Siegel modular forms of degree three and invariants of ternary quartics
aa r X i v : . [ m a t h . N T ] D ec SIEGEL MODULAR FORMS OF DEGREE THREEAND INVARIANTS OF TERNARY QUARTICS
REYNALD LERCIER AND CHRISTOPHE RITZENTHALER
Abstract.
We determine the structure of the graded ring of Siegel modular forms of degree3. It is generated by 19 modular forms, among which we identify a homogeneous system ofparameters with 7 forms of weights , , , , , and . We also give a completedictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators. Introduction and main results
Let g ≥ be an integer and let R g (Γ g ) denote the C -algebra of modular forms of degree g forthe symplectic group Sp g ( Z ) (see Section 2 for a precise definition). It is a normal and integraldomain of finite type over C , closely related to the moduli space of principally polarized abelianvarieties over C . But even generators of these algebras are only known for small values of g : g = 1 is usually credited to Klein [Kle90, FK65] and Poincaré [Poi05, Poi11], g = 2 to Igusa[Igu62] and g = 3 to Tsuyumine [Tsu86]. In the latter, Tsuyumine gives generators and asks ifthey form a minimal set of generators. We answer in the negative and prove in the present paperthat there exists a subset of of them which still generates the algebra and which is minimal(Theorem 3.1). As a by-product we also exhibit a (possibly incomplete) set of relations anduse them to obtain a homogeneous system of parameters for this algebra (Theorem 3.3).Unlike Tsuyumine, we extensively use computer algebra software since we base our strategy onevaluation/interpolation which leads to computing ranks and invert large dimensional matrices.Still, a naive application of this strategy would have forced us to work with complex numbers,which would have been bad for efficiency but also to certify our computations. Hence, in orderto perform exact arithmetic computations, we make a detour through the beautiful geometryof smooth plane quartics and Weber’s formula [Web76] which allows us to express values (ofquotients) of the theta constants and ultimately modular forms as rational numbers (up to afourth root of unity). The strategy could be interesting for future investigations for g = 4 asthose theta constants can be computed in a similar way [Çel19].We then move on to a second task in the continuation of the famous Klein’s formula, see[Kle90, Eq. 118, p. 462] and [LRZ10, MV13, Ich18a]. This formula relates a certain modularform of weight , namely χ , to the square of the discriminant of plane quartics. A completedictionary between modular forms and invariants was only known for g = 1 and g = 2 . For g = 3 ,these formulas can come in two flavors: restricting to the the image of the hyperelliptic locus inthe Jacobian locus, one gets expressions of the modular forms in terms of Shioda invariants forbinary octics, see [Tsu86] and [LG19]; considering the generic case, one gets expressions in termsof Dixmier-Ohno invariants for ternary quartics, see Proposition 4.3. Extra care was taken in Date : December 18, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Siegel modular forms, plane quartics, invariants, generators, explicit. aking these formulas as normalized as possible using the background of [LRZ10] and also toeliminate parasite coefficients coming from relations between the invariants as much as possible.As a striking example, the modular form χ is equal to − · I I (the exponent of is large because the normalization chosen by Dixmier for I is not optimal at ). We finallygive formulas in the opposite direction and express all Dixmier-Ohno invariants as quotients ofmodular forms by powers of I , see Proposition 4.5. We hope that such formulas may eventuallylead to a set of generators for the ring of invariants of ternary quartics with good arithmeticproperties. Indeed, theta constants have intrinsically good “reduction properties modulo primes”(in the sense that they often have a primitive Fourier expansion) and may help guessing such aset of generators.The full list of expressions for the Siegel modular forms either in terms of the theta constantsor in terms of curve invariants, the expressions of Dixmier-Ohno invariants in terms of Siegelmodular forms and the relations in the algebra, are available at [LR19]. Acknowledgments.
We warmly thank the anonymous referees for carefully reading this workand for suggestions. This work is partially supported by the French National Research Agencyunder the anr - - ce - - cl ap -cl ap project.2. Review of Tsuyumine’s construction of Siegel modular forms
We recall here the definition of the generators for the C -algebra of modular forms of degree built by Tsuyumine. Surprisingly, they all are polynomials in theta constants with rationalcoefficients: one knows that when g ≥ , there exists modular forms which are not in the algebragenerated by theta constants [SM86], while the answer for g = 4 is pending [OPY08]. We takespecial care of the multiplicative constant involved in each expression.2.1. Theta functions and theta constants.
Let g ≥ be an integer and H g = { τ ∈ M g ( C ) , t τ = τ, Im τ > } . Definition 2.1.
The theta function with characteristics [ ε ε ] ∈ M ,g ( Z ) is given, for z ∈ C g and τ ∈ H g , by θ [ ε ε ]( z, τ ) = X n ∈ Z g exp( iπ ( n + ε / τ t ( n + ε /
2) ) exp( 2 iπ ( n + ε / t ( z + ε /
2) ) . The theta constant (with characteristic [ ε ε ] ) is the function of τ defined as θ [ ε ε ]( τ ) = θ [ ε ε ](0 , τ ) . Proposition 2.2.
