2.1 Preliminaries

Let us recall some standard facts related to the classification of Jacobian elliptic fibrations on a K3 surfaces. Given a Jacobian elliptic fibration $\pi _{\mathcal {X}} \colon \mathcal {X} \rightarrow \mathbb {P}^1$ with a section $\sigma _{\mathcal {X}}$ , the cohomology classes of the fiber and section span a rank-two lattice isomorphic to the standard rank-two hyperbolic lattice H. One, therefore, obtains a primitive lattice embedding $ H \hookrightarrow \operatorname {NS}(\mathcal {X})$ . The converse of this result also holds: given a primitive lattice embedding $H \hookrightarrow \operatorname {NS}(\mathcal {X})$ whose image contains a quasi-ample class, it is known (see, for instance, [Reference Clingher and Doran7]) that there exists a Jacobian elliptic fibration on the surface $\mathcal {X}$ whose fiber and section classes span H. Moreover, given $H \hookrightarrow \operatorname {NS}(\mathcal {X})$ , one can always compose the embedding with an isometry of $\mathrm {NS}(\mathcal {X})$ in such a way that the image of the resulting embedding contains a quasi-ample class. One, therefore, has a natural one-to-one correspondence between lattice embeddings $H \hookrightarrow \operatorname {NS}(\mathcal {X})$ , up to isometry, and Jacobian elliptic fibrations on $\mathcal {X}$ , up to automorphism.

For a given lattice embedding $j \colon H \hookrightarrow \mathrm {NS}(\mathcal {X})$ , associated with a Jacobian elliptic fibration, denote by $R(-1)= j(H)^{\perp }$ the orthogonal complement in $\mathrm {NS}(\mathcal {X})$ . The lattice R is known as the frame of the respective Jacobian elliptic fibration. Associated with the frame, denote by $R^{\text {root}}$ the sub-lattice spanned by the roots of R. Consider also the factor group . The pair is the type of the frame, and plays an important role in the classification of Jacobian elliptic fibrations. The lattice $R^{\text {root}}$ is also known as the trivial lattice associated with the elliptic fibration and is a direct sum of ADE type lattices. $R^{\text {root}}$ provides information about the reducible fibers, while the group is isomorphic to the the Mordell–Weil group of the fibration.

Refining the above arguments, one would like to distinguish between non-isomorphic Jacobian elliptic fibrations on $\mathcal {X}$ sharing the same frame, or frame type. A detailed discussion on this aspect may be found in [Reference Braun, Kimura and Watari4]. As established by Festi and Veniani [Reference Festi and Veniani18], for every frame type , there are finitely many Jacobian elliptic fibrations of this type supported on $\mathcal {X}$ , up to the action of the automorphism group $\mathrm {Aut}(\mathcal {X})$ . Moreover, one has a one-to-one correspondence between the possible Jacobian elliptic fibration classes and a certain lattice-theoretic double-coset. The order of this double-coset is known as the multiplicity of the frame type.

A full classification of the possible frame pairs , as well as their corresponding multiplicities, in the case when $\mathrm {Aut}(\mathcal {X})$ is finite, is given in [Reference Clingher and Malmendier13].

2.2 Weierstrass equations for Jacobian elliptic fibrations

Turning to the area of concern for this article, we note that various geometric aspects for K3 surfaces $\mathcal {X}$ polarized by lattices $S = H \oplus R(-1)$ , with R a direct sum of positive definite lattices of ADE type, have been studied in previous works of the first two authors [Reference Clingher and Malmendier13]. The general member in this class carries a special Jacobian elliptic fibration with the associated lattice embedding $H \hookrightarrow S$ as the left-side factor in $H \oplus R(-1)$ . We refer to this elliptic fibration of frame R as standard Footnote ³ . These standard fibrations may be described explicitly via affine Weierstrass forms, that is, equations of type $y^2 = x^3 + f(t) x + g(t)$ , where $f(t), g(t)$ are polynomials with prescribed properties (degrees, root multiplicities, etc), depending on the choice of R.

A special sub-case of the above is given by the situation when one has a canonical primitive embedding $ H \oplus N \hookrightarrow S $ , where N is the rank-ten Nikulin latticeFootnote ⁴ . In such a situation, the lattice embedding $H \hookrightarrow S$ corresponding to the left-side factor of $H \oplus N$ determines a second Jacobian elliptic fibration on the S-polarized K3 surfaces, which will be referred to as alternate. This fibration has the property that, in addition to the base section, it possesses a second section of order two with respect to the Mordell–Weil group. This type of fibration was studied in detail by van Geemen and Sarti in [Reference van Geemen and Sarti50]. The alternate fibration may again be described via an affine Weierstrass form $y^2 = x^3 + f_a(t) x + g_a(t)$ . However, a more effective description is given by a slight modification – the affine van Geemen–Sarti form $y^2= x^3 + A(t) x^2 + B(t) x$ , where here $A(t), B(t)$ are polynomials of degrees at most 4 and 8, respectively, with prescribed properties. Given a van Geemen–Sarti form, one may recover the Weierstrass form of the fibration via the formulas:

$$ \begin{align*} \begin{aligned} f_a(t) & = \frac{1}{3} \left( 3 B(t) - A(t)^2 \right), \qquad g_a(t) = \frac{1}{27} A(t) \left( 2 A(t)^2 - 9 B(t) \right), \\[0.2em] &\Delta_a(t) \ = \ 4 f_a(t)^3 + 27 g_a(t)^2 \ = \ B(t)^2 \left( 4 B(t) - A(t)^2 \right). \end{aligned} \end{align*} $$

For this article, we shall consider two further restrictions of the above discussion. First, we shall assume that the polarizing lattice S is of finite automorphism group typeFootnote ⁵ , in the sense of Nikulin [Reference Nikulin42], [Reference Nikulin44]. Second, we shall impose a condition on the rank of S, namely, $13 \leqslant \rho _S \leqslant 18$ . Lower rank cases will be discussed in subsequent works.

The following result follows via arguments from [Reference Clingher, Malmendier and Poon14].

Proposition 2.1. Let S be an even lattice of rank $\rho _S$ and signature $(1, \rho _S-1)$ , satisfying the following conditions:

1. 1. S admits a primitive embedding in the K3 lattice $\Lambda _{K3}$ ,

2. 2. S admits a direct sum decomposition $ H \oplus R(-1)$ , with R a positive definite lattice of ADE type,

3. 3. there exists a primitive embedding $ H \oplus N \hookrightarrow S $ ,

4. 4. S is of finite automorphism group type, in the sense of Nikulin [Reference Nikulin42], [Reference Nikulin44],

5. 5. $13 \leqslant \rho _S \leqslant 18$ .

Then, S is isomorphic to one of the lattices of the list below:

(2.1) $$ \begin{align} \begin{aligned} H \oplus E_8(-1)^{\oplus 2}, \ H \oplus E_8(-1) \oplus E_7(-1), \\ H \oplus E_8(-1) \oplus D_6(-1), \ H \oplus E_8(-1) \oplus D_4(-1) \oplus A_1(-1), \\ H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}, \ H \oplus D_8(-1) \oplus D_4(-1), \ H \oplus E_7(-1) \oplus A_1(-1)^{\oplus 4}. \end{aligned} \end{align} $$

We also note that, via classical results by Nikulin [Reference Nikulin40], [Reference Nikulin41], [Reference Nikulin43], Vinberg [Reference Vinberg51], and Kondo [Reference Kondo28], one obtains the following.

The following observation, which follows from [Reference Clingher, Malmendier and Poon14], is important for the considerations of this article.

Remark 2.5. Re-framing Proposition 2.4 within the standard/alternate terminology established above, we conclude that, for a general S-polarized K3 surface $\mathcal {X}$ , with S in (2.1), while multiple non-isomorphic elliptic Jacobian fibrations with standard frame $R_{\text {std}}$ may occur, one always has a unique (canonical) alternate fibration with root lattice of the frame $R^{\text {root}}_{\text {alt}}$ and . All possible standard/alternate framesFootnote ⁷ are detailed in Table 1.

Table 1

Lattices in Remark 2.2.

2.3 Van Geemen–Sarti forms

The canonical alternate fibrations $\pi _{\mathcal {X}}\colon \mathcal {X} \rightarrow \mathbb {P}^1$ of Proposition 2.4 may be described explicitly. We shall give these descriptions here and construct the corresponding van Geemen–Sarti form

(2.2) $$ \begin{align} \overline{\mathcal{X}}\colon \quad Y^2 Z = X^3 + A(u, v) \, X^2 Z + B(u, v) \, X Z^2, \end{align} $$

with elliptic fibers in $\mathbb {P}^2 = \mathbb {P}(X, Y, Z)$ varying over $\mathbb {P}^1=\mathbb {P}(u, v)$ , for the subset of lattices with Picard rank 13 and higher.

