix T = (cid:16) ab/ b/ c (cid:17) satisfies | a ( T ) | = O ((4 ac − b ) w − / ) if b − ac isnon-zero. Proof.
Using the functional equation for Dirichlet L -series, see e.g. [3, Th. 10.2.6],we bound L D (2 − w ) = O ( D w − / ). The inequalities σ n ( x ) x n = X d | x d n ≤ ∞ X d =1 d n = ζ ( n ) < ∞ give σ n ( x ) = O ( x n ) for n >
1. It follows that the c ( D ′ ) in Theorem 3.4 is of size O ( D ′ w − / ). As P d | n d w − /d w − is finite for w ≥ n → ∞ , the corollaryfollows. (cid:3) Remark 3.7.
It is not hard to make the constant c in the O -symbol explicit. Onecan take c = (cid:12)(cid:12)(cid:12)(cid:12) w ( w − ζ ( w − ζ (2 w − ζ ( w − π w − ζ (3 − w ) B w (cid:12)(cid:12)(cid:12)(cid:12) .
4. Computing special values of L -series The hard part in computing Fourier coefficients of Siegel Eisenstein series is com-puting the special values of L -series occuring in Theorem 3.4. If the discriminant ofthe quadratic field Q ( √ b − ac ), corresponding to the matrix (cid:16) ab/ b/ c (cid:17) , is smallthese computations can be efficiently done employing generalized Bernoulli numbersas we now explain.For n ≥
1, we let χ n be the quadratic Dirichlet character modulo n and definethe χ n -Bernoulli numbers B k ( χ n ) by the expansion P nr =1 χ n ( r ) te rt e nt − X k ≥ B k ( χ n ) k ! t k ∈ Q [ t ] . (4 . VALUATING IGUSA FUNCTIONS 11
The generalized Bernoulli numbers B k ( χ n ) equal the ordinary Bernoulli numbers B k for n = 1 and k ≥ Lemma 4.1.
For n ≥ and w ≥ , we have L n (2 − w ) = − B w − ( χ n ) / ( w − . Proof.
See [24, Th. 4.2]. (cid:3)
The values B w − ( χ n ) can easily be computed using the definition (4.1) for small w and n . For evaluating the Igusa functions, we are only interested in the values w = 4 , , ,
12 and by computing B ( χ n ) we get the other values B ( χ n ) , B ( χ n )and B ( χ n ) ‘for free’.To compute the Fourier coefficients of the Eisenstein series E w for large valuesof D = b − ac , we clearly need another method. It is a relatively well-knownfact that Jacobi forms of even weight and index 1 ‘correspond to’ classical modularforms of half-integral weight. Explicitly, for a discriminant D <
0, we define α w ( D ) = 1 ζ (3 − w ) C ( w − , D )= 1 ζ (3 − w ) L D (2 − w ) X d | f µ ( d ) (cid:18) D d (cid:19) d w − σ w − ( f /d )as in Theorem 3.4. Here, D is the discriminant of the quadratic field Q ( √ D ) and f satisfies D f = D . We put α w (0) = 1, and α w ( D ) = 0 if D < H w : H → C defined by H w ( z ) = ∞ X n =0 α w ( − n ) q n ( q = exp(2 πiz ))is known as Cohen’s function . Lemma 4.2.
Let H w be defined as above. Then H w is a modular form of weight w − / for the congruence subgroup Γ (4) . Proof.
See [4, Th. 3.1], or an alternate proof in [13, Prop. IV.6]. (cid:3)
Remark.
The bound α w ( n ) = O ( n w − / ) from the proof of Corollary 3.6 is in niceaccordance with the general result that the Fourier coefficients of a modular formof weight k are of size O ( n k − ) . As the C -vector space of modular forms of fixed (half-integral) weight is finitedimensional, we can easily compute coefficients of H w given a basis for the vectorspace. It is not hard to show that the function θ ( z ) = X n ∈ Z q n = 1 + 2 ∞ X n =1 q n ( q = e πiz ) is a modular form of weight 1 / (4). The function e θ ( z ) = θ ( z + 1 /
2) = (cid:16) ∞ X n =1 ( − n q n (cid:17) ( q = e πiz )is therefore a modular form of weight 2. Analogous to the proof of [13, Prop. IV.4],it follows that θ and e θ generate the C - algebra of all modular forms. The mainadvantage of choosing this basis is that θ is very lacunary. Proposition 4.3.