Let z ∈ C g , τ ∈ H g , [ ε ε ] ∈ M ,g ( Z ) , then θ [ ε ε ]( − z, τ ) = θ (cid:2) − ε − ε (cid:3) ( z, τ ) , (2.1) and ∀ (cid:2) δ δ (cid:3) ∈ M ,g (2 Z ) , θ (cid:2) ε + δ ε + δ (cid:3) ( z, τ ) = exp( iπ ε δ / θ [ ε ε ]( z, τ ) . (2.2)Combining these two equations shows that z θ [ ε ε ]( z, τ ) is even if ε ε ≡ , andodd otherwise. The characteristics [ ε ε ] are then said to be even and odd, respectively.The modular group Γ g := Sp g ( Z ) acts on H g by τ → M.τ := ( A τ + B ) ( C τ + D ) − for M = ( A BC D ) , (2.3)and on characteristics by [ ε ε ] → M. [ ε ε ] = ( ε ⌢ ε ) M + ( t A C ) ∆ ⌢ ( t B D ) ∆ . ere, “ ⌢ ” denotes the concatenation of two row vectors, and “ ( . ) ∆ ” denotes the row vector equalto the diagonal of the square matrix given in argument. These result in the following action of Γ g on theta constants. Proposition 2.3 (Transformation formula [Igu72, Chap. 5, Th. 2][SM89, p.442] [Cos11, Prop. 3.1.24]) . Let τ ∈ H g , [ ε ε ] ∈ M ,g ( R ) and M ∈ Γ g , then θ [ ε ε ]( M.τ ) = ζ M p det( C τ + D ) exp( − iπ σ/ θ (cid:2) δ δ (cid:3) ( τ ) (2.4) with (cid:2) δ δ (cid:3) = M. [ ε ε ] , ζ M an eighth root of unity depending only on M and σ = ε A t B t ε + 2 ε B t C t ε + ε C t D t ε + ( 2 ε A + 2 ε C + ( t A C ) ∆ ) t ( t B D ) ∆ . In the following, we only make use of theta constants with characteristics with coefficients in { , } . Using Eq. (2.2) in combinaison with Eq. (2.4) allows to have a transformation formulapurely between characteristics of this form.To lighten notations, we number the theta constants as in [KLL + θ n := θ h δ δ ... δ g − ε ε ... ε g − i where ≤ n < g − is the integer whose binary expansion is “ δ · · · δ g − ε . . . ε g − ”.In genus 3, there are 36 even theta constants (the odd ones are all ). We give in Table 1 thecorrespondence between their numbering in [Tsu86, pp.789–790] and our binary numbering. Tsuyumine 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Binary θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ Tsuyumine 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36Binary θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ Table 1.
Tsuyumine’s numbering of even theta constants2.2.
Siegel modular forms.
Let Γ g ( ℓ ) denote the principal congruence subgroup of level ℓ , i.e. { M ∈ Γ g | M ≡ g mod ℓ } , and let Γ g ( ℓ, ℓ ) denote the congruence subgroup { M ∈ Γ g ( ℓ ) | ( t A C ) ∆ ≡ ( t B D ) ∆ ≡ ℓ } . For a congruence subgroup Γ ⊂ Γ g , let R g,h (Γ) be the C -vector space of analytic Siegelmodular forms of weight h and degree g for Γ , consisting of complex holomorphic functions f on H g satisfying f ( M.τ ) = det( C τ + D ) h · f ( τ ) for all M ∈ Γ . For g = 1 , one also requires that f is holomorphic at “infinity” but we will notlook at this case here. We also denote the C -algebra of Siegel modular forms of degree g for Γ by R g (Γ) := L R g,h (Γ) . The modular group acts on R g,h (Γ g ) by f → M.f := det( C τ + D ) − h · f ( M.τ ) . In particular, f ∈ R g,h (Γ) if and only if M.f = f for all M ∈ Γ .We now restrict to g = 3 . A strategy to build modular forms for Γ is first to construct aform F ∈ R (Γ (2)) , and then average over the finite quotient Γ / Γ (2) to get a modular form f ∈ R (Γ ) , namely f = X M ∈ Γ / Γ (2) M.F . (2.5) ll forms F which will be considered are polynomials in the theta constants, and are of evenweight. Hence, given an F , a careful application of the transformation formula (Proposition. 2.3)gives all summands, where we do not care about the choice of the square root as it is raised toan even power.Tsuyumine gives some of the building blocks F s in terms of maximal syzygetic sets of evencharacteristics [Tsu86, Sec. 21]. Multiplying the theta constants in a given set is an elementof R (Γ(2)) . The quotient Γ / Γ (2) acts transitively on these sets numbered from ((1)) to ((135)) by Tsuyumine. Among them, are actually used to define a set of generators for R (Γ ) . We give their expressions in Table 2. θ -monomial ((1)) θ θ θ θ θ θ θ θ ((2)) - θ θ θ θ θ θ θ θ ((3)) - θ θ θ θ θ θ θ θ ((4)) θ θ θ θ θ θ θ θ ((5)) θ θ θ θ θ θ θ θ ((18)) - θ θ θ θ θ θ θ θ ((31)) θ θ θ θ θ θ θ θ ((32)) - θ θ θ θ θ θ θ θ ((34)) θ θ θ θ θ θ θ θ ((36)) - θ θ θ θ θ θ θ θ ((37)) - θ θ θ θ θ θ θ θ θ -monomial ((38)) θ θ θ θ θ θ θ θ ((39)) - θ θ θ θ θ θ θ θ ((43)) θ θ θ θ θ θ θ θ ((45)) θ θ θ θ θ θ θ θ ((47)) - θ θ θ θ θ θ θ θ ((51)) - θ θ θ θ θ θ θ θ ((54)) - θ θ θ θ θ θ θ θ ((55)) θ θ θ θ θ θ θ θ ((73)) θ θ θ θ θ θ θ θ ((85)) θ θ θ θ θ θ θ θ ((89)) - θ θ θ θ θ θ θ θ θ -monomial ((90)) θ θ θ θ θ θ θ θ ((99)) - θ θ θ θ θ θ θ θ ((103)) θ θ θ θ θ θ θ θ ((111)) θ θ θ θ θ θ θ θ ((115)) - θ θ θ θ θ θ θ θ ((118)) - θ θ θ θ θ θ θ θ ((119)) - θ θ θ θ θ θ θ θ ((131)) θ θ θ θ θ θ θ θ ((132)) - θ θ θ θ θ θ θ θ ((133)) θ θ θ θ θ θ θ θ ((135)) θ θ θ θ θ θ θ θ Table 2.