Proposition 2.6. For S in (1.2) with $\rho _S=10+k$ for $4 \le k \le 8$ and $L = S^{\perp }_{\Lambda _{K3}}$ , the defining polynomials in Equation (2.2) can be arranged to have the form

(2.3) $$ \begin{align} A = \big(u^3 + a_4 u v^2 + a_6 v^3\big) v, \qquad B = \big(b_{2(k-2)} u^{8-k} + \dots + b_{12} v^{8-k}\big) v^k = \sum_{l=0}^{8-k} b_{2(6-l)} u^{l} v^{8-l} \end{align} $$

for $L = H^{\oplus 2} \oplus A_1(-1)^{\oplus 8-k}$ , and

(2.4) $$ \begin{align} A = 2\Big(c_2 u^2 + c_6 u v + c_{10} v^2\Big)v^2, \qquad B = \Big(u^5 + d_8 u^3 v^2 + d_{12} u^2v^3 + d_{16} u v^4 + d_{20} v^5\Big)v^3 \end{align} $$

for $L = H \oplus H(2) \oplus D_4(-1)$ . The coefficients in Equation (2.3) satisfy $( b_{2(k-2)}, \dots , b_{12} ) \neq 0$ for $k=5, 6, 7, 8$ and

(2.5) $$ \begin{align} \Big(\, 2 a_4 b_4 + b_8, \ a_4 b_6 + 3 b_{10}, \ a_4 b_6 - 6 a_6 b_4, \ a_4 b_8 + 18 b_{12}, \ 8 b_4 b_8 - 3 b_6^2, \ 4 a_4 b_8 - 9 a_6 b_6 \, \Big) \ \neq \ 0 \end{align} $$

for $k=4$ , and the coefficients in Equation (2.4), satisfy

(2.6) $$ \begin{align} \begin{aligned} \Big( \, c_2 d_8 + 10 c_{10}, \ 4 c_2 c_{10} - c_6^2, \ c_2 d_{12} - c_6 d_8, \ 3 d_8^2 + 20 d_{16}, \ & \\ 8 c_2 d_{16} - 3 c_6 d_{12}, \ d_8 d_{12} + 50 d_{20}, \ 8 d_8 d_{16} -3 d_{12}^2 \, \Big) & \ \neq \ 0. \end{aligned} \end{align} $$

Proof. Proposition 2.4 shows that a general S-polarized K3 surface $\mathcal {X}$ has a Jacobian elliptic fibration $\pi _{\mathcal {X}}\colon \mathcal {X} \rightarrow \mathbb {P}^1$ with $\operatorname {MW}(\mathcal {X}, \pi _{\mathcal {X}})\cong \mathbb {Z}/2\mathbb {Z}$ . This yields Equation (2.2), if we move the zero-section to $[X: Y: Z] = [0:1:0]$ and the 2-torsion section to $[X: Y: Z] = [0:0:1]$ .

In the first case, it was proved in [Reference Clingher, Malmendier and Poon14] that the elliptic fibration has the singular fibers $I^*_{2(k-2)} + (8-k) I_2+6 I_1$ . The van Geemen–Sarti form (2.2) with a singular fiber of type $I^*_{2(k-2)}$ at $v=0$ has $A=(a_0 u^3 + \dots + a_6 v^3)v$ and $B=(b_{2(k-2)} u^{8-k} + \dots + b_{12} v^{8-k})v^k$ . It follows that $B \neq 0$ since otherwise the surface has non-isolated singularities. Moreover, in order for the elliptic surface to have at worst a rational double point singularity at $v=0$ , we must have $a_0 \neq 0$ . This can be seen by converting Equation (2.2) into its standard Weierstrass form

(2.7) $$ \begin{align} y^2 z = x^3 - \frac{1}{3} \Big( A(u, v)^2 - 3 B(u, v) \Big) \, xz^2 + \frac{1}{27} A(u, v) \Big(2 A(u, v)^2 - 9 B(u, v)\Big) \, z^3. \end{align} $$

For $a_0=0,$ we obtain a $(4,6, 12)$ -point at $v=0$ whence a non-minimal Weierstrass equation. Thus, we must have $a_0 \neq 0$ . For $a_0 \neq 0$ the coefficient $a_0$ can be rescaled to equal one, and $a_2$ can be subsequently eliminated by a shift in u. We, then obtain Equation (2.2) with defining polynomials given by Equation (2.3). Moreover, any $(4,6, 12)$ -point with $v \neq 0$ requires $A = (u+\beta v)^2 (u -2 \beta v) v$ and $B = b_4 ( u + \beta v)^4 v^4$ for some $\beta \in \mathbb {C}$ and $b_4 \neq 0$ . This is only possible for $k\le 4$ , and Equation (2.5) follows by computing a Gröbner basis for the intersection of the corresponding elimination ideal and the ideal generated by $b_4 = \dots = b_{12}=0$ . The second case is analogous: the singular fibers are $III^*+ 5 I_2+5 I_1$ (see [Reference Clingher, Malmendier and Roulleau15]). We, then moved the fiber of type $III^*$ to $v=0$ . Here, a $(4,6, 12)$ -point with $v\neq 0$ requires $A = 2c_2 (u+\beta v)^2v^2$ and $B = (u - 4 \beta v) ( u + \beta v)^4v^3$ . The left side of Equation (2.6) is obtained as Gröbner basis of the corresponding elimination ideal.

The following is immediate.

In the remaining case in Equation (1.2), we have the following.

Proposition 2.9. For $S=H \oplus E_7(-1) \oplus A_1(-1)^{\oplus 4}$ and $L = S^{\perp }_{\Lambda _{K3}} =H^{\oplus 2} \oplus A_1(-1)^{\oplus 5}$ , the defining polynomials in Equation (2.2) can be arranged to have the form

(2.8) $$ \begin{align} \begin{aligned} A = \Big(u^3 + a_4 u v^2 + a_6 v^3\Big) v, \quad B = \Big( b_{2} u^5 + b_4 u^4 v + b_6 u^3 v^2 + b_8 u^2 v^3 + b _{10} u v^4+ b_{12} v^{5} \Big) v^3, \end{aligned} \end{align} $$

or

(2.9) $$ \begin{align} \begin{aligned} A = 2 \Big( c_{-2} u^3 + c_2 u^2 v + c_6 u v^2 + c_{10} v^3\Big)v , \quad B = u^5 + d_8 u^3v^2 + d_{12} u^2v^3 + d_{16} uv^4 + d_{20} v^5. \end{aligned} \end{align} $$

The coefficients in Equation (2.8) satisfy

(2.10) $$ \begin{align} \begin{aligned} \Big(\, 2 a_4 b_4 - 10 a_6 b_2 + b_8, \ a_4 b_6 - 2 a_6 b_4 + 2 b_{10}, \ 10 a_4^2 b_2 - 3 a_4 b_6 + 18 a_6 b_4, \ a_4 b_8 - a_6 b_6 + 10 b_{12}, \\ 8 a_4^2 b_4 - 30 a_4 a_6 b_2 + 9 a_6 b_6, \ 20 b_2 b_{10} - 8 b_4 b_8 + 3 b_6^2, \ 4 a_4^3 b_2 + 27 a_6^2 b_2 \, \Big) \ \neq \ 0 , \end{aligned} \end{align} $$

and the coefficients in Equation (2.9) satisfy

(2.11) $$ \begin{align} \begin{aligned} \Big( \, c_{-2} d_{12} - c_2 d_8 - 10 c_{10}, \ 3 c_{-2}^2 d_{16} - 3 c_{-2} c_2 d_{12} + 2 c_2^2 d_8 + 5 c_6^2, \ 2 c_{-2} d_{16} - c_2 d_{12} + c_6 d_8, \\ 3 d_8^2 + 20 d_{16}, \ 45 c_{-2} d_{20} - 8 c_2 d_{16} + 3 c_6 d_{12}, \ d_8 d_{12} + 50 d_{20}, \ 8 d_8 d_{16} -3 d_{12}^2 \,\Big) \ \neq \ 0. \end{aligned} \end{align} $$

We have the following.

Lemma 2.10. For $c_{-2} \neq 0,$ the coefficients in (2.9) can be transformed into coefficients in (2.8), with $b_2 = 1/(2c_{-2})$ and

(2.12) $$ \begin{align} \begin{aligned} & a_4 = 4 c_{-2} c_6 - \frac{4}{3} c_2^2, \ \ b_4 = - \frac{5}{3} \frac{c_2}{c_{-2}}, \ \ a_6 = 8 c_{-2}^2 c_{10} - \frac{8}{3} c_{-2} c_2 c_6 + \frac{16}{27} c_2^3, \ \ b_6 = \frac{20}{9} \frac{c_2^2}{c_{-2}} + 2 c_{-2} d_8,\\ & b_8 = - \frac{40}{27} \frac{c_2^3}{c_{-2}} - 4 c_{-2} c_2 d_8 + 4 c_{-2}^2 d_{12},\ \ b_{10} = \frac{40}{81} \frac{c_2^4}{c_{-2}} + \frac{8}{3} c_{-2} c_2^2 d_8 - \frac{16}{3} c_{-2}^2 c_2 d_{12} + 8 c_{-2}^3 d_{16}, \\ & b_{12} = - \frac{16}{243} \frac{c_2^5}{c_{-2}} - \frac{16}{27} c_{-2} c_2^3 d_8 + \frac{16}{9} c_{-2}^2 c_2^2 d_{12} - \frac{16}{3} c_{-2}^3c_2 d_{16} + 16 c_{-2}^4 d_{20}. \end{aligned} \end{align} $$

A similar statement holds for $b_2 \neq 0$ .

3.1 Modular forms on orthogonal groups

Let L be an even integral lattice of signature $(2, n)$ and define $L_{\mathbb {R}} = L \otimes \mathbb {R}$ and $L_{\mathbb {C}} = L \otimes \mathbb {C}$ .

The Hermitian symmetric domain $\mathcal {D}$ for $\mathrm {O}(L)$ is the space

$$ \begin{align*}\mathcal{D} = \Big\{ [z] \in \mathbb{P}(L_{\mathbb{C}}): \; \langle z, z \rangle = 0, \; \langle z, \overline{z} \rangle> 0, \; \{\mathrm{re}[z], \mathrm{im}[z]\} \; \text{oriented} \Big\}.\end{align*} $$

In other words, if $z = x+iy$ then $[z] \in \mathcal {D}$ if and only if, up to scalar, $(x, y)$ is an oriented orthonormal basis of a positive-definite plane in $L_{\mathbb {R}}$ . The affine cone over $\mathcal {D}$ is the set

$$ \begin{align*}\mathcal{A} = \Big\{ z \in L_{\mathbb{C}}: \; [z] \in \mathcal{D} \Big\}\end{align*} $$

of points that span a line in $\mathcal {D}$ .

Let $\mathrm {O}(L)$ be the orthogonal group of L. There is an index two subgroup $\mathrm {O}^+(L)$ (the spinor kernel) of $\mathrm {O}(L)$ defined by the fact that it preserves the set $\mathcal {D}$ under multiplication. It is also characterized by the following fact: for $r \in L_{\mathbb {R}}$ , the reflection

$$ \begin{align*}\sigma_r \colon v \mapsto v - \frac{\langle v, r \rangle}{\langle r, r \rangle} r\end{align*} $$

belongs to $\mathrm {O}^+(L_{\mathbb {R}})$ if and only if $r^2> 0$ .