The following equalities hold: H = θ + 7 θ e θ H = − θ + 22 θ e θ + 11 θ e θ H = − θ + 725876 θ e θ + 12824886 θ e θ + 8845412 θ e θ + 107597 θ e θ H = 77683 θ + 212405 θ e θ + 38627902 θ e θ + 100820362 θ e θ θ e θ + 42481 θ e θ . Proof.
Using Lemma 4.1, we compute the first few Fourier coefficients of H w for w = 4 , , ,
12. With the obervation that H w equals an isobaric polynomial in θ and e θ , we have to solve a system of w/ w/ (cid:3) This theorem allows us to compute the first N coefficients of H w in time O ( N o (1) )using fast multiplication techniques. This leads to the theorem stated in the intro-duction. An important conclusion is that it is much faster to compute L -values simultaneously than to compute them individually. Corollary 4.4.
For
A, C ∈ Z ≥ and B ∈ Z with B ≤ AC , the Fourier co-efficients of the Siegel Eisenstein series E w for all matrices (cid:16) ab/ b/ c (cid:17) satisfying ≤ a ≤ A , ≤ c ≤ C , | b | ≤ B can be computed in time O (( ABC ) ε ) for every ε > . The constant in the O -symbol depends on the weight w .
5. Cusp forms
The techniques explained in Sections 3 and 4 allow us to efficiently compute theFourier coefficients of Siegel Eisenstein series. This suffices for evaluating Igusafunctions, since these functions are rational expressions in E w for w = 4 , , , χ directly via Jacobi forms. We explain this method in this Section. VALUATING IGUSA FUNCTIONS 13
Let M w be the vector space of classical modular forms of integral weight w ,and let M = ` w ≥ M w be the space of all classical modular forms. It is wellknown that we have M ∼ = C [ E , E ], with E w the classical Eisenstein series ofweight w , see [21, Cor. 2 to Th. VII.4]. We define the Siegel operator S : M → M as follows. For a Siegel modular form f : H → C with Fourier expansion f ( τ ) = P T a ( T ) exp(2 πi Tr(
T τ )) we put S ( f ) = X n ≥ a (cid:18)(cid:16) n (cid:17)(cid:19) e πinτ , with τ = (cid:16) τ ε ετ (cid:17) . The Siegel operator is a ring homomorphism M → M , and it maps Eisensteinseries to Eisenstein series. In fact, for the Eisenstein series E w , it is the compositionof the maps M w −→ Y m ≥ J w,m pr − ։ J w, −→ M w , introduced in Section 2.A Siegel modular form f is called a cusp form if it satisfies S ( f ) = 0. Equiva-lently, f is a cusp form if and only if the Fourier coefficients a ( T ) are zero for allsemi-definite T that are not definite. It follows from well-known identities betweenclassical Eisenstein series that χ = − · − · − · − · − · − ( E E − E )and χ = 131 · · − · − · − · − · − (3 · E + 2 · E − E ) , are cusp forms. The constants in χ and χ should be regarded as ‘normalizationfactors’. Lemma 5.1.
The ideal of cusp forms in M e is generated by χ and χ . The idealof cusp forms in M is generated by χ , χ and a modular form χ of weight 35corresponding to X in Lemma 3.1 Proof.
See [12, Th. 3]. (cid:3)
It is well-known that the cusp forms χ and χ are contained in the MaaßSpezialschar Ψ( J k, ), the gest of the proof being [7, Th. 6.3]. A Jacobi form g ∈ J w,m is called a cusp form if its Fourier coefficients c ( n, r ) are zero for 4 nm − r = 0. Inparticular, the map M w → Y m ≥ J w,m pr − ։ J w, maps Siegel cusp forms to Jacobi cusp forms. In weight 10 and 12 we have theJacobi cusp forms ϕ , = 1144 ( E E , − E E , ) and ϕ , = 1144 (( E ) E , − E E , ) , with E = 1 + 240 P n> σ ( n ) q n and E = 1 − P n> σ ( n ) q n the classicalEisenstein series. The factor 144 should again be regarded as a normalization factor. Lemma 5.2.
