Differential Galois group of the level-three cubic-theta differential field over Ramanujan's differential field

Matsuda (Osaka J. Math. 2020) derived closed polynomial differential systems for the level-three cubic theta functions $a(q)$, $b^3(q)$, $c^3(q)$ together with the Eisenstein series $E_2(q)$ and $E_2(q^3)$, and proved that the differential fields they generate under $q\,d/dq$ all coincide and contain Ramanujan's differential field $\mathbb{C}(E_2,E_4,E_6)$. At level two the corresponding structure is completely understood: Ohyama proved that Halphen's differential field is a finite extension of Ramanujan's field whose differential Galois group is the symmetric group $S_3$. The level-three analogue is open: determining the differential Galois group of $K_3=\mathbb{C}\langle a,E_2,b^3\rangle$ over Ramanujan's field. No determination of this group appears in the literature. Relevant context for the determination includes the degree-four covering $X_0(3)\to X(1)$ of modular curves (Galois closure group $\mathrm{PSL}_2(\mathbb{F}_3)\cong A_4$), the nontrivial weights and characters of the generators, and the quasi-modular $E_2$.

Background

Let $q=\exp(2\pi i\tau)$ with $\tau$ in the upper half-plane, and let $\delta=q\,\frac{d}{dq}=\frac{1}{2\pi i}\frac{d}{d\tau}$ be the natural derivation acting on $q$-series. The classical Eisenstein series

$E_2(q)=1-24\sum_{n\ge1}\sigma_1(n)q^n,\qquad E_4(q)=1+240\sum_{n\ge1}\sigma_3(n)q^n,\qquad E_6(q)=1-504\sum_{n\ge1}\sigma_5(n)q^n,$

where $\sigma_k(n)$ denotes the sum of the $k$-th powers of the positive divisors of $n$, satisfy Ramanujan's system of ordinary differential equations

$\delta E_2=\frac{E_2^2-E_4}{12},\qquad \delta E_4=\frac{E_2E_4-E_6}{3},\qquad \delta E_6=\frac{E_2E_6-E_4^2}{2}.$

Here $E_4$ and $E_6$ are modular forms of weights 4 and 6 (a modular form of weight $k$ for a group $\Gamma\subset SL_2(\mathbb{Z})$ is a holomorphic function on the upper half-plane with $f(\frac{a\tau+b}{c\tau+d})=(c\tau+d)^k f(\tau)$ for $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$), while $E_2$ is only quasi-modular: it satisfies the weight-2 transformation law up to an additive correction. The differential field $\mathcal{R}=\mathbb{C}\langle E_2,E_4,E_6\rangle$ generated by these three functions under $\delta$ is Ramanujan's differential field. Throughout, the level of a form refers to the smallest $N$ such that it is modular for the congruence subgroup $\Gamma_0(N)$ (matrices of $SL_2(\mathbb{Z})$ whose lower-left entry is divisible by $N$).

The level-two member of this hierarchy is governed by Jacobi's theta constants $\vartheta_2,\vartheta_3,\vartheta_4$ (values at $z=0$ of the Jacobi theta function), whose logarithmic derivatives give the solutions of Halphen's nonlinear system $X'+Y'=2XY$, $Y'+Z'=2YZ$, $Z'+X'=2ZX$. Ohyama (1995) proved that Halphen's differential field is a finite algebraic extension of Ramanujan's differential field whose differential Galois group is the symmetric group $S_3$. Here the differential Galois group means the group of automorphisms of the extension that commute with the derivation (the Kolchin group); for a finite algebraic extension of differential fields this is a finite group, and in the modular setting it realizes the deck transformations of the underlying covering of modular curves — at level two, the covering $X(2)\to X(1)$ of compactified modular curves, whose deck group is the anharmonic group of permutations of the modular lambda function. For a finite algebraic extension of differential fields, the extension is called differential-Galois when the base field is the fixed field of the differential Galois group — equivalently, when the order of that group equals the degree of the extension. Ohyama's theorem is the level-two prototype of the problem considered here.

The level-three (cubic) member involves the Borwein cubic theta functions

$a(q)=\sum_{m,n\in\mathbb{Z}}q^{m^2+mn+n^2},\qquad b(q)=\sum_{m,n\in\mathbb{Z}}\omega^{\,n-m}q^{m^2+mn+n^2},\qquad c(q)=\sum_{m,n\in\mathbb{Z}}q^{(n+\frac13)^2+(n+\frac13)(m+\frac13)+(m+\frac13)^2},$

