Siegel modular forms of degree three and invariants of ternary quartics

TL;DR

This paper characterizes the structure of the graded ring of Siegel modular forms of degree 3, identifying generators and linking them to invariants of ternary quartics, thus advancing understanding of modular forms and algebraic invariants.

Contribution

It explicitly determines the generators of the graded ring of degree 3 Siegel modular forms and establishes a correspondence with ternary quartic invariants.

Findings

The graded ring is generated by 19 modular forms.

A homogeneous system of parameters with 7 specific forms is identified.

A complete dictionary between invariants and generators is provided.

Abstract

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

Tables

Table 1. Table 2. Tsuyumine’s maximal syzygetic sequences

Table 2. Table 4. number of relations of a given weight in 𝐑 3 ​ ( Γ 3 ) subscript 𝐑 3 subscript Γ 3 \mathbf{R}_{3}(\Gamma_{3})

Weight	32	34	36	38	40	42	44	46	48	50	52	54	56	58
Number	1	1	2	4	5	5	7	6	8	6	5	2	2	1

Table 3. Table 5. Polynomial expressions of the modular forms from Theorem  3.1 in terms of Dixmier-Ohno invariants: their content, their number of monomials, and the number of digits of the largest coefficient of their primitive part.

Table 4. Table 6. Polynomial expressions of the Dixmier-Ohno in terms of the 19 19 19 generators from Theorem  3.1 : the content, the number of monomials, and the number of digits of the largest coefficient of the primitive parts.

Equations

$$\theta \left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right](z,\tau )=\sum _{n\in {\mathbb{Z}}^g}\exp (i\pi (n+{\epsilon }_1/2)\tau \text{ltrans}(n+{\epsilon }_1/2))\exp (2i\pi (n+{\epsilon }_1/2)\text{ltrans}(z+{\epsilon }_2/2)).$$

$$\theta \left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right](-z,\tau )=\theta \left[\begin{array}{c}-{\epsilon }_1\\ -{\epsilon }_2\end{array}\right](z,\tau ),$$

$$\forall \left[\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right]\in {\mathbf{M}}_{2,g}(2\mathbb{Z}),\theta \left[\begin{array}{c}{\epsilon }_1+{\delta }_1\\ {\epsilon }_2+{\delta }_2\end{array}\right](z,\tau )=\exp (i\pi {\epsilon }_1\text{ltrans}{\delta }_2/2)\theta \left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right](z,\tau ).$$

$$\tau \to M.\tau :=(\text{A}\tau +\text{B})(\text{C}\tau +\text{D}{)}^{-1}\text{for}M=\left(\begin{array}{cc}\text{A}& \text{B}\\ \text{C}& \text{D}\end{array}\right),$$

$$\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right]\to M.\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right]=({\epsilon }_1{}^{⌢}{\epsilon }_2)M+(\text{ltrans}\text{A}\text{C}{)}_{\Delta }{}^{⌢}(\text{ltrans}\text{B}\text{D}{)}_{\Delta }.$$

$$\theta \left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right](M.\tau )={\zeta }_M\sqrt{\det (\text{C}\tau +\text{D})}\exp (-i\pi \sigma /4)\theta \left[\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right](\tau )$$

$$\sigma ={\epsilon }_1\text{A}\text{ltrans}\text{B}\text{ltrans}{\epsilon }_1+2{\epsilon }_1\text{B}\text{ltrans}\text{C}\text{ltrans}{\epsilon }_2+{\epsilon }_2\text{C}\text{ltrans}\text{D}\text{ltrans}{\epsilon }_2+(2{\epsilon }_1\text{A}+2{\epsilon }_2\text{C}+(\text{ltrans}\text{A}\text{C}{)}_{\Delta })\text{ltrans}(\text{ltrans}\text{B}\text{D}{)}_{\Delta }.$$

$$f(M.\tau )=\det (\text{C}\tau +\text{D}{)}^h\cdot f(\tau )$$

$$f\to M.f:=\det (\text{C}\tau +\text{D}{)}^{-h}\cdot f(M.\tau ).$$

$$f=\sum _{M\in {\Gamma }_3/{\Gamma }_3(2)}M.F.$$

$$\frac{(1+T^2)N(T)}{(1-T^4)(1-T^{12}{)}^2(1-T^{14})(1-T^{18})(1-T^{20})(1-T^{30})},$$

$$N(T)=1-T^2+T^4+T^{10}+3T^{16}-T^{18}+3T^{20}+2T^{22}+2T^{24}+3T^{26}+4T^{28}+2T^{30}+7T^{32}+3T^{34}+7T^{36}+5T^{38}+9T^{40}+6T^{42}+10T^{44}+8T^{46}+10T^{48}+9T^{50}+12T^{52}+7T^{54}+14T^{56}+7T^{58}+12T^{60}+9T^{62}+10T^{64}+8T^{66}+10T^{68}+6T^{70}+9T^{72}+5T^{74}+7T^{76}+3T^{78}+7T^{80}+2T^{82}+4T^{84}+3T^{86}+2T^{88}+2T^{90}+3T^{92}-T^{94}+3T^{96}+T^{102}+T^{108}-T^{110}+T^{112}.$$

$$c(f)=\frac{\#{\Gamma }_3/{\Gamma }_3(2)}{\#\{\text{summands of }f\}}=\frac{2^9\cdot 3^4\cdot 5\cdot 7}{\#\{\text{summands of }f\}}\in \mathbb{Z}.$$

$${\chi }_{28}:=2^{-10}\cdot 3^{-2}\cdot 5^{-1}\cdot 7^{-1}\sum _{M\in {\Gamma }_g/{\Gamma }_g(2)}M.({\chi }_{18}^2/((131){)}^2),$$

$$f=\sum _{M^{\prime }\in {\Gamma }_3/\Theta }{M}^{\prime }.\sum _{M^{\prime \prime }\in \Theta /{\Gamma }_3(2)}{M}^{\prime \prime }.F$$

$$\frac{{\theta }_{\sigma (i)}({\tau }^{\prime }{)}^8}{{\theta }_{i_0}({\tau }^{\prime }{)}^8}=\frac{{\theta }_i(\tau {)}^8}{{\theta }_0(\tau {)}^8}.$$