Let $\Gamma \le \mathrm {O}^+(L)$ be a finite-index subgroup. A meromorphic function $f\colon \mathcal {A} \rightarrow \mathbb {C}$ is called a meromorphic modular form of weight k and level $\Gamma $ if

1. 1. $f(\gamma z) = f(z)$ for all $\gamma \in \Gamma $ ,

2. 2. $f(\lambda z) = \lambda ^{-k} f(z)$ for $\lambda \in \mathbb {C}^{\times }$ .

The most important such $\Gamma $ (besides $\mathrm {O}^+(L)$ itself) is the discriminant kernel

$$ \begin{align*}\widetilde{\mathrm{O}}(L) = \Big \{ \gamma \in \mathrm{O}^+(L): \; \gamma x - x \in L \; \text{for all} \; x \in L' \Big\}.\end{align*} $$

3.2 Jacobi forms and lifts

Throughout this section, we assume that we have fixed a splitting $L = H \oplus H \oplus K(-1)$ with a positive-definite lattice K. The dual lattice is labeled

$$ \begin{align*}K' = \{x \in K \otimes \mathbb{Q}: \; \langle x, y \rangle \in \mathbb{Z} \; \text{for all} \; y \in K\}.\end{align*} $$

Definition 3.1. A Jacobi form of weight k and index K is a holomorphic function

$$ \begin{align*}\phi \colon \mathbb{H} \times K_{\mathbb{C}} \longrightarrow \mathbb{C}\end{align*} $$

that satisfies the functional equations:

$$ \begin{align*}\phi\left( \frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d} \right) = (c \tau + d)^k \exp \Big( \pi i \frac{c \cdot \langle z, z \rangle}{c \tau + d} \Big) \cdot \phi(\tau, z)\end{align*} $$

and

$$ \begin{align*}\phi \Big( \tau, z + \lambda \tau + \mu \Big) = \exp \Big( -\pi i \langle \lambda, \lambda \rangle \tau - 2\pi i \langle \lambda, z \rangle \Big) \cdot \phi(\tau, z)\end{align*} $$

for any $\left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ) \in \mathrm {SL}_2(\mathbb {Z})$ and $\lambda , \mu \in K$ .

Jacobi forms are represented by their Fourier series

$$ \begin{align*}\phi(\tau, z) = \sum_{n = -\infty}^{\infty} \sum_{r \in K'} c(n, r) q^n \zeta^r,\end{align*} $$

where $q^n := e^{2\pi i n \tau }$ and $\zeta ^r := e^{2\pi i \langle z, r \rangle }$ . The Jacobi form $\phi $ is called

• • weakly holomorphic if $c(n, r) = 0$ for sufficiently small n and every r;

• • weak if $c(n, r) = 0$ for every $n < 0$ and every r;

• • holomorphic if $c(n, r) = 0$ whenever $n - \langle r, r \rangle / 2 < 0$ ;

• • a cusp form if $c(n, r) = 0$ whenever $n - \langle r, r \rangle / 2 \le 0$ .

The value $n - \langle r, r \rangle / 2$ is also called the hyperbolic norm of the tuple $(n, r)$ .

$G = \mathrm {Aut}(K'/K)$ operates on Jacobi forms of index K via $\phi ^g(\tau , z) := \phi (\tau , gz).$

If $f\colon \mathcal {A} \rightarrow \mathbb {C}$ is a modular form on the discriminant kernel $\widetilde {\mathrm {O}}(L),$ then f can be expressed as a Fourier–Jacobi series as follows. Write elements of L as tuples in the form $(a, b, x, c, d)$ , with $a, b, c, d \in \mathbb {Z}$ and $x \in K$ , where the norm of the tuple is

$$ \begin{align*}\langle (a, b, x, c, d) , (a, b, x, c, d) \rangle / 2 = ad - bc - \langle x, x \rangle / 2,\end{align*} $$

and $\langle -, - \rangle $ also denotes the (positive definite) inner product on K.

For any elements $\tau , w \in \mathbb {H}$ and any $z \in K_{\mathbb {C}}$ , the element

$$ \begin{align*}Z = (1, \tau, z, w, \tau w + \langle z,z \rangle / 2)\end{align*} $$

lies in $\mathcal {A}$ if and only if it is oriented (i.e., $\mathrm {im}(z)$ lies in the so-called positive cone) and if it satisfies

$$ \begin{align*} 0 &< \langle Z, \overline{Z} \rangle \\ &= \tau w + \langle z,z \rangle /2 + \overline{\tau w} + \langle \overline{z}, \overline{z} \rangle / 2 - \tau \overline{w} - \overline{\tau} w - \langle z, \overline{z} \rangle \\ &= -4 \mathrm{im}(\tau) \mathrm{im}(w) - 2 \langle \mathrm{im}(z), \mathrm{im}(z) \rangle, \end{align*} $$

that is, we have $\langle \mathrm {im}(z), \mathrm {im}(z) \rangle / 2 < \mathrm {im}(\tau ) \cdot \mathrm {im}(w)$ .

The Fourier–Jacobi expansion of f is defined by

$$ \begin{align*}f(Z) = \sum_{n=0}^{\infty} \phi_n(\tau, z) e^{2\pi i n w}.\end{align*} $$

Each $\phi _n$ is a Jacobi form of index $K(n)$ , that is, K with its inner product multiplied by n.

The Gritsenko lift is defined in terms of the Hecke raising operators

$$ \begin{align*}V_N \colon J_{k, K} \longrightarrow J_{k, K(N)},\end{align*} $$

which are lattice-index generalizations of the operators introduced in [Reference Eichler and Zagier17]. For $N \ge 1,$ we define

$$ \begin{align*}\phi \Big| V_N(\tau, z) = \frac{1}{N} \sum_{\substack{ad = N \\ a> 0}} \sum_{b \in \mathbb{Z}/d\mathbb{Z}} a^k \phi \Big( \frac{a \tau + b}{d}, a z \Big).\end{align*} $$

If $\phi (\tau , z) = \sum _{n, r} c(n, r) q^n \zeta ^r,$ then

$$ \begin{align*}\phi \Big| V_N(\tau, z) = \sum_{d | (n, r, N)} d^{k-1} c \Big( \frac{Nn}{d^2}, \frac{r}{d} \Big).\end{align*} $$

For $N = 0$ and even $k \ge 4,$ we define $\phi \Big | V_0$ to be the abelian function

$$ \begin{align*}\phi \Big| V_0 = -c(0,0) \cdot \frac{B_k}{2k} E_k(\tau) + \frac{1}{(2\pi i)^k} \sum_{r> 0} c(0, r) \wp^{(k-2)}(\tau, \langle r, z \rangle),\end{align*} $$

where

$$ \begin{align*}\wp(\tau, z) = \frac{1}{z^2} + \sum_{\substack{m,n \in \mathbb{Z} \\ (m, n) \ne (0, 0)}} \Big( \frac{1}{(z - m \tau - n)^2} - \frac{1}{(m \tau + n)^2} \Big)\end{align*} $$

is the Weierstrass $\wp $ -function, where $\wp ^{(k-2)}$ is its $(k-2)$ nd derivative with respect to z, $B_k$ is the kth Bernoulli number, and $E_k$ is the Eisenstein series of weight k for $\mathrm {SL}_2(\mathbb {Z})$ , and where $r> 0$ means that $r|_K$ lies in the positive cone. For $k=2,$ we define

$$ \begin{align*}\phi \Big| V_0 = \frac{1}{(2\pi i)^k} \sum_{r> 0} c(0, r) \wp(\tau, \langle r, z \rangle).\end{align*} $$

Theorem 3.2. Suppose $\phi (\tau , z) = \sum _{n, r} c(n, r) q^n \zeta ^r$ is a weakly holomorphic Jacobi form of weight $k \ge 1$ and index K. Then,

$$ \begin{align*}\mathrm{G}(\phi)(Z) := \sum_{m=0}^{\infty} \Big( \phi \Big| V_m \Big) e^{2\pi i m w}\end{align*} $$

defines a meromorphic modular form of weight k on the discriminant kernel $\mathrm {\widetilde O}(L)$ . The poles of $\mathrm {G}(\phi )$ are all of order k and lie on hyperplanes

$$ \begin{align*}r^{\perp} = \Big\{ Z \in \mathcal{A}: \; \langle Z, r\rangle = 0 \Big\}\end{align*} $$

for positive norm vectors $r \in L'$ .

Moreover, if $\phi $ is invariant under $\mathrm {Aut}(K'/K),$ then $\mathrm {G}(\phi )$ is modular under the larger group $\mathrm {O}^+(L)$ .

This lift is constructed in [Reference Borcherds3, §14]. We call $\mathrm {G}(\phi )$ the Gritsenko lift or additive theta lift of $\phi $ as the case, where $\phi $ is holomorphic is due to Gritsenko [Reference Gritsenko20]. If r is a vector of the form $(0, n, r_K, 1, 0)$ with $r_K = r|_K \in K'$ , then it follows from the results of [Reference Borcherds3] that in a neighborhood of $r^{\perp }$ the Laurent expansion of $\mathrm {G}(\phi )$ begins

(3.1) $$ \begin{align}\mathrm{G}(\phi)(Z) = \left( \sum_{\lambda=1}^{\infty} c \Big( \lambda^2 n, \lambda r_K \Big) \cdot \frac{(k-1)!}{(2\pi i \lambda)^k} \right) \cdot \frac{1}{\langle r, Z \rangle^k} + h(Z),\end{align} $$

where h is holomorphic. Under $\mathrm {\widetilde O}(L),$ every hyperplane $r^{\perp }$ is equivalent to one of this form. The only coefficients that appear in Equation (3.1) are singular coefficients of $\phi $ , and the sum is finite because $c(n, r)$ is zero whenever $(n, r)$ has sufficiently negative hyperbolic norm.

The more famous construction of [Reference Borcherds3] is essentially the exponential of Theorem 3.2 applied to weight $k = 0$ .