We have χ = Ψ( − ϕ , / and χ = Ψ( ϕ , / . Proof.
The cusp forms χ and χ are contained in the Spezialschar and thereforeoccur as images of Jacobi cusp forms. The spaces of Jacobi cusp forms of weight 10and 12 are 1-dimensional by [7, Th. 3.5]. Using Theorem 3.4, we compute a (cid:18)(cid:16)
10 01 (cid:17)(cid:19) = 12for a Fourier coefficient of χ and c (1 ,
0) = − ϕ , . The result for χ follows. The computa-tion for χ yields a (cid:18)(cid:16)
10 01 (cid:17)(cid:19) = 56 and c (1 ,
0) = 10 . The lemma follows. (cid:3)
To compute the Fourier coefficients of ϕ , and ϕ , we note that the coefficients c ϕ , ( n, r ) and c ϕ , ( n, r ) only depend on the value of 4 n − r ≥
0. Furthermore,the functions K = X k ≥ c ϕ , ( k ) q k and K = X k ≥ c ϕ , ( k ) q k are classical modular cusp forms , for the group Γ (4), of weight 9 and 11 respec-tively by [7, Th. 5.4]. Proposition 5.3.
Let θ and e θ be as in Section 4. Then we have K = θ e θ − θ e θ + 3 θ e θ − θ e θ K = 5 θ e θ − θ e θ + 18 θ e θ − θ e θ + θ e θ . Proof.
Analagous to the proof of Theorem 4.3. (cid:3)
It should come as no surprise that there are no terms θ and θ occuring inProposition 5.3. Indeed, the forms K and K are cusp forms and therefore vanishat q = 0 whereas θ does not vanish at q = 0.Proposition 5.3 allows us to evaluate the Siegel cusp forms χ and χ at arbi-trary points τ ∈ H . For the proof of Theorem 1.2, we need a bound on the size of VALUATING IGUSA FUNCTIONS 15 the Fourier coefficients of χ and χ as well. We need an explicit bound, like thebound in Remark 3.7.The ‘Resnikoff-Salda˜na conjecture’ ([20]) | a ( T ) | = O ((det T ) w/ − / ε )for the size of a Fourier coefficient a ( T ) of a Siegel modular cusp form of weight w is known to be false in general. At this moment, the best known result is | a ( T ) | = O ((det T ) w/ − / ε )for every ε >
0, see [15]. We will prove in the remainder of this section that theFourier coefficients of g and g satisfy | a ( T ) | = O ((det T ) w/ − / ε ) , and we will make the constant in the O -symbol explicit. The reason that ourbound is better than the bound in [15] is that our Siegel modular forms lie inthe Maaß Spezialschar, and this allows us to give a stronger bound. In fact, wewill show that if the Lindel¨of-hypothesis is true, the Fourier coefficients are ofsize O ((det T ) w/ − / ε ).First we will bound the Fourier coefficients of K and K explicitly. One ap-proach would be to adapt ‘Hecke’s proof’ [21, Th. VII.5] for cusp forms. This tech-nique would yield a bound of O ( n . ) for g , where we can make the constant inthe O -symbol explicit. However, our modular forms have considerably more struc-ture and we will use a variant of Waldspurger’s formula to obtain a better bound.The modular forms K and K have the property that their Fourier coefficients a ( n ) are zero for n ≡ , f of weight w − /
2, Shimuraconstructs, see e.g. [7, Sec. 5], an integral weight cuspform g of weight 2 w − | a ( D ) | = h g, g ih f, f i ( w − π w − L ( g, χ D , w − | D | w − / (5 . a ( D ) of f . In formula (5.1), known as Waldspurger’sformula, h· , ·i denotes the usual Petersson inner product, and L ( g, χ D , s ) is the L -series associated to g , twisted by the quadratic Dirichlet character χ D . Lemma 5.4.
Let g , g be the modular forms associated to K , K respectivelyunder Shimura’s construction. Then we have h g , g ih K , K i ≤ and h g , g ih K , K i ≤ . Proof.