$$2^5\cdot 3^4\cdot 5\cdot 7^2\cdot 11{\mathbf{\beta }}_{26}=7{\mathbf{\alpha }}_6{\mathbf{\alpha }}_{10}^2-3080{\mathbf{\alpha }}_6^2{\mathbf{\beta }}_{14}-145530{\mathbf{\alpha }}_{12}{\mathbf{\beta }}_{14}+194040{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\beta }}_{14}-11760{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_{16}-7040{\mathbf{\alpha }}_4{\mathbf{\alpha }}_6{\mathbf{\beta }}_{16}+16660{\mathbf{\alpha }}_{10}{\mathbf{\beta }}_{16}-20824320{\mathbf{\alpha }}_4^2{\mathbf{\chi }}_{18}-4435200{\mathbf{\alpha }}_6{\mathbf{\alpha }}_{20}+2822512{\mathbf{\alpha }}_6{\mathbf{\gamma }}_{20}-55440{\mathbf{\alpha }}_4{\mathbf{\beta }}_{22}+36960{\mathbf{\alpha }}_4{\mathbf{\beta }}_{22}^{\prime }-105557760{\mathbf{\gamma }}_{26},$$

$$2^8\cdot 3^4\cdot 5^2\cdot 7^4{\mathbf{\beta }}_{28}=-105{\mathbf{\alpha }}_4^2{\mathbf{\alpha }}_{10}^2-42000{\mathbf{\alpha }}_4^2{\mathbf{\alpha }}_6{\mathbf{\beta }}_{14}+66885{\mathbf{\alpha }}_4{\mathbf{\alpha }}_{10}{\mathbf{\beta }}_{14}+129654{\mathbf{\beta }}_{14}^2-96000{\mathbf{\alpha }}_4^3{\mathbf{\beta }}_{16}+77792400{\mathbf{\alpha }}_{12}{\mathbf{\beta }}_{16}+207446400{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\beta }}_{16}+5399533440{\mathbf{\alpha }}_4{\mathbf{\alpha }}_6{\mathbf{\chi }}_{18}-9996323400{\mathbf{\alpha }}_{10}{\mathbf{\chi }}_{18}-4321800{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_{18}+320544000{\mathbf{\alpha }}_4^2{\mathbf{\alpha }}_{20}+82576256{\mathbf{\alpha }}_4^2{\mathbf{\gamma }}_{20}-12965400{\mathbf{\alpha }}_6{\mathbf{\beta }}_{22}-17287200{\mathbf{\alpha }}_6{\mathbf{\beta }}_{22}^{\prime }-666792000{\mathbf{\alpha }}_4{\mathbf{\alpha }}_{24}-700378560{\mathbf{\alpha }}_4{\mathbf{\gamma }}_{24}-442172001600{\mathbf{\chi }}_{28},$$

$$2^3\cdot 3^4\cdot 5\cdot 7^4{\mathbf{\delta }}_{30}=-37044{\mathbf{\beta }}_{14}{\mathbf{\beta }}_{16}+23040{\mathbf{\alpha }}_4^3{\mathbf{\chi }}_{18}+987840{\mathbf{\alpha }}_6^2{\mathbf{\chi }}_{18}+47508930{\mathbf{\alpha }}_{12}{\mathbf{\chi }}_{18}+133358400{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\chi }}_{18}-1568{\mathbf{\alpha }}_4{\mathbf{\alpha }}_6{\mathbf{\gamma }}_{20}+46305{\mathbf{\alpha }}_{10}{\mathbf{\gamma }}_{20}-246960{\mathbf{\alpha }}_6{\mathbf{\gamma }}_{24}+282240{\mathbf{\alpha }}_4{\mathbf{\gamma }}_{26},$$

$$2\cdot 3\cdot 5\cdot 7{\mathbf{\gamma }}_{32}^{\prime }={\mathbf{\chi }}_{18}({\mathbf{\alpha }}_4{\mathbf{\alpha }}_{10}-252{\mathbf{\beta }}_{14}).$$

$$0=-25226544365568{\mathbf{\beta }}_{16}^2+50854572195840{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{16}-25092716544000{\mathbf{\alpha }}_{16}^2+13916002383360{\mathbf{\alpha }}_{18}{\mathbf{\beta }}_{14}-18410871153185280{\mathbf{\chi }}_{18}{\mathbf{\beta }}_{14}+1109304189987840{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_{12p}-1951854879744000{\mathbf{\alpha }}_{20}{\mathbf{\alpha }}_{12p}-413549252645760{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_{12}+1463891159808000{\mathbf{\alpha }}_{20}{\mathbf{\alpha }}_{12}+474409172160{\mathbf{\beta }}_{22p}{\mathbf{\alpha }}_{10}+355806879120{\mathbf{\beta }}_{22}{\mathbf{\alpha }}_{10}+8471592360{\mathbf{\alpha }}_{12p}{\mathbf{\alpha }}_{10}^2-3882813165{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_{10}^2+14993672601600{\mathbf{\gamma }}_{26}{\mathbf{\alpha }}_6-1800579432960{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{12p}{\mathbf{\alpha }}_6+559752621120{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_6+14755739264{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6-25299240960{\mathbf{\alpha }}_{16}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6-4775514472960{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_6^2+10174277836800{\mathbf{\alpha }}_{20}{\mathbf{\alpha }}_6^2-43285228{\mathbf{\alpha }}_{10}^2{\mathbf{\alpha }}_6^2+7065470720{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_6^3+779296133468160{\mathbf{\chi }}_{28}{\mathbf{\alpha }}_4-530133424128{\mathbf{\beta }}_{14}^2{\mathbf{\alpha }}_4-2857212610560{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{12p}{\mathbf{\alpha }}_4+1510363895040{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_4-5020202880{\mathbf{\alpha }}_{18}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4+59052646477440{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4-104866460160{\mathbf{\beta }}_{22p}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4-38488222080{\mathbf{\beta }}_{22}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4+16149647360{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_6^2{\mathbf{\alpha }}_4+642585968640{\mathbf{\gamma }}_{24}{\mathbf{\alpha }}_4^2+516363724800{\mathbf{\alpha }}_{24}{\mathbf{\alpha }}_4^2+1529966592{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4^2-5609877504000{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4^2-130817347584{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_4^3-154557849600{\mathbf{\alpha }}_{20}{\mathbf{\alpha }}_4^3-1036728{\mathbf{\alpha }}_{10}^2{\mathbf{\alpha }}_4^3+97574400{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4^3+223027200{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_4^4,$$