Definition 3.3. Let $\phi (\tau , z) = \sum _{n, r} c(n, r) q^n \zeta ^r$ be a weakly holomorphic Jacobi form of weight $0$ and index K in which all Fourier coefficients $c(n, r)$ are integers. The theta block associated with $\phi $ is the infinite product

$$ \begin{align*} \Theta_{\phi}(\tau, z) &= \eta(\tau)^{c(0, 0)} \cdot \prod_{r> 0} \Big( \frac{\theta_{11}(\tau, \langle r, z \rangle)}{\eta(\tau)} \Big)^{c(0, r)} \\ &= q^{\frac{1}{24} \sum_{r \in K'} c(0, r)} \prod_{r > 0} (\zeta^{r/2} - \zeta^{-r/2}) \times \prod_{r \in K'} \prod_{n=1}^{\infty} (1 - q^n \zeta^r)^{c(0, r)}, \end{align*} $$

where

$$ \begin{align*}\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n)\end{align*} $$

is the Dedekind eta function and where

$$ \begin{align*}\theta_{11}(\tau, z) = \sum_{n \in \frac{1}{2}+ \mathbb{Z}} (-1)^{n-1/2} q^{n^2 / 2} e^{2\pi i n z}\end{align*} $$

is the odd Jacobi theta function.

Theorem 3.4. Suppose $\phi (\tau , z) = \sum _{n, r} c(n, r) q^n \zeta ^r$ is a weakly holomorphic Jacobi form of weight $0$ and index K in which all Fourier coefficients $c(n, r)$ are integers.

Then,

$$ \begin{align*}\Psi(Z) := \Theta_{\phi}(\tau, z) e^{2\pi i C w} \times \exp \Big( \sum_{m=1}^{\infty} \phi \Big| V_m \cdot e^{2\pi i m w} \Big)\end{align*} $$

converges on an open subset of $\mathcal {A}$ and it is the Fourier–Jacobi expansion of a meromorphic modular form $\mathrm {B}(\phi )$ of weight $\frac {1}{2}c(0, 0)$ on the discriminant kernel $\mathrm {\widetilde O}(L)$ . The zeros and poles of $\mathrm {B}(\phi )$ all lie on hyperplanes $r^{\perp }$ . Here, C is the constant

$$ \begin{align*}C = \frac{1}{\mathrm{rank}(K)} \cdot \sum_{r \in K'} c(0, r) \frac{\langle r, r\rangle}{2}.\end{align*} $$

If r is a vector of the form $(0, n, r_K, 1, 0),$ then the order of the zero or pole of $\Psi $ along $r^{\perp }$ is

$$ \begin{align*}\mathrm{ord} \Big( \Psi; r^{\perp} \Big) = \sum_{\lambda =1}^{\infty} c \Big( \lambda^2 n, \lambda r_K \Big).\end{align*} $$

This form is constructed in [Reference Borcherds3, §13] and it is called the Borcherds lift of $\phi $ . The setup in terms of Jacobi forms follows the discussion of [Reference Wang and Williams57].

5.1 Rank 18

Since the generic transcendental lattice in this family is

$$ \begin{align*}L = L_0 := H \oplus H,\end{align*} $$

the Jacobi forms that lift to modular forms on the orthogonal group are Jacobi forms of index $\{0\}$ , that is, elliptic modular forms of level one.

The lattice L can be identified with the lattice of matrices

$$ \begin{align*}\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad a, b, c, d \in \mathbb{Z}\end{align*} $$

with quadratic form given by the determinant. The spinor kernel $\mathrm {O}^+(L)$ contains two copies of $\mathrm {SL}_2(\mathbb {Z})$ , one acting by multiplication from the left, the other from the right, and in fact, it is generated by these two copies of $\mathrm {SL}_2(\mathbb {Z})$ together with the transpose (cf. [Reference Freitag and Hermann19]).

This implies that modular forms for $F \in M_k(\mathrm {O}^+(L))$ are linear combinations of products

$$ \begin{align*}f(\tau_1) g(\tau_2), \quad f, g \in M_k(\mathrm{SL}_2(\mathbb{Z}))\end{align*} $$

that are symmetric in the two variables $\tau _1, \tau _2$ , that is,

$$ \begin{align*}M_*(\mathrm{O}^+(L)) = \mathrm{Sym}^2(M_*(\mathrm{SL}_2(\mathbb{Z}))).\end{align*} $$

It is not hard to see that the graded ring $M_*(\mathrm {O}^+(L))$ is, therefore, a free algebra on three generators,

$$ \begin{align*}M_*(\mathrm{O}^+(L)) = \mathbb{C} \Big[ E_4 \otimes E_4, E_6 \otimes E_6, \Delta \otimes \Delta \Big].\end{align*} $$

The general K3 surface in the family – according to Proposition 2.6 – is described by the equation

(5.1) $$ \begin{align} y^2 = x^3 + A(t) x^2 + B(t) x, \end{align} $$

where $A(t) = t^3 + a_4 t + a_6$ and where in this case $B = b_{12}$ is constant. The discriminant of the right-hand side of Equation (5.1) is $B^2 (A^2 - 4B)$ , which itself is a polynomial of t with discriminant

$$ \begin{align*}D = 4096 b_{12}^3 \cdot \Big( 16a_4^6 + 216 a_4^3 a_6^2 + 729 a_6^4 + 864 a_4^3 b_{12} - 5832a_6^2 b_{12} + 11664 b_{12}^2 \Big).\end{align*} $$

Here, $a_4$ , $a_6$ and $b_{12}$ are holomorphic modular forms whose weight is indicated by the subscript, and due to the ring structure they are constant multiples of $E_4 \otimes E_4$ , $E_6 \otimes E_6,$ and $\Delta \otimes \Delta $ , respectively (as $a_4$ and $a_6$ live in a one-dimensional space of modular forms, and $B = b_{12}$ is never allowed to vanish).

The weight 24 form $D / b_{12}^3$ is a modular form that vanishes on the discriminant locus

$$ \begin{align*}\Big\{ (\tau_1, \tau_2): \; \mathrm{SL}_2(\mathbb{Z}) \cdot \tau_1 = \mathrm{SL}_2(\mathbb{Z}) \cdot \tau_2 \Big\}\end{align*} $$

and is, therefore, a constant multiple of the form

$$ \begin{align*}F(\tau_1, \tau_2) := \Delta(\tau_1)^2 \Delta(\tau_2)^2 \Big( j(\tau_1) - j(\tau_2) \Big)^2,\end{align*} $$

where j is the j-invariant. (Note that $F = \mathrm {B}(\psi )$ for the Jacobi form

$$ \begin{align*}\psi(\tau) = 2j(\tau) - 1440 = 2q^{-1} + 48 + 2 \cdot 196884q + O(q^2)\end{align*} $$

of weight zero.) By comparing coefficients between that form and

$$ \begin{align*}16a_4^2 + 216a_4^3 a_6^2 + 729a_6^4 + 864a_4^3 b_{12} - 5832 a_6^2 b_{12} + 11664b_{12}^2,\end{align*} $$

we find that if $b_{12} = C \cdot \Delta \otimes \Delta $ then we must have

$$ \begin{align*}a_4 = -\frac{1}{48} C^{1/3} \cdot E_4 \otimes E_4 \quad \text{and} \quad a_6 = \frac{1}{864} C^{1/2} \cdot E_6 \otimes E_6.\end{align*} $$

The parameter C is arbitrary, but it is natural to fix $C = 12^6$ such that the van Geemen–Sarti equation simplifies to

(5.2) $$ \begin{align} y^2 = x^3 + \Big( t^3 - 3 (E_4 \otimes E_4) t + 2 (E_6 \otimes E_6) \Big)x^2 + 12^6 (\Delta \otimes \Delta ) x. \end{align} $$

5.2 Rank 17

The transcendental lattice of the general member of the family is now

$$ \begin{align*}L = L_1 := H \oplus H \oplus A_1(-1).\end{align*} $$

There is a well-known exceptional isomorphism

$$ \begin{align*}\mathrm{Spin}(L) = \mathrm{Sp}_4(\mathbb{Z})\end{align*} $$

that makes it possible to identify modular forms on $\mathrm {O}^+(L)$ with Siegel modular forms of degree two and even weight (and modular forms on $\widetilde {\mathrm {O}}(L)$ with Siegel modular forms of any weight). By a famous theorem of Igusa [Reference Igusa24], the ring structure is

$$ \begin{align*}M_*(\mathrm{O}^+(L)) = \mathbb{C}[E_4, E_6, \chi_{10}, \chi_{12}],\end{align*} $$

where $E_4, E_6$ are the Siegel Eisenstein series and where the cusp forms $\chi _{10}, \chi _{12}$ are easiest to describe as Gritsenko lifts:

$$ \begin{align*}\chi_{10} = \mathrm{G}(\Delta \cdot \phi_{-2, 1}), \quad \chi_{12} = \mathrm{G}(\Delta \cdot \phi_{0, 1}).\end{align*} $$

(Note that these Gritsenko lifts differ from Igusa’s definitions [Reference Igusa24] by multiples of 4 and 12, respectively.)