As the space of weight 18 cusp forms is one-dimensional, we have g = ∆ E .Likewise, g = ∆ E . By formula (5.1) we have h g , g ih K , K i = L ( g , χ D , | a ( D ) | · π | D | . for every discriminant D for which the Fourier coefficient a ( D ) of K is nonzero.Since we can compute the Fourier coefficients of K , it suffices to explicitly evaluatethe L -series at the center of the critical strip.Since g is a Hecke eigenform, the formula(2 π ) − s Γ( s ) L ( g , s ) = Z ∞ g ( iy ) y s dy/y is valid for all s ∈ C . Analogous to the example in [16], we derive the relation L ( g , χ D ,
9) = 2Γ(9) (2 π/ | D | ) ∞ X n =1 (cid:18) Dn (cid:19) c ( D ) φ (2 πn/ | D | ) (5 . g = P n c ( n ) n − s . Here, we write φ ( x ) = Z ∞ y exp( − xy ) dy = 8! x exp( − x ) (cid:0) x + x /
2! + . . . + x / (cid:1) . The right hand side of (5.2) converges exponentially fast, and since we know theFourier coefficients of g we easily compute the first bound of the Lemma.Since the space of weight 22 cusp forms is also one-dimensional, the bound for K follows analogously. (cid:3) Lemma 5.5.
For every ε > , the twisted L -series associated to the cusp forms g and g satisfy | L ( g , χ D , | ≤ B ( ε, | D | . ε | L ( g , χ D , | ≤ B ( ε, | D | . ε for all discriminants D < . Here, B is defined by B ( ε, n ) = 1 √ π max (cid:26) ζ (1 + ε ) , ζ (1 + ε ) Γ( n + 1 / ε )Γ( n − / − ε ) (cid:27) . Proof.
Let g = P m a ( m ) q m be either g or g , and let 2 w be the weight of g .With Λ( s, χ D ) = (cid:0) D π (cid:1) s Γ( s + w − / L ( s + w − / , g, χ D ), the twisted L -seriesfor g satisfies the functional equationΛ( s, χ D ) = Λ(1 − s, χ D ) VALUATING IGUSA FUNCTIONS 17 for all s ∈ C . We will bound L ( s, g, χ D ) on a vertical line to the right of the criticalstrip, which by the functional equation gives a bound on a vertical line to the leftof the critical strip. A variant of the Phragmen-Lindel¨of theorem will then give theresult.We put P ( s ) = (cid:0) D π (cid:1) s L ( s + w − / , g, χ D ) and A ( m ) = c ( m ) /m w − / . We have L ( s + w − / , g , χ D ) = P m A ( m ) χ D ( m ) m s , and the coefficients A ( m ) are boundedby σ ( m ) = P d | m ε > t ∈ R ,we bound | L (1 + ε + w − / , g , χ D ) | ≤ X m | A ( m ) | m ε ≤ X m σ ( m ) m ε = ζ (1 + ε ) . We get | P (1 + ε + it ) | ≤ | D | ε ζ (1+ ε ) π = C ( ε ) | D | ε . Using the functional equation,we bound | P ( − ε + it ) | = (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + ε − it + w − / − ε + it + w − / ζ (1 + ε − it ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ε )(1 + | t | ) ε C ( ε ) | D | ε , where the last inequality follows from Stirling’s formula.By the Phragmen-Lindel¨of theorem, see e.g. [5, Sec. VI.4], we can bound | P ( σ + it ) | ≤ C ( ε )(1 + | t | ) M ( σ ) | D | ε for all σ ∈ [ − ε, ε ] , where C ( ε ) = max { C ( ε ) , C ( ε ) C ( ε ) } is the maximum of the two ε -dependentbounds on the vertical lines, and M ( σ ) = 1 + ε − σ takes the values M ( − ε ) = 1 + 2 ε and M (1 + ε ) = 0. Taking σ = 1 / t = 0, we derive | L ( w, g, χ D ) | ≤ √ πC ( ε ) | D | / ε , which yields the lemma. (cid:3) Corollary 5.6.
Let the notation be as in Lemma 5.5. Then, for every ε > , thecoefficients c , ( n ) and c , ( n ) of K and K satisfy c , ( n ) ≤ p B ( ε, n . / ε and c , ≤ p B ( ε, n . / ε for all n ≥ . Proof.