$$0=-113265734400{\mathbf{\alpha }}_{18}{\mathbf{\beta }}_{16}-107036119008000{\mathbf{\chi }}_{18}{\mathbf{\beta }}_{16}+130691232000{\mathbf{\alpha }}_{18}{\mathbf{\alpha }}_{16}+123503214240000{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_{16}+711613758240{\mathbf{\gamma }}_{20}{\mathbf{\beta }}_{14}+242595599400{\mathbf{\beta }}_{22}{\mathbf{\alpha }}_{12}^{\prime }-121297799700{\mathbf{\beta }}_{22}^{\prime }{\mathbf{\alpha }}_{12}-107820266400{\mathbf{\gamma }}_{24}{\mathbf{\alpha }}_{10}+670881657600{\mathbf{\chi }}_{28}{\mathbf{\alpha }}_6-399334320{\mathbf{\beta }}_{14}^2{\mathbf{\alpha }}_6+2662228800{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\alpha }}_6+3993343200{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_6+80673600{\mathbf{\alpha }}_{18}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6+699198091200{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6-221852400{\mathbf{\beta }}_{22}^{\prime }{\mathbf{\alpha }}_6^2-665557200{\mathbf{\beta }}_{22}{\mathbf{\alpha }}_6^2+657308736{\mathbf{\beta }}_{16}{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_4-4978713600{\mathbf{\alpha }}_{16}{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_4+37811907302400{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\alpha }}_4+16298463535200{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_4+5427686880{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4+21254365440{\mathbf{\gamma }}_{24}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4-2545060{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4-194295615360{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_6^2{\mathbf{\alpha }}_4-2123573760{\mathbf{\gamma }}_{26}{\mathbf{\alpha }}_4^2+27165600{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{12}^{\prime }{\mathbf{\alpha }}_4^2-50935500{\mathbf{\beta }}_{14}{\mathbf{\alpha }}_{12}{\mathbf{\alpha }}_4^2-7299040{\mathbf{\beta }}_{16}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4^2+9466800{\mathbf{\alpha }}_{16}{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_4^2-1339238208{\mathbf{\gamma }}_{20}{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4^2+5145{\mathbf{\alpha }}_{10}^2{\mathbf{\alpha }}_6{\mathbf{\alpha }}_4^2+5174400{\mathbf{\beta }}_{22}^{\prime }{\mathbf{\alpha }}_4^3-19404000{\mathbf{\beta }}_{22}{\mathbf{\alpha }}_4^3-4011279360{\mathbf{\chi }}_{18}{\mathbf{\alpha }}_4^4.$$

$$\begin{gathered} 1-T^{32}-T^{34}-2T^{36}-4T^{38}-5T^{40}-5T^{42}-7T^{44}-6T^{46}-8T^{48}-5T^{50}-4T^{52} \\ +4T^{56}+9T^{58}+15T^{60}+22T^{62}+27T^{64}+32T^{66}+36T^{68}+39T^{70}+36T^{72}+34T^{74}+26T^{76}+\dots \\ \dots -5T^{296}-8T^{298}-6T^{300}-7T^{302}-5T^{304}-5T^{306}-4T^{308}-2T^{310}-T^{312}-T^{314}+T^{346}. \end{gathered}$$

$$Q\mapsto {\Phi }_3(f)(Q)={\left(\frac{(2i\pi {)}^3}{\det {\Omega }_2}\right)}^h\cdot f(\tau )$$

$${\Phi }_3({\mathbf{\chi }}_{18})=-2^{28}\cdot D_{27}^2=-2^{28}\cdot (2^{40}{I}_{27}{)}^2.$$

$${\mathbf{\beta }}_{16}+9{\mathbf{\alpha }}_{16}+3{\mathbf{\alpha }}_{10}{\mathbf{\alpha }}_6=0\mathrm{mod}11.$$

$$\frac{(f/{\theta }_0^{2h}{)}^{p/h}}{(f_0/{\theta }_0^{2h_0}{)}^{p/h_0}}=\frac{({\mu }^{h}f{)}^{p/h}}{({\mu }^{h_0}{f}_0{)}^{p/h_0}}=\frac{{\Phi }_3(f{)}^{p/h}}{{\Phi }_3(f_0{)}^{p/h_0}},$$

$${\Phi }_3(f)=P_f(I_3,I_6,\dots ,I_{27}).$$

$${\Phi }_3({\mathbf{\alpha }}_4)=2^{20}\cdot 3^3\cdot 7(486I_{12}-155520I_6^2-423J_9{I}_3+117I_9{I}_3+14418I_6{I}_3^2+8I_3^4),$$

$$5\cdot 7{\Phi }_3({\mathbf{\alpha }}_6)=\text{-}{2}^{28}\cdot 3^4(40415760J_{18}-1224720I_{18}-2664900J_9^2-8323560J_9{I}_9+2506140I_9^2-76982400J_{12}{I}_6-1143538560I_{12}{I}_6+135992908800I_6^3-40041540J_{15}{I}_3+2143260I_{15}{I}_3+247160160J_9{I}_6{I}_3+289325520I_9{I}_6{I}_3+400950J_{12}{I}_3^2-6206220I_{12}{I}_3^2-7357573440I_6^2{I}_3^2+1527453J_9{I}_3^3-266481I_9{I}_3^3-36764280I_6{I}_3^4-62720I_3^6),$$

Full text

1. Introduction and main results

Let $g\geq 1$ be an integer and let $\mathbf{R}_{g}(\Gamma_{g})$ denote the $\mathbb{C}$-algebra of modular forms of degree $g$ for the symplectic group $\operatorname{Sp}_{2g}(\mathbb{Z})$ (see Section 2 for a precise definition). It is a normal and integral domain of finite type over $\mathbb{C}$, closely related to the moduli space of principally polarized abelian varieties over $\mathbb{C}$. 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 $34$ generators and asks if they form a minimal set of generators. We answer in the negative and prove in the present paper that there exists a subset of $19$ 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 $55$ relations and use 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 on evaluation/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 order to perform exact arithmetic computations, we make a detour through the beautiful geometry of smooth plane quartics and Weber’s formula [Web76] which allows us to express values (of quotients) of the theta constants and ultimately modular forms as rational numbers (up to a fourth root of unity). The strategy could be interesting for future investigations for $g=4$ as those 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 modular form of weight $18$, namely $\bm{\chi}_{18}$, to the square of the discriminant of plane quartics. A complete dictionary 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 in the Jacobian locus, one gets expressions of the modular forms in terms of Shioda invariants for binary octics, see [Tsu86] and [LG19]; considering the generic case, one gets expressions in terms of Dixmier-Ohno invariants for ternary quartics, see Proposition 4.3. Extra care was taken in making these formulas as normalized as possible using the background of [LRZ10] and also to eliminate parasite coefficients coming from relations between the invariants as much as possible. As a striking example, the modular form $\bm{\chi}_{28}$ is equal to $-2^{171}\cdot 3^{3}\>I_{27}^{3}\,I_{3}$ (the exponent of $2$ is large because the normalization chosen by Dixmier for $I_{27}$ is not optimal at $2$). We finally give formulas in the opposite direction and express all Dixmier-Ohno invariants as quotients of modular forms by powers of $I_{27}$, see Proposition 4.5. We hope that such formulas may eventually lead to a set of generators for the ring of invariants of ternary quartics with good arithmetic properties. 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 a set of generators.