Tsuyumine’s maximal syzygetic sequencesThen Tsuyumine considers F s written as combinations of • χ = Q θ i even θ i , • a rational function of the 36 non-zero θ i , • the monomials (( i )) defined in Table 2 , and • the squares of the gcd between two such (( i )) .Using the map from modular forms to invariants of binary octics introduced by Igusa [Igu67],he proves the following result. Theorem 2.4 (Tsuyumine [Tsu86, Sec. 20] ) . The graded algebra R (Γ ) is generated by the34 modular forms defined in Table 3. Its Hilbert–Poincaré series is generated by the rational See [Tsu89, p. 44] for the (1 − T ) misprint in the denominator of Equation (2.6) in [Tsu86]. ame [Tsu86] Coeff. F ∈ R (Γ(2)) sum. χ / (2 · · · Q θ i even θ i χ / (2 · · · χ / ((131)) α / (2 · ·
7) gcd( ((131)) , ((132)) ) α / (2 · · θ · ((131)) 1080 α - / (2 · · ·
11) ( θ θ θ θ θ θ ) · ((131)) 30240 α / (2 · ·
5) ( θ θ θ θ θ θ ) α ′ / ((85)) · ((119)) / ( θ θ ) α - / ((85)) · ((119)) · ((131)) 3780 α - / θ ((85)) · ((119)) · ((131)) 7560 α / (2 ·
5) (((85)) · ((119)) · ((131)) / ( θ θ ) α / θ ((85)) · ((119)) · ((131)) / θ α / (2 ·
5) (((85)) · ((119)) · ((131)) / ( θ θ ) β / (2 · · θ χ / ( ((5)) · ((54)) ) 4320 β / (2 ·
3) ((31)) · ((43)) · ((47)) · ((51)) 7560 β - / (2 ·
3) ( θ θ θ θ θ θ ) χ / ( ((2)) · ((54)) ) 30240 β ′ χ ((119)) · ((133)) / ( θ θ ((18)) · ((34)) ) 90720 β - / ((32)) · ((36)) · ((37)) · ((45)) · ((90)) · ((111)) · ((135)) / θ β - / ((32)) · ((36)) · ((37)) · ((45)) · ((90)) · ((111)) · ((135)) 362880 β / χ ((85)) · ((89)) · ((90)) · ((111)) · ((135)) / ( θ θ θ ((4)) · ((99)) ) 362880 β / (2 · θ χ ((90)) · ((111)) · ((135)) / ( θ θ ((3)) · ((31)) ) 120960 γ / (2 · θ χ ((135)) / ((1)) 7560 γ / θ χ / ( ((4)) · ((5)) · ((47)) · ((54)) ) 11340 γ / ( θ θ ) χ ((38)) · ((135)) / ((1)) 22680 γ / (2 ·
3) ( θ θ θ θ θ θ θ ) χ ((135)) / ((1)) 120960 c ′ - / θ χ ((90)) · ((111)) · ((135)) / ( ( θ θ ) ((1)) ) 30240 γ - / ( θ θ ) χ ((38)) · ((90)) · ((111)) · ((135)) / ( θ ((1)) ) 181440 γ / θ χ ((31)) · ((39)) · ((43)) / ( θ ((4)) · ((5)) · ((47)) · ((54)) ) 90720 c ′ / θ χ ((38)) · ((90)) · ((111)) · ((135)) / ( θ ((1)) ) 362880 γ / χ (((38)) · ((85)) · ((90)) · ((111)) · ((119)) · ((135)) / ( ( θ θ ) ((1)) ) 181440 γ / χ θ ((45)) · ((55)) · ((103)) / ( ( θ θ ) ((4)) · ((5)) · ((47)) · ((54)) ) 90720 δ / θ θ ) χ ((47)) · ((115)) · ((118)) / ((1)) 90720 δ / ( θ θ θ ) χ ((31)) · ((38)) · ((118)) · ((135)) / ((1)) 181440 δ - / θ θ ) χ ((31)) · ((38)) · ((90)) · ((111)) · ((118)) · ((135)) / ((1)) 725760 c / θ χ ((31)) · ((38)) · ((90)) · ((111)) · ((118)) · ((135)) / ((1)) 725760 Table 3.