Substitute Lemmas 5.4 and 5.5 into Waldspurger’s formula (5.1). (cid:3)
Remark 5.7.
The only room for improvement in Lemma 5.5, and hence in Corol-lary 5.6, is in the use of the Phragmen-Lindel¨of theorem. This theorem yields afactor n / in the bound. We can use stronger results to lower the exponent, but itis harder to make the constants explicit. If the Lindel¨of hypothesis is true, then thefactor n / can be replaced by n ε (but the constant could in theory be not explicitlycomputable). Theorem 5.8.
Let the function B be as in Lemma 5.5, and define B ( x ) =exp(2 /x / ( x log 2)) . Then, for every ε > and any η > , the Fourier coefficients a ( T ) and a ( T ) of χ and χ satisfy | a ( T ) | ≤ B ( η ) p B ( ε, T ) . / ε + η | a ( T ) | ≤ B ( η ) p B ( ε, T ) . / ε + η . Proof.
The Fourier coefficient of χ for the matrix T is bounded by X d | (4 det T ) d c , (cid:18) Td (cid:19) ≤ p B ( ε, X d | (4 det T ) (4 det T ) . ε/ d ε . The sum on the right hand side is bounded by(4 det T ) . ε/ X d | (4 det T ) ≤ B ( η )(4 det T ) . ε/ η , see e.g. [9, Sec. 18.1.]. The proof for χ is similar. (cid:3) Remark 5.9.
If the Lindel¨of-hypothesis is true, then we get a bound | a ( T ) | = O ( n w/ − / ε ) for the Fourier coefficients of χ and χ . This bound is optimalin the sense of the Resnikoff-Salda˜na conjecture [20].
6. Speed of convergence
In section we carefully analyse the speed of convergence of the Siegel Eisensteinseries occuring in (1.2), and this will yield Theorem 1.2 without too much effort. Toanalyse the convergence of a Siegel modular function we a priori have to consider three variables. We begin by showing that it suffices to look at a ‘one-dimensional’convergence problem.The imaginary part Im( τ ) of a matrix τ ∈ H is positive definite. Hence, thereexists δ ∈ R > with Im( τ ) ≥ δ , meaning that Im( τ ) − δ is positive semi-definite.We define δ ( τ ) = sup { δ ′ ∈ R | Im( τ ) ≥ δ ′ } to be the ‘largest’ of all these values. With this notation, we have the followinglemma. Lemma 6.1.
Let T = (cid:16) ab/ b/ c (cid:17) ∈ Mat ( Z ) be positive semi-definite and let τ ∈ H . Then the inequality | exp(2 πi Tr(
T τ )) | ≤ exp( − π Tr( T ) δ ( τ )) holds. VALUATING IGUSA FUNCTIONS 19
Proof.
We have an equality | exp(2 πi Tr(
T τ )) | = exp( − π Tr( T Im( τ ))). Since T ispositive semi-definite, we have T Im( τ ) ≥ T δ ( τ ). The lemma follows. (cid:3) We have(6 . E w ( τ ) = X T a ( T ) exp(2 πi Tr(
T τ )) = ∞ X t =0 X T ∈ S ( t ) a ( T ) exp(2 πi Tr(
T τ ))where S ( t ) is the set of all 2 × t with non-negativeinteger entries on the diagonal and half-integer entries on the off-diagonal. The set S ( t ) clearly has at most 2( t + 1) elements for which a ( T ) is non-zero.The technique of ‘splitting up’ the evaluation of a Siegel modular form as in equa-tion (6.1) enables us to find a lower bound for | χ ( τ ) | . The idea is that if wehave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X T ∈ S ( t ) t ≤ B a ( T ) exp(2 πi Tr(
T τ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X T ∈ S ( t ) t>B a ( T ) exp(2 πi Tr(
T τ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6 . | χ ( τ ) | is roughly equal to the left hand side of (6.2). Further-more, we can apply the upper bound for the Fourier coefficients of χ given byTheorem 5.8 to bound the right hand side of (6.2). Taking B = 2 yields the followinglemma. Lemma 6.2.