The full list of expressions for the $19$ Siegel modular forms either in terms of the theta constants or in terms of curve invariants, the expressions of Dixmier-Ohno invariants in terms of Siegel modular forms and the $55$ relations in the algebra, are available at [LR19].

Acknowledgments

We warmly thank the anonymous referees for carefully reading this work and for suggestions. This work is partially supported by the French National Research Agency under the anr-18-ce40-0026-01 clap-clap project.

2. Review of Tsuyumine’s construction of Siegel modular forms

We recall here the definition of the $34$ generators for the $\mathbb{C}$-algebra of modular forms of degree $3$ built by Tsuyumine. Surprisingly, they all are polynomials in theta constants with rational coefficients: one knows that when $g\geq 5$, there exists modular forms which are not in the algebra generated by theta constants [SM86], while the answer for $g=4$ is pending [OPY08]. We take special care of the multiplicative constant involved in each expression.

2.1. Theta functions and theta constants

Let $g\geq 1$ be an integer and $\mathbb{H}_{g}=\{\tau\in\mathbf{M}_{g}(\mathbb{C}),\;{{}^{t}}{\tau}=\tau,\ \operatorname{Im}\tau>0\}$.

Definition 2.1**.**

The theta function with characteristics $\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]\in\mathbf{M}_{2,g}(\mathbb{Z})$ is given, for $z\in\mathbb{C}^{g}$ and $\tau\in\mathbb{H}_{g}$, by

[TABLE]

The theta constant (with characteristic $\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]$) is the function of $\tau$ defined as $\theta\,{\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]}(\tau)=\theta\,{\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]}(0,\tau)$.

Proposition 2.2**.**

Let $z\in\mathbb{C}^{g}$, $\tau\in\mathbb{H}_{g}$, $\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]\in\mathbf{M}_{2,g}(\mathbb{Z})$, then

[TABLE]

and

[TABLE]

Combining these two equations shows that $z\mapsto\theta\,{\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]}(z,\tau)$ is even if $\varepsilon_{1}\>\ltrans{\,\varepsilon_{2}}\equiv 0\pmod{2}$, and odd otherwise. The characteristics $\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]$ are then said to be even and odd, respectively.

The modular group $\Gamma_{g}:=\operatorname{Sp}_{2g}(\mathbb{Z})$ acts on $\mathbb{H}_{g}$ by

[TABLE]

and on characteristics by

[TABLE]

Here, “$\,{}^{\frown}$” denotes the concatenation of two row vectors, and “$(.)_{\Delta}$” denotes the row vector equal to the diagonal of the square matrix given in argument. These result in the following action of $\Gamma_{g}$ on theta constants.

Proposition 2.3** **(Transformation formula [Igu72, Chap. 5,

Th. 2][SM89, p.442] [Cos11, Prop. 3.1.24]).

Let $\tau\in\mathbb{H}_{g}$, $\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]\in\mathbf{M}_{2,g}(\mathbb{R})$ and $M\in\Gamma_{g}$, then

[TABLE]

with $\left[{\begin{subarray}{c}\delta_{1}\\ \delta_{2}\end{subarray}}\right]=M.\left[{\begin{subarray}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{subarray}}\right]$ ,  $\zeta_{M}$ an eighth root of unity depending only on $M$ and

[TABLE]

In the following, we only make use of theta constants with characteristics with coefficients in $\{0,1\}$. Using Eq. (2.2) in combinaison with Eq. (2.4) allows to have a transformation formula purely between characteristics of this form.

To lighten notations, we number the theta constants as in [KLL*+*18]. We write $\theta_{n}:=\theta\,{\left[{\begin{subarray}{c}\delta_{0}\>\delta_{1}\>\ldots\>\delta_{g-1}\\ \varepsilon_{0}\>\varepsilon_{1}\>\ldots\>\varepsilon_{g-1}\end{subarray}}\right]}$ where $0\leq n<2^{2g-1}$ is the integer whose binary expansion is “$\delta_{0}\cdots\delta_{g-1}\varepsilon_{0}\ldots\varepsilon_{g-1}$”. In genus 3, there are 36 even theta constants (the odd ones are all [math]). We give in Table 1 the correspondence between their numbering in [Tsu86, pp.789–790] and our binary numbering.

2.2. Siegel modular forms

Let $\Gamma_{g}(\ell)$ denote the principal congruence subgroup of level $\ell$, i.e. $\{M\in\Gamma_{g}\>|\>M\equiv\mathbf{1}_{2g}\bmod\ell\},$ and let $\Gamma_{g}(\ell,2\ell)$ denote the congruence subgroup $\{M\in\Gamma_{g}(\ell)\>|\>(\ltrans{\,{\text{\tt\bf A}}}\,{\text{\tt\bf C}})_{\Delta}\equiv(\ltrans{\,{\text{\tt\bf B}}}\,{\text{\tt\bf D}})_{\Delta}\equiv 0\bmod 2\,\ell\}.$

For a congruence subgroup $\Gamma\subset\Gamma_{g}$, let $\mathbf{R}_{g,h}(\Gamma)$ be the $\mathbb{C}$-vector space of analytic Siegel modular forms of weight $h$ and degree $g$ for $\Gamma$, consisting of complex holomorphic functions $f$ on $\mathbb{H}_{g}$ satisfying

[TABLE]

for all $M\in\Gamma$. For $g=1$, one also requires that $f$ is holomorphic at “infinity” but we will not look at this case here. We also denote the $\mathbb{C}$-algebra of Siegel modular forms of degree $g$ for $\Gamma$ by $\mathbf{R}_{g}(\Gamma):=\bigoplus\mathbf{R}_{g,h}(\Gamma)$ . The modular group acts on $\mathbf{R}_{g,h}(\Gamma_{g})$ by

[TABLE]

In particular, $f\in\mathbf{R}_{g,h}(\Gamma)$ if and only if $M.f=f$ for all $M\in\Gamma$.

We now restrict to $g=3$. A strategy to build modular forms for $\Gamma_{3}$ is first to construct a form $F\in\mathbf{R}_{3}(\Gamma_{3}(2))$, and then average over the finite quotient $\Gamma_{3}/\Gamma_{3}(2)$ to get a modular form $f\in\mathbf{R}_{3}(\Gamma_{3})$, namely

[TABLE]

All forms $F$ which will be considered are polynomials in the theta constants, and are of even weight. 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 to an even power.

Tsuyumine gives some of the building blocks $F$s in terms of maximal syzygetic sets of even characteristics [Tsu86, Sec. 21]. Multiplying the theta constants in a given set is an element of $\mathbf{R}_{3}(\Gamma(2))$. The quotient $\Gamma_{3}/\Gamma_{3}(2)$ acts transitively on these $135$ sets numbered from $((1))$ to $((135))$ by Tsuyumine. Among them, $33$ are actually used to define a set of generators for $\mathbf{R}_{3}(\Gamma_{3})$. We give their expressions in Table 2.