Tsuyumine’s generators (the index is their weight), Tsuyumine’s nor-malization constant, the form F and the number of summands of the polynomialin the theta constants function ( 1 + T ) N ( T )(1 − T ) (1 − T ) (1 − T ) (1 − T ) (1 − T ) (1 − T ) , (2.6) here N ( T ) = 1 − T + T + T + 3 T − T + 3 T + 2 T + 2 T + 3 T + 4 T + 2 T + 7 T + 3 T + 7 T + 5 T + 9 T + 6 T + 10 T + 8 T + 10 T + 9 T + 12 T + 7 T + 14 T + 7 T + 12 T + 9 T + 10 T + 8 T + 10 T + 6 T + 9 T + 5 T + 7 T + 3 T + 7 T + 2 T + 4 T + 3 T + 2 T + 2 T + 3 T − T + 3 T + T + T − T + T . The modular forms f defined in Table 3 are all polynomials in the theta constants whoseprimitive part has all its coefficients equal to ± and whose content is c ( f ) = / Γ (2) { summands of f } = 2 · · · { summands of f } ∈ Z . In order to get simpler expressions when restricting to the hyperelliptic locus or to the decom-posable one, Tsuyumine multiplies each f by an additional normalization constant ( nd columnof Table 3). For instance, as defined by Tsuyumine, χ := 2 − · − · − · − X M ∈ Γ g / Γ g (2) M. ( χ / ((131)) ) , and therefore the 135 summands are each a (monic) monomial in the theta constants times ± (2 − · − · − · − ) · c ( χ ) = ± / (the sign depends on the monomial).Having in mind possible applications of our results to fields of positive characteristic, wereplace the multiplication by Tsuyumine’s constant by a multiplication by /c ( f ) . In this way, f is a sum of (monic) monomials in the theta constants with coefficients ± . To avoid confusionwith Tsuyumine’s notation, our modular forms will be denoted with bold font. Typically, χ :=30 χ , α := 112 α , α := α , α := 165 α , etc.Still driven by the link with the hyperelliptic locus, Tsuyumine adds to c ′ (resp. c ′ and c ) some polynomials in modular forms of smaller weights and denote the result γ (resp. γ and δ ). Theorem 2.4 as stated in [Tsu86] considers modular forms γ , γ and δ , instead of c ′ , c ′ and c . The two theorems are obviously equivalent. Here, we choose instead to define γ := c ′ / , γ := c ′ and δ := c . Remark . Some of the modular forms in Table 3 have a large number of summands. While itwould be cumbersome to store them, evaluating them is relatively quick as it basically consistsin permuting theta constants up to some eighth roots of unity according to Eq. (2.4). FollowingTsuyumine, the sum is computed in two steps. Let Θ be the subgroup of Γ conjugate to Γ (1 , that stabilizes θ ( Γ (1 , stabilizes θ ). Tsuyumine gives explicit coset representativesfor Γ / Θ (36 elements) and Θ / Γ (2) ( elements) and splits the sum in Eq. (2.5) as f = X M ′ ∈ Γ / Θ M ′ . X M ′′ ∈ Θ / Γ (2) M ′′ .F We use this approach in order to perform the computation of the summands . In order to dothat, we also need the eighth roots of unity ζ M ′ and ζ M ′′ from Proposition. 2.3. One approach isto precompute them using a fixed chosen matrix in H . A better solution is, with the notationof Eq. (2.3), to make use of the relation ζ M = ( − tr( B t C ) [Igu72, Chap. 5]. Since the modularforms have even weight, the degree of F in the theta constants is a multiple of 4, as well as thepowers of ζ M ′ and ζ M ′′ . There are two small typos in [Tsu86, pp. 842–846], the (3 , -th coefficients of “ M ” must be -1 instead of 1,and the (2 , -th coefficients of “ M ” must be 1 instead of . This modification makes M and M symplectic. . A minimal set of generators for modular forms of degree Fundamental set of modular forms.