There is a restriction map,

$$ \begin{align*}\mathrm{res} \colon \ M_*(\mathrm{O}^+(L_1)) \longrightarrow M_*(\mathrm{O}^+(L_0))\end{align*} $$

which through the interpretations $M_*(\mathrm {O}^+(L_1)) = M_{2*}(\mathrm {Sp}_4(\mathbb {Z}))$ and $M_*(\mathrm {O}^+(L_0)) = \mathrm {Sym}^2(M_*(\mathrm {SL}_2(\mathbb {Z}))$ is the map

$$ \begin{align*}\mathrm{res}(F)(\tau_1, \tau_2) = F \left( \begin{pmatrix} \tau_1 & 0 \\ 0 & \tau_2 \end{pmatrix} \right), \quad \tau_1, \tau_2 \in \mathbb{H}.\end{align*} $$

The action of the restriction map on the generators is

(5.3) $$ \begin{align} \mathrm{res}(E_4) = E_4 \otimes E_4, \quad \mathrm{res}(E_6) = E_6 \otimes E_6, \quad \mathrm{res}(\chi_{10}) = 0, \quad \mathrm{res}(\chi_{12}) = 12 \cdot (\Delta \otimes \Delta). \end{align} $$

The general K3 surface in the family – according to Proposition 2.6 – is described by the equation

$$ \begin{align*}y^2 = x^3 + A(t) x^2 + B(t) x\,,\end{align*} $$

where $A(t) = t^3 + a_4 t + a_6$ and $B(t) = b_{10} t + b_{12}$ and the previous family is cut out by the equation $b_{10} = 0$ . In view of Equations (5.2) and (5.3), we have

$$ \begin{align*}a_4 = -3E_4, \quad a_6 = 2 E_6, \quad b_{12} = 12^5 \cdot \chi_{12},\end{align*} $$

and there is a constant C such that $b_{10} = C \cdot \chi _{10}$ . We compare the first Fourier coefficients between the discriminant of $A^2 - 4B$ and the reflective product

$$ \begin{align*}\Psi_{60}\left( \begin{pmatrix} \tau & z \\ z & w \end{pmatrix} \right) = q^3 s^3 (q - s)^2 (r^{-1} + 2 + r) + O(q, s)^9, \quad q = e^{2\pi i \tau}, r = e^{2\pi i z}, s = e^{2\pi i w},\end{align*} $$

which is the Borcherds lift of the following weakly holomorphic Jacobi form of weight 0 and index 1,

$$ \begin{align*}\psi = \frac{29}{24} \cdot \frac{E_4^2 E_{4, 1}}{\Delta} + \frac{19}{24} \cdot \frac{E_6 E_{6, 1}}{\Delta} = 2q^{-1} + (2 \zeta^{-2} - 2 \zeta^{-1} + 120 - 2 \zeta + 2 \zeta^2) + O(q).\end{align*} $$

We conclude that $C = -12^5$ . Hence, we have derived the van Geemen–Sarti equation

(5.4) $$ \begin{align} y^2 = x^3 + \Big( t^3 - 3 E_4 t + 2 E_6 \Big) x^2 + 12^5 \cdot \Big( -\chi_{10} t + \chi_{12} \Big) x.\end{align} $$

5.3 Rank 16

The transcendental lattice of the general K3 surface in this family is

$$ \begin{align*}L = L_2 := H \oplus H \oplus A_1(-1)^{\oplus 2},\end{align*} $$

so the Jacobi forms that are lifted to produce modular forms on the orthogonal group have lattice index $A_1^{\oplus 2}$ .

By Theorem 4.1, the algebra of modular forms of level $\mathrm {O}^+(L)$ is

$$ \begin{align*}M_*(B_2) = \mathbb{C}[E_4, E_6, \chi_8, \chi_{10}, \chi_{12}],\end{align*} $$

where $E_4$ and $E_6$ are the Eisenstein series and where

$$ \begin{align*}\chi_8 = \mathrm{G} \Big( \Delta \phi_{-2,1} \otimes \phi_{-2, 1} \Big), \quad \chi_{10} = \mathrm{G} \Big( \Delta \cdot \phi_{-2, 1} \otimes \phi_{0, 1} + \Delta \cdot \phi_{0, 1} \otimes \phi_{-2, 1} \Big),\end{align*} $$

and

$$ \begin{align*}\chi_{12} = \mathrm{G} \Big( \Delta \cdot \phi_{0,1} \otimes \phi_{0, 1} \Big).\end{align*} $$

There are two obvious embeddings of $A_1$ into $A_1^{\oplus 2}$ and they induce the same restriction map

$$ \begin{align*}\mathrm{res}\colon \ M_*(\mathrm{O}^+(L_2)) \longrightarrow M_*(\mathrm{O}^+(L_1)).\end{align*} $$

The images of the generators are

$$ \begin{align*}\mathrm{res}(E_4^{L_2}) = E_4^{L_1}, \; \mathrm{res}(E_6^{L_2}) = E_6^{L_1}, \; \mathrm{res}(\chi_8^{L_2}) = 0, \; \mathrm{res}(\chi_{10}^{L_2}) = 12 \cdot \chi_{10}^{L_1}, \; \mathrm{res}(\chi_{12}^{L_2}) = 12 \cdot \chi_{12}^{L_1},\end{align*} $$

where the superscript indicates to which algebra the form belongs.

We will need the following weak Jacobi forms of weight two which are not Weyl-invariant and whose Gritsenko lifts are modular only under the discriminant kernel:

$$ \begin{align*} f_1(\tau, z_1, z_2) &:= \phi_{-2, 1}(\tau, z_1) E_{4, 1}(\tau, z_2) = (\zeta_1^{-1} - 2 + \zeta_1) + O(q); \\ f_2(\tau, z_1, z_2) &:= E_{4, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) = (\zeta_2^{-1} - 2 + \zeta_2) + O(q); \\ \end{align*} $$

where $\zeta _j = e^{2\pi i z_j}$ and $q = e^{2\pi i \tau }$ .

By Theorem 3.2, the Fourier–Jacobi series of the meromorphic Gritsenko lifts $\mathrm {G}(f_i)$ begin

$$ \begin{align*}\mathrm{G}(f_i) = -\frac{1}{4\pi^2} \wp(\tau, z_i) + f_i(\tau, z_1, z_2) s + O(s^2).\end{align*} $$

The general K3 surface in the family – according to Proposition 2.6 – is described by the equation

$$ \begin{align*}y^2 = x^3 + (t^3 + a_4 t + a_6) x^2 + (b_8 t^2 + b_{10} t + b_{12}) x,\end{align*} $$

where the previous family is cut out by $b_8 = 0$ . In view of the ring structure of $M_*(\mathrm {O}^+(L)),$ the forms $a_4, a_6, b_{10}$ are uniquely determined by their values (5.4) when $b_8 = 0$ . We obtain

$$ \begin{align*}a_4 = -3 E_4, \quad a_6 = 2 E_6, \quad b_{10} = -12^4 \cdot \chi_{10},\end{align*} $$

and moreover $b_8$ is a multiple of $\chi _8$ ; say $b_8 = A \cdot \chi _8$ . Also, when we factor

$$ \begin{align*}B(t) = b_8 (t - \beta_1) (t - \beta_2) = b_4 t^2 - b_8 (\beta_1 + \beta_2) t + b_4 \beta_1 \beta_2,\end{align*} $$

then $\beta _1$ and $\beta _2$ define meromorphic modular forms of weight two on the discriminant kernel with double poles on rational quadratic divisors on which $b_4$ vanishes. Since there are no nonzero holomorphic modular forms of weight two, we obtain

$$ \begin{align*}\beta_1 = C \cdot \mathrm{G}(f_1) \quad \text{and} \quad \beta_2 = C \cdot \mathrm{G}(f_2)\end{align*} $$

for some other constant C.

By comparing the leading Fourier–Jacobi coefficients in

$$ \begin{align*} & -b_8 (\beta_1 + \beta_2) = -AC \cdot \chi_8 \cdot (\mathrm{G}(f_1) + \mathrm{G}(f_2)) \\ &\quad = -AC \Big(\Delta \phi_{-2, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) s + O(s^2) \Big) \cdot \Big( -\frac{1}{4\pi^2} \wp(\tau, z_1) - \frac{1}{4\pi^2} \wp(\tau, z_2) + O(s) \Big) \\ & \quad = -\frac{AC}{12} \Delta(\tau) \Big(\phi_{-2,1}(\tau, z_1) \phi_{0,1}(\tau, z_2) + \phi_{0,1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \Big) s + O(s^2) \end{align*} $$

and

$$ \begin{align*}b_{10} = -12^4 \cdot \chi_{10} = -12 \Delta(\tau) \cdot \Big( \phi_{-2, 1}(\tau, z_1) \phi_{0, 1}(\tau, z_2) + \phi_{0, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \Big) s + O(s^2),\end{align*} $$

we conclude that $AC = 12^5$ , and by comparing

$$ \begin{align*} b_8 \beta_1 \beta_2 &= A C^2 \chi_8 \mathrm{G}(f_1) \mathrm{G}(f_2) \\ &= A C^2 \Big(\Delta \phi_{-2,1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) s + O(s^2) \Big) \Big( -\frac{1}{4\pi^2} \wp(\tau, z_1) + f_1(\tau, z_1, z_2) s + O(s^2) \Big) \\ &\quad \quad \times \Big(-\frac{1}{4\pi^2} \wp(\tau, z_2) + f_2(\tau, z_1, z_2) s + O(s^2) \Big) \\ &= \frac{A C^2}{144} \Delta(\tau) \phi_{0, 1}(\tau, z_1) \phi_{0, 1}(\tau, z_2) s \\ &= \frac{AC^2}{144} \phi_{12} \end{align*} $$

with $b_{12}$ , we conclude that

$$ \begin{align*}b_{12} = \frac{A C^2}{144}\chi_{12} = 12^3 C \cdot \chi_{12}.\end{align*} $$

When $b_8 = 0$ , the result of (5.4) implies

$$ \begin{align*}12^5 \chi_{12} = \mathrm{res}(b_{12}) = 12^3 C \cdot 12 \chi_{12},\end{align*} $$

that is, $C = 12$ and, therefore, $A = 12^4$ . Hence, we have derived the van Geemen–Sarti equation

(5.5) $$ \begin{align} y^2 = x^3 + (t^3 - 3 E_4 t + 2 E_6) x^2 + 12^4 \cdot (\chi_8 t^2 - \chi_{10} t + \chi_{12}) x. \end{align} $$

5.4 Rank 15

The transcendental lattice of the general surface in the family is

$$ \begin{align*}L = L_3 := H \oplus H \oplus A_1(-1)^{\oplus 3}.\end{align*} $$

For the full orthogonal group, we have the graded ring structure

$$ \begin{align*}M_*(\mathrm{O}^+(L)) = \mathbb{C}[E_4, E_6, \chi_6, \chi_8, \chi_{10}, \chi_{12}]\end{align*} $$

as a special case of Theorem 4.1, where $E_4$ and $E_6$ are the Eisenstein series and where $\chi _6, \chi _8, \chi _{10}, \chi _{12}$ are defined as Gritsenko lifts:

$$ \begin{align*}\chi_{6 + 2k} = \mathrm{G}(\varphi_{6 + 2k}), \quad k = 0, 1, 2, 3,\end{align*} $$

where $\varphi _{6 + 2k}$ is the Jacobi cusp form

$$ \begin{align*}\varphi_{6+2k}(\tau, z_1, z_2, z_3) = \Delta(\tau) \phi_{-2, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \phi_{-2, 1}(\tau, z_3) \sigma_k \Big( \frac{3}{\pi^2} \wp(\tau, z_1),\dots, \frac{3}{\pi^2} \wp(\tau, z_3) \Big),\end{align*} $$

and $\sigma _0,\dots ,\sigma _3$ are the elementary symmetric polynomials

$$ \begin{align*}\sigma_0 = 1, \quad \sigma_1(x,y,z) = x+y+z, \quad \sigma_2(x,y,z) = xy+yz+zx, \quad \sigma_3(x,y,z) = xyz.\end{align*} $$