Let M =
10 01 ! , M =
12 12 ! , M = − − ! , and for ε, η > , put M ( ε, η ) = 320 B ( η ) p B ( ε, , where the notation is as in Theo-rem 5.8. If, for any ε, η > , we have | c | ≥ R ∞ M ( ε, η ) t ε +2 η exp( − πtδ ( τ )) dt for c = 12 exp(2 πi Tr( M τ )) −
14 exp(2 πi Tr( M τ )) −
14 exp(2 πi Tr( M τ )) , then we have | χ ( τ ) | ≥ / | c | . Proof.
Since χ is a cusp form, there are no matrices T ∈ S (0) ∪ S (1) for whichthe Fourier coefficient a ( T ) of χ is non-zero. The only matrices T ∈ S (2) forwhich a ( T ) is nonzero are the matrices M , M , M . These matrices have Fouriercoefficients 1 / , − / , − / c equals the left hand side of (6.2)with B = 2.Using Theorem 5.8, we bound the right hand side of (6.2) from above by10 ∞ X t =3 t (cid:12)(cid:12)(cid:12) max T ∈ S ( t ) (cid:8) a ( T ) exp( − π Tr( T ) δ ( τ )) (cid:9)(cid:12)(cid:12)(cid:12) ≤ Z ∞ M ( ε, η ) t ε +2 η exp( − πtδ ( τ )) dt, where we used the ‘AGM-inequality’ 4 det( T ) ≤ Tr( T ) . The lemma follows. (cid:3) Remark 6.3.
In Lemma 6.2, we can choose any ε and η . The optimal choicedepends on the value of δ ( τ ) . Remark 6.4.
If the condition in Lemma 6.2 does not hold for any ε, δ , we canlook at the contribution of all matrices of trace and . If that majorates thecontribution coming from all matrices of trace and higher, we have found a lowerbound on | χ ( τ ) | . Proof of Theorem 1.2.
The Igusa functions are rational expressions in the Eisen-stein series E , E and the cusp forms χ and χ . The proof consists of 2 parts:first we analyse the ‘loss of precision’ that occurs when applying the formulas (1.2).Knowing the precision to which to evaluate the four Siegel modular forms, we thencarefully analyse the speed of convergence of these series.Using Corollary 3.6, we bound(5 . | a ( T ) | ≤ T ) for a Fourier coefficient of E in case det( T ) is non-zero. For det( T ) = 0 andTr( T ) = 0, inequality (5.2) holds by Theorem 3.4. We conclude that | E ( τ ) | isbounded by1 + Z ∞ · t ( t + 1) exp( − πtδ ( τ ))d t ≤ δ ( τ ) + 144 δ ( τ ) + 76 δ ( τ ) , and our assumption δ ( τ ) ≥ E ( τ ) weget the bound | E ( τ ) | ≈ /δ ( τ ) ≤
94. Using Theorem 5.8 with η = 1 .
37 and ε = 0 .
28, we derive the bounds | χ ( τ ) | ≤ | χ ( τ ) | ≤ k +22 decimal digits, then we know the products χ ( τ ) , E ( τ ) χ ( τ ) and E ( τ ) χ ( τ ) occuring in formula (1.2) up to k decimaldigits precision. Furthermore, we know by assumption that χ ( τ ) does not equalzero. Let n ∈ Z be the smallest n such that | χ ( τ ) | ≥ − n holds. By dividingby χ ( τ ) , we lose max { , n } digits precision. Hence, if we evaluate all the Siegelmodular forms occuring in (1.2) up to l = k + max { , n } digits of precision, weknow the Igusa values j ( τ ) , j ( τ ) , j ( τ ) up to k decimal digits of precision.We evaluate the Siegel modular functions E , E , χ , χ using the sum (6.1),truncated to only include matrices whose trace is below some bound B . It remainsto give a value for B such that the function values are accurate up to l decimal VALUATING IGUSA FUNCTIONS 21 digits. As the speed of convergence of the four series involved is slowest for χ , itsuffices to look at this function. Taking η = 1 .
45 and ε = 0 .