Then Tsuyumine considers $34$ $F$s written as combinations of

• •

$\chi_{18}=\prod_{\theta_{i}\text{ even}}\theta_{i}$,

• •

a rational function of the 36 non-zero $\theta_{i}^{\,4}$ ,

• •

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]111See [Tsu89, p. 44]

for the $(1-T^{12})$ misprint in the denominator of Equation (2.6) in [Tsu86].).

The graded algebra $\mathbf{R}_{3}(\Gamma_{3})$ is generated by the 34 modular forms defined in Table 3. Its Hilbert–Poincaré series is generated by the rational function

[TABLE]

where {dgroup}[style=,spread=-2pt]*

[TABLE]

The modular forms $f$ defined in Table 3 are all polynomials in the theta constants whose primitive part has all its coefficients equal to $\pm 1$ and whose content is

[TABLE]

In order to get simpler expressions when restricting to the hyperelliptic locus or to the decomposable one, Tsuyumine multiplies each $f$ by an additional normalization constant ($2^{\mathrm{nd}}$ column of Table 3). For instance, as defined by Tsuyumine,

[TABLE]

and therefore the 135 summands are each a (monic) monomial in the theta constants times $\pm(2^{-10}\cdot 3^{-2}\cdot 5^{-1}\cdot 7^{-1})\cdot c(\chi_{28})=\pm 1/30$ (the sign depends on the monomial).

Having in mind possible applications of our results to fields of positive characteristic, we replace the multiplication by Tsuyumine’s constant by a multiplication by $1/c(f)$. In this way, $f$ is a sum of (monic) monomials in the theta constants with coefficients $\pm 1$. To avoid confusion with Tsuyumine’s notation, our modular forms will be denoted with bold font. Typically, $\bm{\chi}_{28}:=30\,\chi_{28}$, $\bm{\alpha}_{4}:=112\,\alpha_{4}$, $\bm{\alpha}_{6}:=\alpha_{6}$, $\bm{\alpha}_{10}:=165\,\alpha_{10}$, etc.

Still driven by the link with the hyperelliptic locus, Tsuyumine adds to $c_{32}^{\prime}$ (resp. $c_{38}^{\prime}$ and $c_{48}$) some polynomials in modular forms of smaller weights and denote the result $\gamma_{32}$ (resp. $\gamma_{38}$ and $\delta_{48}$). Theorem 2.4 as stated in [Tsu86] considers modular forms $\gamma_{32}$, $\gamma_{38}$ and $\delta_{48}$, instead of $c_{32}^{\prime}$, $c_{38}^{\prime}$ and $c_{48}$. The two theorems are obviously equivalent. Here, we choose instead to define $\bm{\gamma}_{32}:=c_{32}^{\prime}/6$, $\bm{\gamma}_{38}:=c_{38}^{\prime}$ and $\bm{\delta}_{48}:=c_{48}$ .

Remark 2.5*.*

Some of the modular forms in Table 3 have a large number of summands. While it would be cumbersome to store them, evaluating them is relatively quick as it basically consists in permuting theta constants up to some eighth roots of unity according to Eq. (2.4). Following Tsuyumine, the sum is computed in two steps. Let $\Theta$ be the subgroup of $\Gamma_{3}$ conjugate to $\Gamma_{3}(1,2)$ that stabilizes $\theta_{61}$ ($\Gamma_{3}(1,2)$ stabilizes $\theta_{0}$). Tsuyumine gives explicit coset representatives for $\Gamma_{3}/\Theta$ (36 elements) and $\Theta/\Gamma_{3}(2)$ ($8!$ elements) and splits the sum in Eq. (2.5) as

[TABLE]

We use this approach in order to perform the computation of the summands222There are two small typos in [Tsu86, pp. 842–846], the $(3,6)$-th coefficients of “$M_{\scalebox{0.5}{ \leavevmode\hbox to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip 4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$” must be -1 instead of 1, and the $(2,2)$-th coefficients of “$M_{\scalebox{0.5}{ \leavevmode\hbox to11.48pt{\vbox to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73833pt\lower-5.73833pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.53833pt}{0.0pt}\pgfsys@curveto{5.53833pt}{3.05876pt}{3.05876pt}{5.53833pt}{0.0pt}{5.53833pt}\pgfsys@curveto{-3.05876pt}{5.53833pt}{-5.53833pt}{3.05876pt}{-5.53833pt}{0.0pt}\pgfsys@curveto{-5.53833pt}{-3.05876pt}{-3.05876pt}{-5.53833pt}{0.0pt}{-5.53833pt}\pgfsys@curveto{3.05876pt}{-5.53833pt}{5.53833pt}{-3.05876pt}{5.53833pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.5pt}{-2.25555pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{27}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$” must be 1 instead of [math]. This modification makes $M_{\scalebox{0.5}{ \leavevmode\hbox to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip 4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ and $M_{\scalebox{0.5}{ \leavevmode\hbox to11.48pt{\vbox to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73833pt\lower-5.73833pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.53833pt}{0.0pt}\pgfsys@curveto{5.53833pt}{3.05876pt}{3.05876pt}{5.53833pt}{0.0pt}{5.53833pt}\pgfsys@curveto{-3.05876pt}{5.53833pt}{-5.53833pt}{3.05876pt}{-5.53833pt}{0.0pt}\pgfsys@curveto{-5.53833pt}{-3.05876pt}{-3.05876pt}{-5.53833pt}{0.0pt}{-5.53833pt}\pgfsys@curveto{3.05876pt}{-5.53833pt}{5.53833pt}{-3.05876pt}{5.53833pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.5pt}{-2.25555pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{27}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ symplectic.. In order to do that, we also need the eighth roots of unity $\zeta_{M^{\prime}}$ and $\zeta_{M^{\prime\prime}}$ from Proposition. 2.3. One approach is to precompute them using a fixed chosen matrix in $\mathbb{H}_{3}$. A better solution is, with the notation of Eq. (2.3), to make use of the relation $\zeta_{M}^{4}=(-1)^{\operatorname{tr}({\text{\tt\bf B}}\>\ltrans{{\text{\tt\bf C}}})}$ [Igu72, Chap. 5]. Since the modular forms have even weight, the degree of $F$ in the theta constants is a multiple of 4, as well as the powers of $\zeta_{M^{\prime}}$ and $\zeta_{M^{\prime\prime}}$.