Up to the action of the discriminant kernel, there are three embeddings of $H \oplus H \oplus A_1(-1)^{\oplus 2}$ into L as hyperplanes $r_1^{\perp }$ , $r_2^{\perp }$ , $r_3^{\perp }$ , where

(5.6) $$ \begin{align} r_1 = (1, 0, 0), \quad r_2 = (0, 1, 0), \quad r_3 = (0, 0, 1) \in A_1(-1)^{\oplus 3}.\end{align} $$

The result of restricting the generating forms along any of these embeddings is

$$ \begin{align*}E_4^{L_3} \mapsto E_4^{L_2}, \; E_6^{L_3} \mapsto E_6^{L_2}, \; \chi_6^{L_3} \mapsto 0, \; \chi_8^{L_3} \mapsto 12 \chi_8^{L_2}, \; \chi_{10}^{L_3} \mapsto 12 \chi_{10}^{L_2}, \; \chi_{12}^{L_3} \mapsto 12 \chi_{12}^{L_2}.\end{align*} $$

There are uniquely determined weak Jacobi forms $f_1, f_2, f_3$ of weight two and lattice index $A_1^{\oplus 3}$ whose Fourier expansions begin as

$$ \begin{align*}f_j(\tau, z_1, z_2, z_3) = (\zeta_j^{-1} - 2 + \zeta_j) + O(q),\end{align*} $$

with $q = e^{2\pi i \tau }$ and $\zeta _j = e^{2\pi i z_j}$ . In particular, each $\mathrm {G}(f_j)$ has double poles precisely on the orbit of $r_j^{\perp }$ under the discriminant kernel, with Fourier–Jacobi expansion beginning

$$ \begin{align*}\mathrm{G}(f_j)(\tau, z_1, z_2, z_3, w) = -\frac{1}{4\pi^2} \wp(\tau, z_j) + f_j(\tau, z_1, z_2, z_3) s + O(s^2), \quad s = e^{2\pi i w}.\end{align*} $$

The general K3 surface in the family – according to Proposition 2.6 – is described by the equation

$$ \begin{align*}y^2 = x^3 + A(t) x^2 + B(t)x,\end{align*} $$

where $A(t) = t^3 + a_4 t + a_6$ and $B(t) = b_6 t^3 + b_8 t^2 + b_{10} t + b_{12}$ , and the rank 16 family of the previous section is cut out by the equation $b_6 = 0$ . The forms $a_k$ , $b_k$ are modular forms of weight k whose restrictions to any of the hyperplanes $r_j^{\perp }$ are given by Equation (5.5), and in view of the algebra structure this uniquely determines

$$ \begin{align*}a_2 = -3 E_4, \quad b_4 = 12^3 \chi_8;\end{align*} $$

in addition, $b_3 = A \cdot \chi _6$ for some nonzero constant A.

If we factor

$$ \begin{align*}B(t) = b_6 (t - \beta_1) (t - \beta_2) (t - \beta_3),\end{align*} $$

then $\beta _1, \beta _2, \beta _3$ are meromorphic modular forms of weight two on the discriminant kernel of L with double poles on $r_1^{\perp }, r_2^{\perp },$ and $r_3^{\perp }$ , respectively, and they are permuted under $\mathrm {O}^+(L)$ . This forces (possibly after reordering)

$$ \begin{align*}\beta_1 = C \cdot \mathrm{G}(f_1), \; \beta_2 = C \cdot \mathrm{G}(f_2), \; \beta_3 = C \cdot \mathrm{G}(f_3)\end{align*} $$

with a common nonzero constant C.

Now, we have

$$ \begin{align*} 12^3 \cdot \chi_8 &= b_8 \\ &= -b_6 (\beta_1 + \beta_2 + \beta_3) \\ &= \frac{1}{4\pi^2} AC \chi_6 \sum_{i=1}^3 \wp(\tau, z_i) + O(s) \\ &= -\frac{AC}{12} \Delta \phi_{-2, 1}(\tau, z_1)\phi_{-2,1}(\tau, z_2) \phi_{-2, 1}(\tau, z_3) \sum_{i=1}^3 \frac{\phi_{0,1}(\tau, z_i)}{\phi_{-2, 1}(\tau, z_i)} + O(s), \end{align*} $$

and comparing the constant Fourier–Jacobi coefficients on both sides of this equation yields $AC = -12^4$ . The Fourier–Jacobi expansions of $b_{10}$ and $b_{12,}$ therefore, begin as

$$ \begin{align*} b_{10} &= b_6 (\beta_1 \beta_2 + \beta_1 \beta_3 + \beta_2 \beta_3) \\ &= \frac{AC^2}{144} \cdot \Delta(\tau) \phi_{-2, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \phi_{-2, 1}(\tau, z_3) \sigma_2 \Big(\frac{3}{\pi^2} \wp(\tau, z_1), \frac{3}{\pi^2} \wp(\tau, z_2), \frac{3}{\pi^2} \wp(\tau, z_3) \Big)s\\ &\quad + O(s^2) \\ &= 144 C \cdot \varphi_{10}(\tau, z_1, z_2, z_3) s + O(s^2),\end{align*} $$

which uniquely determines $C = 12$ and $A = -12^3$ and $b_{10} = -1728 \chi _{10},$ and

$$ \begin{align*} b_{12} &= -b_6\beta_1 \beta_2 \beta_3 \\ &= -\frac{AC^3}{1728} \cdot \Delta(\tau) \phi_{-2, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \phi_{-2, 1}(\tau, z_3) \\ & \quad\times\Big( -\frac{3}{\pi^2} \wp(\tau, z_1) \cdot -\frac{3}{\pi^2} \wp(\tau, z_2) \cdot -\frac{3}{\pi^2}\wp(\tau, z_3) \Big)s \\ & \quad - \frac{AC^3}{144} \Delta(\tau) \phi_{-2, 1}(\tau, z_1) \phi_{-2, 1}(\tau, z_2) \phi_{-2, 1}(\tau, z_3) \\ & \quad\times\sum_{i=1}^3 \Big[f_i(\tau, z_1, z_2, z_3) \prod_{j \ne i} -\frac{3}{\pi^2} \wp(\tau, z_j)\Big] s^2 + O(s^3), \\ &= 12 C^2 \cdot \varphi_{12}(\tau, z_1, z_2, z_3)s - 12C^2 \cdot \Big( \varphi_{12} \Big| V_2 - 2592 \varphi_6^2 \Big) s^2 + O(s^3).\end{align*} $$

The coefficients of $B(t)$ are, therefore,

$$ \begin{align*}b_6 = -1728 \chi_6, \quad b_8 = 1728 \chi_8,\end{align*} $$

$$ \begin{align*}b_{10} = -1728 \chi_{10}, \quad b_{12} = 1728 \Big( \chi_{12} - 2592 \cdot \chi_6^2 \Big).\end{align*} $$

Only the coefficient $a_6$ remains to be determined. Since the value at $b_6 = 0$ is determined by Equation (5.5), we have

$$ \begin{align*}a_6 = 2 E_6 + C \cdot \chi_6\end{align*} $$

for some constant C. The contant C can theoretically be determined by computing the discriminant form (as in the derivation of Equation (5.4) but this is computationally difficult. Instead, we use a method that we call paramodular restriction, which will also be useful in the later sections. It is based on the following observation.

Lemma 5.1. Write

$$ \begin{align*}B(t) = b_6 t^3 + b_8 t^2 + b_{10} t + b_{12} = b_6 (t - \beta_1) (t - \beta_2)(t - \beta_3),\end{align*} $$

where each $\beta _i$ has poles exactly on the $\mathrm {\widetilde O}(L)$ -orbit of $r_i^{\perp }$ . Then, $\beta _i - \beta _j$ vanishes precisely on the $\mathrm {\widetilde O}(L)$ -orbit of $(r_i + r_j)^{\perp }$ .

Proof. Since the reflection through the hyperplane $(r_i + r_j)^{\perp }$ swaps $r_i^{\perp }$ and $r_j^{\perp }$ , it maps $\beta _i - \beta _j$ to $\beta _j - \beta _i$ . This implies that $\beta _i - \beta _j$ vanishes on $(r_i + r_j)^{\perp }$ .

One can show that there are unique weak Jacobi forms of weight 0 and lattice index $A_1^{\oplus 3}$ whose Fourier series begin

$$ \begin{align*} b_{12}(\tau, z_1, z_2, z_3) &= (\zeta_1^{-1} - 2 + \zeta_1) (\zeta_2^{-1} - 2 + \zeta_2) + O(q); \\ b_{13}(\tau, z_1, z_2, z_3) &= (\zeta_1^{-1} - 2 + \zeta_1)(\zeta_3^{-1} - 2 + \zeta_3) + O(q); \\ b_{23}(\tau, z_1, z_2, z_3) &= (\zeta_2^{-1} - 2 + \zeta_2)(\zeta_3^{-1} - 2 + \zeta_3) + O(q). \end{align*} $$

The Borcherds products $\psi _{ij} = \mathrm {B}(b_{ij})$ have double poles on $r_i^{\perp }$ and $r_j^{\perp }$ and vanish on $(r_i + r_j)^{\perp }$ by construction. This means that $(\beta _i - \beta _j) / \psi _{ij}$ are holomorphic modular forms of weight $0$ and hence (nonzero) constants, so

$$\begin{align*}\mathrm{div}(\beta_i - \beta_j) = \mathrm{div}(\psi_{ij}) = (r_i + r_j)^{\perp} - 2 r_i^{\perp} - 2r_j^{\perp}. \end{align*}$$

We consider the specialization of the family

$$ \begin{align*}y^2 = x^3 + (t^3 + a_4 t + a_6) x^2 + \Big(b_6 \prod_{i=1}^3 (t - \beta_i) \Big) x\end{align*} $$

to $\beta _1 = \beta _2 = \beta _3 =: \beta $ . In view of Lemma 5.1, the general member of the specialized family has transcendental lattice $H \oplus H \oplus A_1($ - $3).$ The orthogonal group of that lattice is essentially the paramodular group of level 3 (see, e.g., [Reference Gritsenko and Nikulin21, §1.3]).