with $\omega=e^{2\pi i/3}$, introduced in the cubic (signature-3) theory of the Borwein brothers and Garvan; they satisfy the cubic identity $a^3=b^3+c^3$ and the eta-quotient expressions $b^3(q)=\frac{(q;q)_\infty^9}{(q^3;q^3)_\infty^3}$ and $c^3(q)=27q\,\frac{(q^3;q^3)_\infty^9}{(q;q)_\infty^3}$, where $(q;q)_\infty=\prod_{n\ge1}(1-q^n)$ and $\eta(\tau)=q^{1/24}(q;q)_\infty$ is the Dedekind eta function. These are modular forms for $\Gamma_0(3)$: $a$ has weight 1 and transforms with the nontrivial Dirichlet character modulo 3 (the quadratic character $\chi_{-3}$ with $\chi_{-3}(1)=1$, $\chi_{-3}(2)=-1$, $\chi_{-3}(3)=0$), while $b^3$ and $c^3$ have weight 3. Ohyama (2001) derived Halphen-type systems at level three from Picard-Fuchs equations of elliptic modular surfaces; Huber (2011) obtained the differential systems for the cubic theta functions; and Matsuda (2020) — the source paper — rederived and extended them using Farkas and Kra's theory of theta functions with rational characteristics: theta functions $\theta\bigl[\begin{smallmatrix}\epsilon\\ \epsilon'\end{smallmatrix}\bigr](z,\tau)$ indexed by a characteristic $(\epsilon,\epsilon')$, whose values at $z=0$ (theta constants) at rational characteristics generate modular forms of the corresponding level. Matsuda's Theorems 1.1 and 1.2 give the closed polynomial systems

$\delta a=\frac{3a^3+aE_2-4b^3}{12},\quad \delta E_2=\frac{-9a^4+8ab^3+E_2^2}{12},\quad \delta b^3=\frac{-a^2b^3+E_2b^3}{4},$

$\delta a=\frac{-3a^3+3aE_2(q^3)+4c^3}{12},\quad \delta E_2(q^3)=\frac{-9a^4+8ac^3+9E_2(q^3)^2}{36},\quad \delta c^3=\frac{a^2c^3+3E_2(q^3)c^3}{4},$

and, at the heart of the present problem, the structural facts that the resulting differential fields coincide,

$K_3:=\mathbb{C}\langle a,E_2,b^3\rangle=\mathbb{C}\langle a,E_2\rangle=\mathbb{C}\langle a,E_2(q^3)\rangle=\mathbb{C}\langle a,b^3\rangle,$

and that Ramanujan's field is contained in $K_3$, with $E_4$ and $E_6$ rationally expressible through $a$ and $b^3$ (so $\mathcal{R}\subset K_3$ and the extension is finite algebraic). Related systems were developed at level five by Mano and for the Hecke groups $\Gamma_0(N)$, $N=2,3,4$, by Maier; Opanasenko and Ferapontov later gave a general framework of Lie-group-invariant defining equations for modular forms in which the level-three form $a$ appears as an example.

What no published work supplies is the group-theoretic analogue of Ohyama's level-two theorem for this level-three field. The naive guess — that the group should be read off from the covering $X_0(3)\to X(1)$ of modular curves, a degree-4 cover whose Galois closure has deck group $\mathrm{PSL}_2(\mathbb{F}_3)\cong A_4$ — must be reconciled with the facts that the generators carry nontrivial weights and characters (so deck transformations act on them with weight factors and character signs, and a character extension of $A_4$, for instance related to $SL_2(\mathbb{F}_3)$, may occur) and that the base field contains the quasi-modular $E_2$. The actual group, the degree of the extension, and even whether the extension is differential-Galois are undetermined in the literature.

Problem Statement

Let $\delta=q\,\frac{d}{dq}$, let $\mathcal{R}=\mathbb{C}\langle E_2(q),E_4(q),E_6(q)\rangle$ be Ramanujan's differential field, and let

$K_3=\mathbb{C}\langle a(q),E_2(q),b^3(q)\rangle=\mathbb{C}\langle a(q),E_2(q)\rangle=\mathbb{C}\langle a(q),E_2(q^3)\rangle=\mathbb{C}\langle a(q),b^3(q)\rangle$

be the level-three differential field generated by the cubic theta functions together with the Eisenstein series. The equalities of these fields, the containment $\mathcal{R}\subset K_3$, and the closed polynomial differential systems satisfied by the triples $(a,E_2,b^3)$ and $(a,E_2(q^3),c^3)$ (with $c^3=a^3-b^3\in K_3$) are established in the source paper; any of these generating sets may be used. The problem is to determine the differential-algebraic structure of $K_3$ as an extension of $\mathcal{R}$: determine the group of differential automorphisms of $K_3$ over $\mathcal{R}$ — the automorphisms of $K_3$, or of its Galois closure if the extension is not normal, that fix $\mathcal{R}$ pointwise and commute with $\delta$. The answer must specify the isomorphism type of this finite group, the degree $[K_3:\mathcal{R}]$ (and the degree of the Galois closure), whether the extension is differential-Galois, and explicit generators of the group, each automorphism being described by the images it assigns to a chosen set of generators of $K_3$ (for instance $a$ and $b^3$, with $c^3=a^3-b^3$ and $E_2(q^3)$ then determined through the field equalities above; elements of $\mathcal{R}$, in particular $E_2(q)$, are fixed pointwise; in the non-normal case, automorphisms of the Galois closure that do not preserve $K_3$ are described instead by their images on a generating set of the closure) — in the way the anharmonic group $S_3$ permutes the level-two generators — together with a proof that no further differential automorphisms exist. This is the level-three analogue of Ohyama's theorem that the Halphen field extends Ramanujan's field with differential Galois group $S_3$.

References

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