Since we know the dimensions of each R ,h (Γ ) from the generating functions of Theorem 2.4, it is a matter of linear algebra to check thata given subset of Tsuyumine’s generators is enough for generating the full algebra. It wouldinvolve choosing a monomial ordering on the ring of theta constants and computing a Gröbnerbasis of the homogeneous ideal defined by the generating subset given as formal expressions interms of them (see [DK15, Section 1.4.1]). However, it is difficult to perform these computationssince there exist numerous algebraic relations between the theta constants. Therefore we favoran interpolation/evaluation strategy as follows.Suppose that we want to prove that a given form f of weight h , given as a polynomial in thetheta constants, can be obtained from a given set { f , . . . , f m } . This set produces F , . . . , F n ,homogeneous polynomials in the f i of weight h . If n < d = dim R ,h (Γ ) , then all forms ofweight h cannot be obtained. Assume that n ≥ d . Then, if we can find ( τ i ) i =1 ,...,d ∈ H dg suchthat the matrix ( F i ( τ j )) ≤ i,j ≤ d is of rank d , we know that f can be written in terms of the f i ,and even find such a relation. Equivalently, we will actually find a polynomial relation between f /θ h and the f i /θ w i where w i denotes the weight of f i .By Remark 2.5, the evaluation of a form f ( τ ) /θ h ( τ ) boils down to the computation ofquotients ( θ i /θ )( τ ) . A naive approach would be to use an arbitrary matrix τ ∈ H . But thenthe theta constants would in general be transcendental complex numbers which would makethe computations much more costly and the final result hard to certify. We therefore prefer toconsider a complex torus Jac C attached to a smooth plane quartic C given by an Aronholdsystem. Indeed (see for instance [Web76, Rit04, NR17]), general lines in ¶ form an Aronholdsystem of bitangents for a unique plane quartic C . Then, one can easily recover the equations ofthe other bitangents and an expression of the quotients ( θ i /θ ) ( τ ) in terms of the coefficientsof the linear forms defining the bitangents (see for instance [NR17, Theorems 2 and 3]). Notethat we do not explicitly know the Riemann matrix τ here, since it depends not only on C butalso on the choice of a symplectic basis for H ( C, Z ) . But when each of the bitangents in theAronhold system is defined over Q , all computations can be performed over Q and ( θ i /θ ) ( τ ) is a rational number.To remove the fourth root of unity ambiguity that remains, we start by computing indepen-dently an approximation over C of an explicit Riemann matrix τ ′ for the curve C . We needto do it only at very low precision (a typical choice is 20 decimal digits) and this can be doneefficiently either in maple (package algcurves by Deconinck et al. [DvH01]) or in magma (package riemann surfaces by Neurohr [Neu18]). Then, we can calculate an approximationof the theta constants at τ ′ .To conclude, note that [NR17, Theorem 3.1] shows that the set n θ j /θ i o running throughevery even theta constants θ i , θ j depends only on C and not on the Riemann matrix. Indeed,the dependence on this matrix relies only on the quadratic form q (in the notation of loc. cit. )whose contribution disappears in the eighth power. Therefore, there exist an integer i and apermutation σ such that θ σ ( i ) ( τ ′ ) θ i ( τ ′ ) = θ i ( τ ) θ ( τ ) . e simply enumerate all the possible candidates for i until we find a suitable σ that gives i and σ . Then, since we know θ σ ( i ) ( τ ′ ) /θ i ( τ ′ ) with small precision and its eighth power exactly,it is possible to obtain the exact value of θ i ( τ ) /θ ( τ ) .Using this method extensively leads to a set of generators for R (Γ ) . Moreover it is easy toprove, by the same algorithms, that this set is fundamental , i.e. one cannot remove any elementand still generate the algebra R (Γ ) . Theorem 3.1.
The Siegel modular forms α , α , α , α , α ′ , β , α , β , χ , α , α , γ , β , β ′ , α , γ , γ , χ and α define a fundamental set of generators for R (Γ ) .Remark . Note that [Run95] proved that R (Γ (2)) has a fundamental set of generators of elements.A word on the complexity. The proof mainly consists in checking for all the even weight h between 4 and 48 that there exists an evaluation matrix of rank dim R ,h (Γ ) for this set of modular forms. It is a matter of few hours for the largest weight to perform this calculation in magma . Most of the time is spent on the evaluation of the forms f i at a matrix τ j , whichtakes about 1 minute on a laptop.Additionally, we find the expressions of the remaining 15 modular forms given in Table 3.The first ones are · · · · β = 7 α α − α β − α β + 194040 α ′ β − α α − α α β + 16660 α β − α χ − α α + 2822512 α γ − α β + 36960 α β ′ − γ , · · · β = − α α − α α β + 66885 α α β + 129654 β − α β + 77792400 α β + 207446400 α ′ β + 5399533440 α α χ − α χ − α α + 320544000 α α + 82576256 α γ − α β − α β ′ − α α − α γ − χ , · · · δ = − β β + 23040 α χ + 987840 α χ + 47508930 α χ + 133358400 α ′ χ − α α γ + 46305 α γ − α γ + 282240 α γ , · · · γ ′ = χ ( α α − β ) . The last ones, for instance γ , δ and δ , tend to be heavily altered with the relations thatexist between these 19 modular forms, and have huge coefficients (thousands of digits).3.2. Module of relations between the generators.