3. A minimal set of generators for modular forms of degree $3$

3.1. Fundamental set of modular forms

Since we know the dimensions of each $\mathbf{R}_{3,h}(\Gamma_{3})$ from the generating functions of Theorem 2.4, it is a matter of linear algebra to check that a given subset of Tsuyumine’s generators is enough for generating the full algebra. It would involve choosing a monomial ordering on the ring of theta constants and computing a Gröbner basis of the homogeneous ideal defined by the generating subset given as formal expressions in terms of them (see [DK15, Section 1.4.1]). However, it is difficult to perform these computations since there exist numerous algebraic relations between the theta constants. Therefore we favor an interpolation/evaluation strategy as follows.

Suppose that we want to prove that a given form $f$ of weight $h$, given as a polynomial in the theta constants, can be obtained from a given set $\{f_{1},\ldots,f_{m}\}$. This set produces $F_{1},\ldots,F_{n}$, homogeneous polynomials in the $f_{i}$ of weight $h$. If $n<d=\dim\mathbf{R}_{3,h}(\Gamma_{3})$, then all forms of weight $h$ cannot be obtained. Assume that $n\geq d$. Then, if we can find $(\tau_{i})_{i=1,\dots,d}\in\mathbb{H}_{g}^{d}$ such that the matrix $(F_{i}(\tau_{j}))_{1\leq i,j\leq d}$ 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 $f/\theta_{0}^{2h}$ and the $f_{i}/\theta_{0}^{2w_{i}}$ where $w_{i}$ denotes the weight of $f_{i}$.

By Remark 2.5, the evaluation of a form $f(\tau)/\theta_{0}^{2h}(\tau)$ boils down to the computation of quotients $(\theta_{i}/\theta_{0})(\tau)$. A naive approach would be to use an arbitrary matrix $\tau\in\mathbb{H}_{3}$. But then the theta constants would in general be transcendental complex numbers which would make the computations much more costly and the final result hard to certify. We therefore prefer to consider a complex torus $\operatorname{Jac}C$ attached to a smooth plane quartic $C$ given by an Aronhold system. Indeed (see for instance [Web76, Rit04, NR17]), $7$ general lines in $\P^{2}$ form an Aronhold system of $7$ bitangents for a unique plane quartic $C$. Then, one can easily recover the equations of the $21$ other bitangents and an expression of the quotients ${(\theta_{i}/\theta_{0})^{4}(\tau)}$ in terms of the coefficients of the linear forms defining the bitangents (see for instance [NR17, Theorems 2 and 3]). Note that we do not explicitly know the Riemann matrix $\tau$ here, since it depends not only on $C$ but also on the choice of a symplectic basis for $H_{1}(C,\mathbb{Z})$. But when each of the bitangents in the Aronhold system is defined over $\mathbb{Q}$, all computations can be performed over $\mathbb{Q}$ and ${(\theta_{i}/\theta_{0})^{4}(\tau)}$ is a rational number.

To remove the fourth root of unity ambiguity that remains, we start by computing independently an approximation over $\mathbb{C}$ of an explicit Riemann matrix $\tau^{\prime}$ for the curve $C$. We need to do it only at very low precision (a typical choice is 20 decimal digits) and this can be done efficiently either in maple (package algcurves by Deconinck et al. [DvH01]) or in magma (package riemann surfaces by Neurohr [Neu18]). Then, we can calculate an approximation of the theta constants at $\tau^{\prime}$.

To conclude, note that [NR17, Theorem 3.1] shows that the set $\left\{{\theta_{j}^{\,8}}/{\theta_{i}^{\,8}}\right\}$ running through every even theta constants $\theta_{i},\theta_{j}$ depends only on $C$ and not on the Riemann matrix. Indeed, the dependence on this matrix relies only on the quadratic form $q_{{}_{0}}$ (in the notation of loc. cit.) whose contribution disappears in the eighth power. Therefore, there exist an integer $i_{0}$ and a permutation $\sigma$ such that

[TABLE]

We simply enumerate all the possible candidates for $i_{0}$ until we find a suitable $\sigma$ that gives $i_{0}$ and $\sigma$. Then, since we know ${\theta_{\sigma(i)}(\tau^{\prime})}/{\theta_{i_{0}}(\tau^{\prime})}$ with small precision and its eighth power exactly, it is possible to obtain the exact value of ${\theta_{i}(\tau)}/{\theta_{0}(\tau)}$.

Using this method extensively leads to a set of generators for $\mathbf{R}_{3}(\Gamma_{3})$. Moreover it is easy to prove, by the same algorithms, that this set is fundamental, i.e. one cannot remove any element and still generate the algebra $\mathbf{R}_{3}(\Gamma_{3})$.

Theorem 3.1**.**

The $19$ Siegel modular forms $\bm{\alpha}_{4}$, $\bm{\alpha}_{6}$, $\bm{\alpha}_{10}$, $\bm{\alpha}_{12}$, $\bm{\alpha}_{12}^{\prime}$, $\bm{\beta}_{14}$, $\bm{\alpha}_{16}$, $\bm{\beta}_{16}$, $\bm{\chi}_{18}$, $\bm{\alpha}_{18}$, $\bm{\alpha}_{20}$, $\bm{\gamma}_{20}$, $\bm{\beta}_{22}$, $\bm{\beta}_{22}^{\prime}$, $\bm{\alpha}_{24}$, $\bm{\gamma}_{24}$, $\bm{\gamma}_{26}$, $\bm{\chi}_{28}$ and $\bm{\alpha}_{30}$ define a fundamental set of generators for $\mathbf{R}_{3}(\Gamma_{3})$.

Remark 3.2*.*

Note that [Run95] proved that $\mathbf{R}_{3}(\Gamma_{3}(2))$ has a fundamental set of generators of $30$ 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 $\dim\mathbf{R}_{3,h}(\Gamma_{3})$ for this set of $19$ 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 $19$ forms $f_{i}$ at a matrix $\tau_{j}$, which takes 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{dgroup*}[style=,spread=-2pt]

[TABLE]

{dsuspend}

[TABLE]

{dsuspend}

[TABLE]

{dsuspend}

[TABLE]

The last ones, for instance $\bm{\gamma}_{44}$, $\bm{\delta}_{46}$ and $\bm{\delta}_{48}$, tend to be heavily altered with the relations that exist 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 relations defining the algebra $\mathbf{R}_{3}(\Gamma_{3})$. With the same techniques, involving modular forms up to weight $70$ (see Remark 4.4 for speeding up the computations), we find a (possibly incomplete) list of $55$ relations for our generators of $\mathbf{R}_{3}(\Gamma_{3})$ given by weighted polynomials of degree $32$ to $58$ (cf. Table 4).