The discriminant of

$$ \begin{align*}y^2 = x^3 + (t^3 + a_4 t + a_6) x^2 + b_6 (t - \beta)^3 x\end{align*} $$

is given by

$$ \begin{align*}b_6^2 (t - \beta)^6 \Big( t^6 + 2 a_4 t^4 + 2(a_6 - 2 b_6) t^3 + (a_4^2 + 12 b_6 \beta) t^2 + (2 a_4 a_6 - 12 b_6 \beta^2) t + (4 b_6 \beta^3 + a_6^2) \Big),\end{align*} $$

and the irreducible factor of degree six of that has discriminant

$$ \begin{align*}D := b_6^3 \cdot (\beta^3 + a_4 \beta + a_6)^3 \cdot P\end{align*} $$

with a modular form P of weight $36$ whose expression is too long to write out here.

The presence of $b_6 \cdot (\beta ^3 + a_4 \beta + a_6)$ in the discriminant D implies that it is a reflective Borcherds product of weight $12$ with a double zero on the paramodular Humbert surface of discriminant $4$ in the sense of [Reference Gritsenko and Nikulin21], and this in fact determines it uniquely: we obtain

$$ \begin{align*}b_6 \cdot (\beta^3 + a_4 \beta + a_6) = b_6 \beta^3 + a_4 \beta + \Big( \frac{6}{17} b_6^2 + 2 E_6 b_6 \Big)\end{align*} $$

and, therefore,

$$ \begin{align*}a_6 = 2 E_6 + \frac{6}{17} b_6 = 2 E_6 + \frac{10368}{17} \chi_6.\end{align*} $$

Altogether, we have derived the van Geemen–Sarti equation

(5.7) $$ \begin{align} y^2 = x^3 + \Big( t^3 - 3 E_4 t + 2 E_6 + \frac{10368}{17} \chi_6 \Big) x^2 + 1728 \Big( -\chi_6 t^3 + \chi_8 t^2 - \chi_{10} t + \chi_{12} - 2592 \chi_6^2 \Big)x. \end{align} $$

5.5 Rank 14

The transcendental lattice of the generic K3 surface in the family is

$$ \begin{align*}L = L_4 := H \oplus H \oplus A_1(-1)^{\oplus 4}.\end{align*} $$

By Theorem 4.1, the algebra structure of modular forms for the full orthogonal group is

$$ \begin{align*}M_*(\mathrm{O}^+(L)) = \mathbb{C}[E_4, E_6, \chi_4, \chi_6, \chi_8, \chi_{10}, \chi_{12}],\end{align*} $$

where $E_4, E_6$ are Eisenstein series and where $\chi _{4+2k} = \mathrm {G}(\varphi _{4+2k})$ for the Jacobi form

$$ \begin{align*} & \varphi_{4+2k}(\tau, z_1,\dots,z_4) \\ & \quad = \Delta(\tau) \cdot \Big( \prod_{i=1}^4 \phi_{-2, 1}(\tau, z_i) \Big) \cdot \sigma_k \Big( \frac{3}{\pi^2} \wp(\tau, z_1),\dots, \frac{3}{\pi^2} \wp(\tau, z_4) \Big), \quad k=0,1,2,3,4.\end{align*} $$

Here, $\sigma _k$ is the usual elementary symmetric polynomial. In contrast to the earlier sections, $\varphi _{4+2k}$ and, therefore, $\chi _{4+2k}$ are not cusp forms.

Similarly to the previous section, there are four natural embeddings of $H \oplus H \oplus A_1(-1)^{\oplus 3}$ into L as the hyperplanes $r_i^{\perp }, i=1,\dots ,4,$ where

$$ \begin{align*}r_i = (0,\dots, 1,\dots,0) \in A_1(-1)^{\oplus 4}\end{align*} $$

has $1$ in the $i^{\text {th}}$ entry and $0$ elsewhere, and along any of the embeddings the generating modular forms have the following restrictions:

$$ \begin{align*}E_4^{L_4} \mapsto E_4^{L_4}, \quad E_6^{L_3} \mapsto E_6^{L_3} + \frac{1512}{17} \chi_6^{L_3};\end{align*} $$

$$ \begin{align*}\chi_4^{L_4} \mapsto 0, \; \chi_6^{L_4} \mapsto 12 \chi_6^{L_3}, \; \chi_8^{L_4} \mapsto 12 \chi_8^{L_3}, \; \chi_{10}^{L_4} \mapsto 12 \chi_{10}^{L_3}, \; \chi_{12}^{L_4} \mapsto 12 \chi_{12}^{L_3}.\end{align*} $$

The general K3 surface in the family – according to Proposition 2.6 – is described by the equation

$$ \begin{align*}y^2 = x^3 + A(t) x^2 + B(t) x,\end{align*} $$

where $A(t) = t^3 + a_4 t + a_6$ and $B(t) = b_4 t^4 + b_6 t^3 + b_8 t^2 + b_{10} t + b_{12}$ .

Using the ring structure and the values from Equation (5.7), we see immediately that

$$ \begin{align*}a_6 = 2 E_6 + 36 \chi_6 \quad \text{and} \quad b_6 = -144 \chi_6.\end{align*} $$

Arguing as in the previous sections, we factor

$$ \begin{align*}B(t) = b_4 (t - \beta_1) (t - \beta_2) (t - \beta_3) (t - \beta_4),\end{align*} $$

where $\beta _i$ are meromorphic modular forms on the discriminant kernel of L with double poles on the hyperplane $r_i^{\perp }$ . There are unique weak Jacobi forms of weight two with Fourier series beginning

$$ \begin{align*}f_i = (\zeta_i^{-1} - 2 + \zeta_i) + O(q), \quad i=1,\dots,4,\end{align*} $$

and whose Gritsenko lifts have Fourier–Jacobi expansions beginning

$$ \begin{align*}\mathrm{G}(f_i) = -\frac{1}{4\pi^2} \wp(\tau, z_i) + f_i(\tau, z_1,\dots,z_4) s + O(s^2);\end{align*} $$

we must have $\beta _i = C \cdot \mathrm {G}(f_i)$ for some constant C. Also, there is a constant A such that $b_4 = A \cdot \chi _4$ .

Now, we consider the Fourier–Jacobi expansion of $b_6$ :

$$ \begin{align*} -144 \chi_6 &= -b_4 (\beta_1 + \beta_2 + \beta_3 + \beta_4) = \frac{1}{4\pi^2} AC \chi_4 \cdot \Big( \sum_{i=1}^4 \wp(\tau, z_i) + O(s) \Big) \\ &= -\frac{AC}{12} \Delta \Big( \prod_{i=1}^4 \phi_{-2, 1}(\tau, z_i) \Big) \sum_{i=1}^4 \frac{\phi_{0, 1}(\tau, z_i)}{\phi_{-2, 1}(\tau, z_i)} + \mathrm{O}(s). \end{align*} $$

By comparing constant terms we obtain $AC = 12^3$ . From the Fourier–Jacobi expansions of the other terms, we obtain

$$ \begin{align*}b_8 = b_4 \prod_{i<j} \beta_i \beta_j = \frac{AC^2}{12^2} (\chi_8 - 432 \chi_4^2) = 12C \cdot (\chi_8 - 432 \chi_4^2)\end{align*} $$

and

$$ \begin{align*}b_{10} = -b_4 \prod_{i < j < k} \beta_i \beta_j \beta_k = -\frac{AC^3}{12^3} (\chi_{10} - 432 \chi_4 \chi_6)\end{align*} $$

and finally

$$ \begin{align*}b_{12} = b_2 \beta_1 \beta_2 \beta_3 \beta_4 = \frac{AC^4}{12^4} \Big( \chi_{12} + 360 \chi_4 \chi_8 - 216 \chi_6^2 - 235008 \chi_4^3 - 648 E_4 \chi_4^2 \Big).\end{align*} $$

By comparing this with the expressions in Equation (5.7), we obtain $A = 12^2$ and $C = 12$ and, therefore,

$$ \begin{align*}b_8 = 144 (\chi_8 - 432 \chi_4^2),\end{align*} $$

$$ \begin{align*}b_{10} = -144(\chi_{10} - 432 \chi_4 \chi_6)\end{align*} $$

and

$$ \begin{align*}b_{12} = 144 (\chi_{12} + 360 \chi_4 \chi_8 - 216 \chi_6^2 - 235008 \chi_4^3 - 648 E_4 \chi_4^2).\end{align*} $$

The only coefficient left undetermined is $a_4$ : we have

$$ \begin{align*}a_4 = -3 E_4 + C \cdot \chi_4\end{align*} $$

for some constant C. The remaining constant can be determined by paramodular restriction as in the previous section. Setting $\beta _1 = \beta _2 = \beta _3 = \beta _4 =: \beta $ specializes the transcendental lattice to $H \oplus H \oplus A_1(-4)$ , and the specialized coefficients become paramodular forms on the group $K(4)$ . The irreducible degree six factor of the discriminant of the van Geemen–Sarti equation

$$ \begin{align*}y^2 = x^3 + \big(t^3 + a_4 t + a_6\big) x^2 + \big(t - \beta\big)^4 x\end{align*} $$

itself has discriminant

$$ \begin{align*}D = b_4^3 (a_6 + a_4 \beta + \beta^3)^4 \cdot P\end{align*} $$

with a factor P of weight $24$ . The expression $a_6 + a_4 \beta + \beta ^3$ is therefore a reflective paramodular form of weight $6$ , and in fact

$$ \begin{align*}a_6 + a_4 \beta + \beta^3 = \frac{\mathrm{B}(\psi_{9/2})^2}{\mathrm{B}(\psi_{1/2})^6},\end{align*} $$

where $\mathrm {B}(\psi _{1/2})$ and $\mathrm {B}(\psi _{9/2})$ are (the unique) reflective Borcherds products with simple zeros on the Heegner divisors of discriminant $1/16$ and $1/4$ , respectively. Using the formulas for $a_6$ and $\beta $ that were determined earlier, we find

$$ \begin{align*}a_4 = -3E_4 - 144 \chi_4.\end{align*} $$

Altogether, we have derived the van Geemen–Sarti equation

$$ \begin{align*} y^2 &= x^3 + \Big( t^3 - 3 E_4 t - 144 \chi_4 t + 2 E_6 + 432 \chi_6 \Big) x^2 + 144 \Big( \chi_4 t^4 - \chi_6 t^3 + (\chi_8 - 432 \chi_4^2) t^2\\ &\quad - (\chi_{10} - 432 \chi_4 \chi_6) t + \chi_{12} + 360 \chi_4 \chi_8 - 216 \chi_6^2 - 235008 \chi_4^3 - 648 E_4 \chi_4^2 \Big) x. \end{align*} $$