We now quickly deal with the relationsdefining the algebra R (Γ ) . With the same techniques, involving modular forms up to weight (see Remark 4.4 for speeding up the computations), we find a (possibly incomplete) list of relations for our generators of R (Γ ) given by weighted polynomials of degree to ( cf. Table 4). Weight 32 34 36 38 40 42 44 46 48 50 52 54 56 58Number 1 1 2 4 5 5 7 6 8 6 5 2 2 1
Table 4. number of relations of a given weight in R (Γ ) he relations of weight and are relatively small, − β + 50854572195840 β α − α + 13916002383360 α β − χ β + 1109304189987840 γ α p − α α p − γ α + 1463891159808000 α α + 474409172160 β p α + 355806879120 β α + 8471592360 α p α − α α + 14993672601600 γ α − β α p α + 559752621120 β α α + 14755739264 β α α − α α α − γ α + 10174277836800 α α − α α + 7065470720 β α + 779296133468160 χ α − β α − β α p α + 1510363895040 β α α − α α α + 59052646477440 χ α α − β p α α − β α α + 16149647360 β α α + 642585968640 γ α + 516363724800 α α + 1529966592 β α α − χ α α − γ α − α α − α α + 97574400 β α α + 223027200 β α , − α β − χ β + 130691232000 α α + 123503214240000 χ α + 711613758240 γ β + 242595599400 β α ′ − β ′ α − γ α + 670881657600 χ α − β α + 2662228800 β α ′ α + 3993343200 β α α + 80673600 α α α + 699198091200 χ α α − β ′ α − β α + 657308736 β β α − α β α + 37811907302400 χ α ′ α + 16298463535200 χ α α + 5427686880 γ α α + 21254365440 γ α α − β α α α − χ α α − γ α +27165600 β α ′ α − β α α − β α α +9466800 α α α − γ α α + 5145 α α α + 5174400 β ′ α − β α − χ α . Runge [Run93, Cor.6.3] shows that R (Γ ) is a Cohen-Macaulay algebra. There exists a stronglink between a minimal free resolution of a Cohen-Macaulay algebra and its Hilbert series. Letus rewrite Equation (2.6) as a rational fraction with denominator Q d i (1 − T d i ) where the degrees d i run through the weights of the fundamental set of generators. We obtain a numerator with140 non-zero coefficients, the first and last ones of which are − T − T − T − T − T − T − T − T − T − T − T + 4 T + 9 T + 15 T + 22 T + 27 T + 32 T + 36 T + 39 T + 36 T + 34 T + 26 T + . . .. . . − T − T − T − T − T − T − T − T − T − T + T . The coefficients of the numerator give information on the weights and numbers of relations.They are consistent with Table 4 up to weight . The drop from (relations) to a coefficient in weight indicates that there is a first syzygy ( i.e. a relation between the relations) ofweight 50.3.3. A homogeneous system of parameters.
Having these relations, one can also try towork out a homogeneous system of parameters ( hsop ) for R (Γ ) . Recall that this is a set ofelements ( f i ) ≤ i ≤ m of the algebra, which are algebraically independent, and such that R (Γ ) is a C [ f , . . . , f m ] -module of finite type. Equation (2.6) suggests that a hsop of weight , , , , , and may exist. An easy Gröbner basis computation made in magma with thelexicographic order α < α < . . . < γ < χ shows that when we set to zero α , α , α ′ , β , χ , α and α in the relations of Table 4, the remaining Siegel modular forms ofthe generating set of Theorem 3.1 must be zero as well. As it is well known that the dimensionof Proj ( R (Γ )) is , this yields the following theorem. Theorem 3.3.
A homogeneous system of parameters for R (Γ ) is given by the forms α , α , α ′ , β , χ , α and α . . A dictionary between modular forms and invariants of quartics
In [Dix87], Dixmier gives a homogeneous system of parameters for the graded C -algebra I of invariants of ternary quartic forms under the action of SL ( C ) . They are denoted I , I , I , I , I , I and I . This list is completed by Ohno with six invariants, J , J , J , J and J , into a list of 13 generators for I , the so-called Dixmier-Ohno invariants [Ohn07, Els15].Note that · I = D where D denotes the normalized discriminant of plane quartics inthe sense of [GKZ94, p.426] or [Dem12, Prop.11].Using the morphism ρ defined in [Igu67], Tsuyumine in [Tsu86, pp. 847–864] relates each ofthe Siegel modular forms given in Table 3 with an invariant for the graded ring of binary octicsunder the action of SL ( C ) . He uses this key argument to prove Theorem 2.4. More generally,there is a way to canonically associate an invariant to a modular form. After briefly recallingthe way to do so when g = 3 , we establish a complete dictionary between R (Γ ) and I .4.1. Modular forms in terms of invariants.