The relations of weight $32$ and $34$ are relatively small, {dgroup*}[style=,spread=-2pt]

[TABLE]

{dsuspend}

[TABLE]

Runge [Run93, Cor.6.3] shows that $\mathbf{R}_{3}(\Gamma_{3})$ is a Cohen-Macaulay algebra. There exists a strong link between a minimal free resolution of a Cohen-Macaulay algebra and its Hilbert series. Let us rewrite Equation (2.6) as a rational fraction with denominator $\prod_{d_{i}}(1-T^{d_{i}})$ where the degrees $d_{i}$ run through the weights of the fundamental set of generators. We obtain a numerator with 140 non-zero coefficients, the first and last ones of which are

[TABLE]

The coefficients of the numerator give information on the weights and numbers of relations. They are consistent with Table 4 up to weight $48$. The drop from $6$ (relations) to a coefficient $5$ in weight $50$ indicates that there is a first syzygy (i.e. a relation between the relations) of weight 50.

3.3. A homogeneous system of parameters

Having these relations, one can also try to work out a homogeneous system of parameters (hsop) for $\mathbf{R}_{3}(\Gamma_{3})$. Recall that this is a set of elements $(f_{i})_{1\leq i\leq m}$ of the algebra, which are algebraically independent, and such that $\mathbf{R}_{3}(\Gamma_{3})$ is a $\mathbb{C}[f_{1},\ldots,f_{m}]$-module of finite type. Equation (2.6) suggests that a hsop of weight $4$, $12$, $12$, $14$, $18$, $20$ and $30$ may exist. An easy Gröbner basis computation made in magma with the lexicographic order $\bm{\alpha}_{6}<\bm{\alpha}_{10}<\ldots<\bm{\gamma}_{26}<\bm{\chi}_{28}$ shows that when we set to zero $\bm{\alpha}_{4}$, $\bm{\alpha}_{12}$, $\bm{\alpha}_{12}^{\prime}$, $\bm{\beta}_{14}$, $\bm{\chi}_{18}$, $\bm{\alpha}_{20}$ and $\bm{\alpha}_{30}$ in the $55$ relations of Table 4, the remaining $12$ Siegel modular forms of the generating set of Theorem 3.1 must be zero as well. As it is well known that the dimension of $\textrm{Proj}(\mathbf{R}_{3}(\Gamma_{3}))$ is $6$, this yields the following theorem.

Theorem 3.3**.**

A homogeneous system of parameters for $\mathbf{R}_{3}(\Gamma_{3})$ is given by the $7$ forms $\bm{\alpha}_{4}$, $\bm{\alpha}_{12}$, $\bm{\alpha}_{12}^{\prime}$, $\bm{\beta}_{14}$, $\bm{\chi}_{18}$, $\bm{\alpha}_{20}$ and $\bm{\alpha}_{30}$.

4. A dictionary between modular forms and invariants of quartics

In [Dix87], Dixmier gives a homogeneous system of parameters for the graded $\mathbb{C}$-algebra $\mathbf{I}_{3}$ of invariants of ternary quartic forms under the action of $\operatorname{SL}_{3}(\mathbb{C})$. They are denoted $I_{3}$, $I_{6}$, $I_{9}$, $I_{12}$, $I_{15}$, $I_{18}$ and $I_{27}$. This list is completed by Ohno with six invariants, $J_{9}$, $J_{12}$, $J_{15}$, $J_{18}$ and $J_{21}$, into a list of 13 generators for $\mathbf{I}_{3}$, the so-called Dixmier-Ohno invariants [Ohn07, Els15]. Note that $2^{40}\cdot I_{27}=D_{27}$ where $D_{27}$ denotes the normalized discriminant of plane quartics in the sense of [GKZ94, p.426] or [Dem12, Prop.11].

Using the morphism $\rho_{3}$ defined in [Igu67], Tsuyumine in [Tsu86, pp. 847–864] relates each of the Siegel modular forms given in Table 3 with an invariant for the graded ring of binary octics under the action of $\operatorname{SL}_{2}(\mathbb{C})$. 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 recalling the way to do so when $g=3$, we establish a complete dictionary between $\mathbf{R}_{3}(\Gamma_{3})$ and $\mathbf{I}_{3}$.

4.1. Modular forms in terms of invariants

Let us recall from [LRZ10, 2.2] how to associate an element of $\mathbf{I}_{3}$ to $f\in\mathbf{R}_{3,h}(\Gamma_{3})$. This morphism only depends on the choice of a universal basis of regular differentials $\omega$ which can be fixed “canonically” for smooth plane quartics (in the sense that it is a basis of regular differentials over $\mathbb{Z}$). Let $Q\in\mathbb{C}[x_{1},x_{2},x_{3}]$ be a ternary quartic form such that $C:Q=0$ is a smooth genus $3$ curve. Let $\Omega=\left[\begin{smallmatrix}\Omega_{1}\\ \Omega_{2}\end{smallmatrix}\right]$ be the $6\times 3$ period matrix of $C$ defined by integrating $\omega_{C}$ with respect to an arbitrary symplectic basis of $H_{1}(C,\mathbb{Z})$. We have $\tau=\Omega_{2}^{-1}\Omega_{1}\in\mathbb{H}_{3}$. The function

[TABLE]

is a homogeneous element of $\mathbf{I}_{3}$ of degree $3h$ (identifying the polynomial with its polynomial function).

Remark 4.1*.*

A similar construction can be worked out with invariants of binary octics (see [IKL*+*19]). Up to a normalization constant, this is actually the same morphism as defined by [Igu67].

Chai’s expansion principle [Cha86] shows that if the Fourier expansion of $f$ has coefficients in a ring $R\subset\mathbb{C}$, then $\Phi_{3}(f)$ is defined over $R$ as well. When $f$ is given by a polynomial in the theta constants with coefficients in $\mathbb{Z}$, we can take $R=\mathbb{Z}$. A particular case is given by the modular form $\bm{\chi}_{18}$ which is the product of the $36$ theta constants. In [LRZ10] (see also [Ich18b]) one shows the following precise form of Klein’s formula [Kle90, Eq. 118, p. 462],

[TABLE]

Remark 4.2*.*

The map (4.1) is obtained by pulling back geometric modular forms to invariants as described in [LRZ10]. Within this background, it is for instance possible to speak about the reduction 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. We will not study this question further here, but for instance, our $19$ generators have a surprising congruence modulo $11$,

[TABLE]

We have seen in Section 3 that we have an evaluation/interpolation strategy to handle quotient of modular forms by a power of $\theta_{0}$. This strategy can also be used to find the relations with invariants. But now, we also need to take care of the transcendental factor $\mu:={(2i\pi)^{3}}/{\det\Omega_{2}}$. This is done in the following way.