1, we have X T ∈ S ( t ) t ≥ B a ( T ) exp(2 πi Tr(
T τ )) ≤ Z ∞ B − t exp( − πtδ ( τ ))d t and if the integral is less than 10 − l then the contribution coming from the matricesof trace larger than B do not alter the first l decimal digits. The theorem follows. (cid:3)
7. Examples
In this section we illustrate the techniques developed in this paper by evaluating j ( τ ) for two choices of τ . We detail the evaluation of the Igusa functions j , j , j at τ = (cid:16) i
13 + 26 i
13 + 26 i
83 + 141 i (cid:17) ∈ H to 500 decimal digits of precision. The Igusa functions are rational expressions inthe Siegel modular forms E , E , χ and χ , cf. Section 1. The idea is to simplyevaluate these series at τ to high enough precision and then apply the formulas (1.2).We have the rather low bound δ ( τ ) ≥ .
15 in this case. However, for the purposeof evaluating Igusa functions, we may replace τ by an Sp ( Z )-equivalent matrix τ ′ .It is straightforward to check that the matrix τ ′ = (cid:16) ii i i (cid:17) = − − − − −
180 0 1 50 0 0 1 ( τ )lies in the fundamental domain for Sp ( Z ) \ H as e.g. described in [8]. We have δ ( τ ′ ) ≥ . | χ ( τ ′ ) | from below, we apply Lemma 6.2. With the notation of thislemma, we compute c ≈ − . · − and the value of the integral is roughly equalto 2 · − for ( η, ε ) = (1 . , . c ′ coming from all matrices of at most 4,then we get c ′ ≈ − . · − ≈ c but we now have20 Z ∞ t . exp( − πt · . t ≈ . · − We conclude that | χ ( τ ′ ) | is bounded from below by 1 . · − .The lower bound on | χ ( τ ′ ) | yields that we lose 6 ·
28 = 168 decimal digits ofprecision in the computation of j ( τ ′ ). However, we also easily bound | χ ( τ ′ ) | ≤ . · − . Hence, we gain 5 ·
29 = 145 decimal digits of precision by multiplyingby χ ( τ ′ ) . The ‘net loss’ of precision is therefore only 168 −
145 = 23 decimaldigits of precision.Putting everything together, we need to evaluate the Siegel modular forms E , E , χ , χ up to 524 decimal digits precision to know the values of the Igusafunctions up to 500 decimal digits precision. The integral Z ∞ B − t . exp( − . πt )d t is less than 10 − for B = 49 and we hence have to consider all matrices of traceup to 49.To compute the Fourier coefficients for all matrices (cid:16) ab/ b/ c (cid:17) of trace at most 49,we compute the Fourier coefficients of all matrices satisfying 4 ac − b ≤ ,with the convention that we only take the matrices of trace at most 49 in the caseof determinant 0. To compute all the coefficients a ( T ) for E , E , χ and χ wecompute the first 2401 terms of the power series θ = 1 + 2 ∞ X n =1 q n and e θ = (cid:16) ∞ X n =1 ( − n q n (cid:17) . Using Proposition 4.3, we compute the first 2401 coefficients of the modular forms H and H : − B H = 240 + 13440 q + 30240 q + 138240 q + 181440 q + 362880 q + O ( q ) − B H = −
504 + 44352 q + 166320 q + 2128896 q + 3792096 q + O ( q ) . The coefficients of the forms H i are the Fourier coefficients of the Jacobi Eisen-stein series E Ji . Using Proposition 5.3 we compute the first 2401 coefficients of themodular forms K and K : − K = − / q + 1 / q + 4 q − q − / q + O ( q ) K = 1 / q + 5 / q − / q − q + 425 / q + O ( q ) . The coefficients of K and K are the Fourier coefficients of the Jacobi cusp forms ϕ , and ϕ , .Since the 4 Siegel modular forms we are interested in lie in the Maaß Spezialschar,the Fourier coefficient a ( T ) of one of them only depends on the determinant of T andthe greatest common divisor of the entries of T . We make an array ‘encoding’ theseFourier coefficients as follows. For every positive integer N ≤ N and write N = N f . For every divisor d | f , we compute andstore the Fourier coefficient belonging to a matrix T = (cid:16) ab/ b/ c (cid:17) with 4 ac − b = N and gcd( a, b, c ) = d . For E and N = 16 we get[997920 , , , VALUATING IGUSA FUNCTIONS 23 for instance. For N = 0 we make a list of all positive integers d ≤ X and store