Let us recall from [LRZ10, 2.2] how to associatean element of I to f ∈ R ,h (Γ ) . This morphism only depends on the choice of a universalbasis of regular differentials ω which can be fixed “canonically” for smooth plane quartics (in thesense that it is a basis of regular differentials over Z ). Let Q ∈ C [ x , x , x ] be a ternary quarticform such that C : Q = 0 is a smooth genus curve. Let Ω = h Ω Ω i be the × period matrixof C defined by integrating ω C with respect to an arbitrary symplectic basis of H ( C, Z ) . Wehave τ = Ω − Ω ∈ H . The function Q Φ ( f )( Q ) = (cid:18) (2 iπ ) det Ω (cid:19) h · f ( τ ) (4.1)is a homogeneous element of I of degree h (identifying the polynomial with its polynomialfunction). Remark . A similar construction can be worked out with invariants of binary octics (see[IKL + f has coefficientsin a ring R ⊂ C , then Φ ( f ) is defined over R as well. When f is given by a polynomial in thetheta constants with coefficients in Z , we can take R = Z . A particular case is given by themodular form χ which is the product of the theta constants. In [LRZ10] (see also [Ich18b])one shows the following precise form of Klein’s formula [Kle90, Eq. 118, p. 462], Φ ( χ ) = − · D = − · ( 2 I ) . (4.2) Remark . The map (4.1) is obtained by pulling back geometric modular forms to invariantsas described in [LRZ10]. Within this background, it is for instance possible to speak about thereduction modulo a prime of modular forms and to consider the algebra that they generate.In small characteristics, one still encounters similar accidents as in the case of invariants. Wewill not study this question further here, but for instance, our generators have a surprisingcongruence modulo , β + 9 α + 3 α α = 0 mod 11 . e have seen in Section 3 that we have an evaluation/interpolation strategy to handle quotientof modular forms by a power of θ . This strategy can also be used to find the relations withinvariants. But now, we also need to take care of the transcendental factor µ := (2 iπ ) / det Ω .This is done in the following way.(i) Assume that a relation Φ ( f ) = I is known for a modular form f of weight h . Thisis the case for χ ( cf. Eq. (4.2)) and we will start with this one, but switch to arelation of lower weight ( i.e. with α or even with χ / α ) after a first round ofthe following steps (this simplifies the last step).(ii) Let now f be one of the generators from Theorem 3.1 of weight h and compute abasis j , . . . , j d of invariants of degree h . We aim at finding a , . . . , a d ∈ Q such that Φ ( f ) = P a i j i . This is done by evaluation/interpolation at Riemann models until onegets a system of d linearly independent equations. More precisely, for a given Q = 0 and an associated τ ∈ H :(a) Compute the values of ( j , . . . , j d ) at Q ;(b) Using the same procedure as in Section 3, compute ( f /θ h )( τ ) and ( f /θ h )( τ ) ;(c) Let p = lcm ( h , h ) . Since ( f /θ h ) p/h ( f /θ h ) p/h = ( µ h f ) p/h ( µ h f ) p/h = Φ ( f ) p/h Φ ( f ) p/h , we get the value of Φ ( f ) p/h . An approximate computation at low precision canthen give the exact value.The above strategy provides explicit expressions for Φ ( f ) where f is any modular form in thefundamental set defined in Theorem 3.1. Proposition 4.3.
Let f be a modular form of weight h from Theorem 3.1. There exists anexplicit polynomial P f of degree h in the Dixmier-Ohno invariants such that Φ ( f ) = P f ( I , I , . . . , I ) . The first ones are Φ ( α ) = 2 · · I − I − J I + 117 I I + 14418 I I + 8 I ) , · ( α ) = - · (40415760 J − I − J − J I + 2506140 I − J I − I I + 135992908800 I − J I + 2143260 I I + 247160160 J I I + 289325520 I I I + 400950 J I − I I − I I + 1527453 J I − I I − I I − I ) , Φ ( α ) = 2 · I J − I I − I I I + 32 I I ) , Φ ( β ) = 2 · ( − I J − I I + 285120 I J I − I I I + 810 I J I +12204 I I I − I I I − I J I +2961 I I I +213912 I I I − I I ) , ( β ) = - · (540 I J − I I − I I + 4005 I J I − I I I − I I I + 56 I I ) . Beside Klein’s formula Φ ( χ ) = - I , one finds a surprisingly compact expression for χ , Φ ( χ ) = - · I I . If we do not not pay attention, the rational coefficients of these formulas tend to have primefactors greater than 7 in their denominators, especially for the forms of higher weight. Wehave eliminated all these “bad primes” using the relations that exist between the Dixmier-Ohno We make available the list of these 19 polynomials at [LR19, file “
SiegelMFfromDO.txt ”]. nvariants. It is also a good way to reduce the size of these expressions significantly. All in all,we gain a factor of 3 in the amount of memory to store the results ( cf. Table 5).
Form Content Terms Digits α · · α - · · - · -
19 12 α - · · - · -
98 23 α · α p · · - · -
200 26 β ·
11 8 α · · - · -
703 35 β · · - · -
29 16 χ - α - · · - · -
813 36
Form Content Terms Digits α · · - · - γ · β - · · - β p · · - · - α · · - · - γ · · - · - γ - · · - · - χ · α · · - · - Table 5.
Polynomial expressions of the modular forms from Theorem 3.1 interms of Dixmier-Ohno invariants: their content, their number of monomials,and the number of digits of the largest coefficient of their primitive part.
Remark . When we deal with the Jacobian of a curve with coefficients in Q , what is a matterof few integer arithmetic operations to evaluate modular forms from invariants is a matter ofhigh precision floating point arithmetic over the complex numbers with analytic computationsof Riemann matrices. In practical calculations, such as the computations in Section 3.2, it isthus much better to use the former, since a calculation that would take the order of the minuteultimately requires only a few milliseconds.4.2. Invariants in terms of modular functions.