1. (i)

Assume that a relation $\Phi_{3}(f_{0})=I_{0}$ is known for a modular form $f_{0}$ of weight $h_{0}$. This is the case for $\bm{\chi}_{18}$ (cf. Eq. (4.2)) and we will start with this one, but switch to a relation of lower weight (i.e. $4$ with $\bm{\alpha}_{4}$ or even $2$ with $\bm{\chi}_{18}/\bm{\alpha}_{4}^{4}$ ) after a first round of the following steps (this simplifies the last step). 2. (ii)

Let now $f$ be one of the generators from Theorem 3.1 of weight $h$ and compute a basis $j_{1},\ldots,j_{d}$ of invariants of degree $3h$. We aim at finding $a_{1},\ldots,a_{d}\in\mathbb{Q}$ such that $\Phi_{3}(f)=\sum a_{i}j_{i}$. This is done by evaluation/interpolation at Riemann models until one gets a system of $d$ linearly independent equations. More precisely, for a given $Q=0$ and an associated $\tau\in\mathbb{H}_{3}$ :

1. (a)

Compute the values of $(j_{1},\ldots,j_{d})$ at $Q$; 2. (b)

Using the same procedure as in Section 3, compute $(f/\theta_{0}^{2h})(\tau)$ and $(f_{0}/\theta_{0}^{2h_{0}})(\tau)$; 3. (c)

Let $p=\textrm{lcm}(h_{0},h)$. Since

[TABLE]

we get the value of $\Phi_{3}(f)^{p/h}$. An approximate computation at low precision can then give the exact value.

The above strategy provides explicit expressions for $\Phi_{3}(f)$ where $f$ is any modular form in the fundamental set defined in Theorem 3.1.

Proposition 4.3**.**

Let $f$ be a modular form of weight $h$ from Theorem 3.1. There exists an explicit polynomial $P_{f}$ of degree $3h$ in the Dixmier-Ohno invariants such that

[TABLE]

The first ones333We make available the list of these 19 polynomials at [LR19, file “SiegelMFfromDO.txt”]. are {dgroup*}[style=,spread=-2pt]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Beside Klein’s formula

$\Phi_{3}(\bm{\chi}_{18})=\text{-}\,2^{108}\>\>I_{27}^{2}$ ,

one finds a surprisingly compact expression for $\bm{\chi}_{28}$ ,

[TABLE]

If we do not not pay attention, the rational coefficients of these formulas tend to have prime factors greater than 7 in their denominators, especially for the forms of higher weight. We have eliminated all these “bad primes” using the relations that exist between the Dixmier-Ohno invariants. 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).

Remark 4.4*.*

When we deal with the Jacobian of a curve with coefficients in $\mathbb{Q}$, what is a matter of few integer arithmetic operations to evaluate modular forms from invariants is a matter of high precision floating point arithmetic over the complex numbers with analytic computations of Riemann matrices. In practical calculations, such as the computations in Section 3.2, it is thus much better to use the former, since a calculation that would take the order of the minute ultimately requires only a few milliseconds.

4.2. Invariants in terms of modular functions

Conversely, we can look for expressions of a generating set of invariants in terms of modular forms. Using [Tsu86, LG19], one obtains such a result for invariants of binary octics. We focus here on the case of Dixmier-Ohno invariants.

Since the locus of plane quartic over $\mathbb{C}$ such that $I_{27}\neq 0$ corresponds to the locus of non-hyperelliptic curve of genus $3$ and then to principally polarized abelian threefolds $\mathbb{C}^{3}/(\tau\mathbb{Z}^{3}+\mathbb{Z}^{3})$ for which $\bm{\chi}_{18}(\tau)\neq 0$ [Igu67, Lem. 10, 11], we see that any invariant in $\mathbf{I}_{3}$ can be obtained as a quotient of a modular form by a power of $I_{27}$.

Proposition 4.5**.**

Let $I$ be a Dixmier-Ohno invariant of degree $3k$. There exist a polynomial $P_{I}$ in the modular forms from Theorem 3.1, of weight $28\,k$, such that

[TABLE]

The first ones444We make available the list of these 13 polynomials at [LR19, file “SiegelMFtoDO.txt”]. are {dgroup*}[style=,spread=-2pt]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In this setting, one can also write $I_{27}^{27}\>I_{27}=\Phi_{3}((2^{-108}\,\bm{\chi}_{18})^{14})$.

Unlike the previous computations, one cannot obtain the above ones by a direct application of the evaluation/interpolation strategy as the degrees (and weights) are sometimes too large. For the invariant $I_{21}$, for instance, one would potentially need to interpolate on a vector space of modular forms of weight 196, which is huge (its dimension is $869\,945$). The trick is to proceed by steps and first look for expressions of a small power of $I_{27}$ by the desired invariant $I$, not only in terms of modular forms, but also in terms of invariants $I_{3k}$ of smaller degrees. For instance in the case of $I_{21}$, {dgroup*}[style=,spread=-2pt]

[TABLE]

Then, mechanically, through a sequence of substitutions of the invariants of smaller degrees by their expression in terms of the modular forms, we arrive to expressions for $I_{27}^{3k}\,I_{3k}$ purely in terms of modular forms. These formulas are very sparse, considering their weight (see Table 5).

Remark 4.6*.*

It is not a coincidence that the power of $I_{27}$ is $3\,k$ in Equation (4.3). Let us consider ternary quartics of the form $Q^{2}+p\,G$, where $p$ is a prime integer, $Q$ is a ternary quadratic form and $G$ is ternary quartic form. Generically, for all but $I_{27}$ the valuation of $p$ of the Dixmier-Ohno invariants of these forms is zero, and $\upsilon_{p}(I_{27})=14$ . And, still generically, we have $\upsilon_{p}(\Phi_{3}(f))=3\,h/2$ where $f$ is any one of the Tsuyumine modular forms, and $h$ is its weight. Thus, if the equation $I_{27}^{\,\kappa}\cdot I=\Phi_{3}(\,P_{I}(\,\bm{\alpha}_{4},\,\bm{\alpha}_{6},\ldots,\bm{\alpha}_{30})\,)$ is satisfied, the power $\kappa$ of $I_{27}$ must be such that the degrees agree, i.e. $27\,\kappa+3\,k=3\,h$, and such that the valuations at $p$ are equal, i.e. $14\,\kappa=3\,h/2$. This yields $\kappa=3\,k$.

Remark 4.7*.*

We are also able to eliminate the primes greater than $7$ in the denominators of the coefficients in these formulas using the relations that exist between Siegel modular forms (cf. Section 3.2), with the notable exception of the primes $11$ and $19$ (cf. Table 5). We suspect that the reason behind this difficulty is that, similarly to the prime $11$ (cf. Remark 4.2), one cannot extend Theorem 3.1 mutatis mutandis to characteristic $19$. Although we do not go further into the topic, it is possible to work directly in these characteristics and find specific formulas valid there.