thecoefficients for the determinant zero matrices with trace d .The computations so far were independent of the choice of τ = (cid:16) τ z zτ (cid:17) ∈ H . Welet q = exp(2 πiτ ), q = exp(2 πiτ ) and q = exp(2 πiz ) be the ‘Fourier variables’of the entries of τ . We compute and store the values q = 1 , q , q , . . . , q X andlikewise for q . For ζ we need to compute both the first X powers of ζ and ζ − because the off-diagonal entries of the matrices can be negative.The precision needed for this computation is easily computed. Indeed, the max-imum bound for a Fourier coefficient is roughly 10 and occurs for χ and atrace 49 matrix. As we need to recognize the values a ( T ) exp(2 πi Tr(
T τ )) up to 524decimal digits precision, we need to compute q , q and q with 524 + 36 = 560decimal digits precision.After making these 4 lists, we now simply loop over a = 0 , . . . , X , c = 0 , . . . , X and b = 0 , . . . , ⌊√ ac ⌋ and for the triples ( a, b, c ) with b − ac ≤ X we computegcd( a, b, c ) and look up the Fourier coefficient in the stored array.We implemented this algorithm in the computer algebra package Magma. Wedid not attempt to be as efficient as possible in our implementation. On our 64-bit,2.1 Ghz computer it took roughly 1 second to compute j ( τ ) , j ( τ ) , j ( τ ) up to 500decimal digits precision. We have j ( τ ) = 17399743914575167430246482183 . . . . for instance. The computation of the Fourier coefficients of the Eisenstein series isnegligible: the bottleneck is the ‘loop’ over all matrices (cid:16) ab/ b/ c (cid:17) satisfying 0 ≤ a ≤ X , 0 ≤ c ≤ X , | b | ≤ ⌊√ ac ⌋ . The evaluation of Igusa functions is a main ingredient in thecomputation of Igusa class polynomials, which is in turn used to construct e.g.hyperelliptic curves with cryptographic properties. We illustrate our algorithm byrecomputing j ( τ ) for a small CM-point τ .Let K = Q ( p − √
5) be a quartic CM field. The extension K/ Q is cyclicand K has class number two. Using [23, Algorithm 1], see also [25, Thm. 3.1], wecompute that τ ′ = (cid:16) . i . i . i . i (cid:17) is an approximation to the matrix τ representing the abelian surface C / Φ( O K ),where Φ is a CM-type for K . We will work with a 50 digit approximation to τ .As shown in [23], the values j i ( τ ) are in fact integers. Hence, we only need onedigit past the decimal place to recognize them and we take k = 1 in Theorem 1.2.The matrix τ already lies in the fundamental domain for Sp ( Z ) \ H , and we have δ ( τ ) ≥ .
66. Just as in the previous example, Lemma 6.2 does not apply directly.Using Remark 6.4, we compute c ≈ − . · − , where we include all matrices of trace up to 6. The corresponding integral is roughly equal to 1 . · − for ε = 0 . η = 1 .
45. We conclude that we may take n = 12 in Theorem 1.2.Just as in Example 7.1, we bound | χ ( τ ) | ≤ . · − . We lose at most 1 + 6 · − ·
12 = 13 digits of precision, and we need to know the evaluations of the fourSiegel modular forms up to precision 10 − . The integral Z ∞ B − t exp( − . πt )d t is less than 10 − for B = 9. To get all matrices of trace at most 9, we take allmatrices (cid:16) ab/ b/ c (cid:17) satisfying 4 ac − b ≤ = 81. We compute j ( τ ) = 6202728393749 . . . . which is accurate enough to derive j ( τ ) = 6202728393750.In this example, it turns out that we only needed to look at the matrices with4 ac − b ≤
6. The fact that our bound of 81 was much higher can be explainedas follows. Firstly, our analysis for the precision loss is for a worst case scenarioand we actually do not lose 14 digits of precision in this example. Secondly, we usethe same bound for all the Fourier coefficients of the matrices of a given trace t ,whereas these coefficients actually vary quite a lot.
8. Acknowledgements
We thank the referee for detailed comments on an earlier draft of this paper, andJeff Hoffstein for helpful discussions.
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