Conversely, we can look for expressions ofa generating set of invariants in terms of modular forms. Using [Tsu86, LG19], one obtains sucha result for invariants of binary octics. We focus here on the case of Dixmier-Ohno invariants.Since the locus of plane quartic over C such that I = 0 corresponds to the locus of non-hyperelliptic curve of genus and then to principally polarized abelian threefolds C / ( τ Z + Z ) for which χ ( τ ) = 0 [Igu67, Lem. 10, 11], we see that any invariant in I can be obtained as aquotient of a modular form by a power of I . Proposition 4.5.
Let I be a Dixmier-Ohno invariant of degree k . There exist a polynomial P I in the modular forms from Theorem 3.1, of weight k , such that I k · I = Φ ( P I ( α , α , . . . , α ) ) . (4.3)The first ones are · I I = Φ ( - χ ) , · · I I = Φ ( χ − · χ γ ) , · · · I I = Φ ( ( − α ′ − α − α − α ) χ + (3259872 β β − γ α + 21732480 γ α − γ α + 137984 γ α α ) χ + 153856080 χ γ χ − χ ) , · · · I J = Φ ( ( − α ′ − α − α − α ) χ + (8594208 β β − γ α + 57294720 γ α − γ α + 363776 γ α α ) χ + 558376560 χ γ χ − χ ) . We make available the list of these 13 polynomials at [LR19, file “
SiegelMFtoDO.txt ”]. n this setting, one can also write I I = Φ ((2 − χ ) ) .Unlike the previous computations, one cannot obtain the above ones by a direct applicationof the evaluation/interpolation strategy as the degrees (and weights) are sometimes too large.For the invariant I , for instance, one would potentially need to interpolate on a vector spaceof modular forms of weight 196, which is huge (its dimension is
869 945 ). The trick is to proceedby steps and first look for expressions of a small power of I by the desired invariant I , not onlyin terms of modular forms, but also in terms of invariants I k of smaller degrees. For instancein the case of I , · · · · I I = 2 · · · · I ( − J J + 5680595070 I J + 109296000 J I − I I − J I + 439538400 I I − J I + 2235454502400 I I + 8070768720 J I − I I − J I + 1754339490 J I I − I I + 70135124400 J I I − I I I − I I − J I + 1352865780 I I + 237928085190 J I I − I I I + 294430290 J I − I I − I I − J I − I I + 4016874680 I I ) + 2 · · · ( − β − α − α α − α α ) . Then, mechanically, through a sequence of substitutions of the invariants of smaller degrees bytheir expression in terms of the modular forms, we arrive to expressions for I k I k purely interms of modular forms. These formulas are very sparse, considering their weight (see Table 5). DO inv. Content Terms Digits I - · - I - · - · I - · - · · -
11 11 J - · - · - · -
11 11 I - · - · - · -
13 13 J - · - · · -
14 13 I - · - · - · - · -
58 17 J - · - · - · - · -
58 17 I - · - · - · - · - · - J - · - · - · - · - · - I - · - · - · - · - · - J - · - · - · - · - · - I - Table 6.
Polynomial expressions of the Dixmier-Ohno in terms of the gener-ators from Theorem 3.1: the content, the number of monomials, and the numberof digits of the largest coefficient of the primitive parts. Remark . It is not a coincidence that the power of I is k in Equation (4.3). Let usconsider ternary quartics of the form Q + p G , where p is a prime integer, Q is a ternaryquadratic form and G is ternary quartic form. Generically, for all but I the valuation of p ofthe Dixmier-Ohno invariants of these forms is zero, and υ p ( I ) = 14 . And, still generically, wehave υ p (Φ ( f )) = 3 h/ where f is any one of the Tsuyumine modular forms, and h is its weight.Thus, if the equation I κ · I = Φ ( P I ( α , α , . . . , α ) ) is satisfied, the power κ of I must be uch that the degrees agree, i.e. κ + 3 k = 3 h , and such that the valuations at p are equal, i.e. κ = 3 h/ . This yields κ = 3 k . Remark . We are also able to eliminate the primes greater than in the denominators of thecoefficients in these formulas using the relations that exist between Siegel modular forms ( cf. Section 3.2), with the notable exception of the primes and ( cf. Table 5). We suspect thatthe reason behind this difficulty is that, similarly to the prime ( cf. Remark 4.2), one cannotextend Theorem 3.1 mutatis mutandis to characteristic . Although we do not go further intothe topic, it is possible to work directly in these characteristics and find specific formulas validthere. References [Çel19] T. Ö. Çelik. A Thomae-like formula: algebraic computations of theta constants, 2019. Available at https://arxiv.org/abs/1901.08459 . 1[Cha86] C.-L. Chai. Siegel moduli schemes and their compactifications over C . In Arithmetic geometry (Storrs,Conn., 1984) , pages 231–251. Springer, New York, 1986. 10[Cos11] R. Cosset.
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Reynald Lercier, DGA & Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
E-mail address : [email protected] Christophe Ritzenthaler, Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
E-mail address : [email protected]@univ-rennes1.fr