5.6 Rank 13

The generic transcendental lattice is

$$ \begin{align*}L = L_5 := H \oplus H \oplus A_1(-1)^{\oplus 5}.\end{align*} $$

The general K3 surface in the family – according to Proposition 2.9 – is described by the equation

(5.8) $$ \begin{align} y^2 = x^3 + A(t) x^2 + B(t)x \end{align} $$

where $A(t) = t^3 + a_4 t + a_6$ and

$$ \begin{align*}B(t) = b_2 t^5 + b_4 t^4 + b_6 t^3 + b_8 t^2 + b_{10} t + b_{12}.\end{align*} $$

Up to the action of the discriminant kernel of L, there are six classes of Heegner divisors of discriminant $1/4$ : they are $r_i^{\perp }, \quad i=1,\dots ,5$ and $r^{\perp }$ , where $r_i = (0, \dots , 1/2, \dots , 0) \in A_1^{\oplus 5}$ is nonzero in the ith component, and where r can be taken to be the vector

(5.9) $$ \begin{align} r = (0, 0) \oplus (1, 1) \oplus (1/2, 1/2, 1/2, 1/2, 1/2) \in H \oplus H \oplus A_1(-1)^{\oplus 5}.\end{align} $$

Along the hyperplane $r^{\perp }$ , the polarizing lattice extends to $H \oplus D_8(-1) \oplus D_4(-1)$ and the description (5.8) is no longer valid. In concrete terms, this means that $a_k, b_k$ define meromorphic modular forms of level $\mathrm {O}^+(L)$ that are allowed to have poles along the rational quadratic divisor $r^{\perp }$ .

By Theorem 4.2, the ring

$$ \begin{align*}M^!_*(\mathrm{O}^+(L)) = \Big\{ \text{meromorphic modular forms, holomorphic away from} \; r^{\perp} \Big\}\end{align*} $$

is a free algebra with generators $F_4, E_6$ and $\chi _{2 + 2k}$ , $k=0,\dots ,5$ , where $E_6$ is an Eisenstein series, $F_4$ is the (unique) holomorphic modular form of weight $4$ with constant Fourier coefficient $1$ , and $\chi _{2 + 2k}$ is the (meromorphic) Gritsenko lift of

$$ \begin{align*}\phi_{2+2k}(\tau, z_1,\dots,z_5) = \Delta(\tau) \prod_{i=1}^5 \phi_{-2, 1}(\tau, z_i) \cdot \sigma_k \Big( \frac{3}{\pi^2} \wp(\tau, z_1),\dots, \frac{3}{\pi^2} \wp(\tau, z_5) \Big).\end{align*} $$

Furthermore, it follows from [Reference Wang and Williams55] that the lowest-weight form $\chi _2$ is also a Borcherds product, with divisor

$$ \begin{align*}\mathrm{div}(\chi_2) = 2 \sum_{i=1}^5 r_i^{\perp} - 2r^{\perp}.\end{align*} $$

Under the restriction map to any of the hyperplanes $r_i^{\perp }$ (i.e., to a sublattice of type $H \oplus H \oplus A_1(-1)^{\oplus 4})$ , the generating forms are mapped as follows:

$$ \begin{align*}F_4 \mapsto E_4 + 48 \chi_4^{L_4}, \quad E_6^{L_5} \mapsto E_6^{L_4}, \quad \chi_2^{L_5} \mapsto 0,\end{align*} $$

$$ \begin{align*}\chi_4^{L_5} \mapsto 12 \chi_4^{L_4}, \; \chi_6^{L_5} \mapsto 12 \chi_6^{L_4}, \; \chi_8^{L_5} \mapsto 12 \chi_8^{L_4}, \; \chi_{10}^{L_5} \mapsto 12 \chi_{10}^{L_4}, \; \chi_{12}^{L_5} \mapsto 12 \chi_{12}^{L_4}.\end{align*} $$

There are again Jacobi forms $f_1,\dots ,f_5$ whose q-expansions begin

$$ \begin{align*}f_j(\tau, z_1,\dots,z_5) = \zeta_j^{-1} - 2 + \zeta_j + O(q).\end{align*} $$

Unlike the previous sections, $f_j$ are no longer uniquely determined; however, there are unique choices of $f_j$ with the property that all of their singular coefficients are represented in the $q^0$ -term of its Fourier expansion. In other words, $f_j$ is unique if we ask for $\mathrm {G}(f_j)$ to be holomorphic along the divisor $r^{\perp }$ .

On the other hand, the polynomial $B(t)$ factors as

$$ \begin{align*}B(t) = (t - \beta_1) \cdot \dots \cdot (t - \beta_5),\end{align*} $$

where each $\beta _i$ has weight two under the discriminant kernel and (after a reordering) has double poles exactly on the orbit of $r_i^{\perp }$ and possibly on $r^{\perp }$ . Since there are no nonzero, holomorphic modular forms of weight two, we obtain

$$ \begin{align*}\beta_i = B \cdot \chi_2 + C \cdot \mathrm{G}(f_i)\end{align*} $$

for common constants $B, C$ . Setting $\chi _2 = 0$ , the result of the previous section implies $C = 12$ but does not determine B. Besides this, the ring structure implies that there are constants $A, C, D, E$ such that

(5.10) $$ \begin{align} a_4 = -3 F_4 + A \cdot \chi_2^2 \; \text{and} \; a_6 = 2 E_6 + 3 \chi_6 + C \cdot F_4 \chi_2 + D \cdot \chi_2^3 + E \cdot \chi_2 \chi_4. \end{align} $$

We will use paramodular restriction as in the previous sections, specializing to $\beta _1 = \beta _2 = \beta _3 = \beta _4 = \beta _5 =: \beta $ , to determine the undetermined constants. Under this specialization, the generic transcendental lattice becomes $H \oplus H \oplus A_1(-5)$ and the coefficients become (meromorphic) paramodular forms of level $5$ . These are considerably more complicated than paramodular forms of level $\le 4$ ; their structure was nevertheless worked out completely in [Reference Williams58]. The discriminant of

$$ \begin{align*}x^3 + (t^3 + a_4 t + a_6) x^2 + b_2 (t - \beta)^5 x\end{align*} $$

factors as $b_2 (t - \beta )^{10} \cdot P(t)$ with an irreducible polynomial P of degree $6$ , whose discriminant itself factors as

$$ \begin{align*}\mathrm{disc}(P) = b_2^3 (\beta^3 + a_4 \beta + a_6)^5 \cdot Q\end{align*} $$

with a factor Q of weight $24$ . In particular, $f := \beta ^3 + a_4 \beta + a_6$ is a reflective paramodular form; more precisely, it is a reflective Borcherds product of weight $6$ , with a pole of order six (coming from $\beta ^3$ ) on the paramodular Humbert surface $H(1/20)$ that is nonvanishing on the paramodular Humbert surface $H(1)$ , and this information determines it uniquely: we have

$$ \begin{align*}\mathrm{div}(f) = -8 H(1/20) + 2 H(1/5) + 4 H(1/4),\end{align*} $$

where $H(D)$ is the (imprimitive) paramodular Heegner divisor of discriminant D. In the notation of [Reference Williams58, Table 1], we have

$$ \begin{align*}f = \text{const} \cdot (b_8 / b_5)^2.\end{align*} $$

Also, the paramodular restriction

$$ \begin{align*}\chi := \mathrm{res}(\chi_2)\end{align*} $$

of the product $\chi _2$ is the reflective form with divisor

$$ \begin{align*}\mathrm{div} \, \chi = 10 H(1/20) - 2 H(1/4),\end{align*} $$

in the notation of [Reference Williams58],

$$ \begin{align*}\chi = \text{const} \cdot b_5^2 / b_8.\end{align*} $$

The paramodular restrictions of the forms $\chi _4$ and $\chi _6$ can be computed to be

$$ \begin{align*}\mathrm{res}\, \chi_4 = 30 \chi \cdot \Big( \chi + 2 G \Big)\end{align*} $$

and

$$ \begin{align*}\mathrm{res}\, \chi_6 = 240 \chi \cdot \Big( 5 \chi^2 + 15 \chi G + 6 G^2 \Big),\end{align*} $$

respectively. Inserting Equation (5.10), we obtain

$$ \begin{align*} f &= \beta^3 + a_4 \beta + a_6 \\ &= (B^3 + AB + D + 30E + 3600) \cdot \chi^3 + (36B^2 + 12A + 60E + 10800) G \chi^2 \\ &\quad + (432B + 4320)G^2 \chi + (C - 3B) F_4 \chi + 1728 G^3 - 36 F_4G + 2E_6. \end{align*} $$

The paramodular forms $\chi , E_6, F_4, G$ are algebraically independent, and comparing a few Fourier coefficients shows that the unique expression for f in terms of them is

$$ \begin{align*}f = 10 \chi (432 G^2 - 3 F) + 1728 G^3 - 36 F G + 2 E_6.\end{align*} $$

This allows us to read off the missing coefficients

$$ \begin{align*}A = 0, \; B = 0, \; C = -30, \; D = 1800, \; E = -180,\end{align*} $$

and, therefore, we obtain

(5.11) $$ \begin{align} a_4 &= -3 F_4, \end{align} $$

(5.12) $$ \begin{align} a_6 &= 2 E_6 + 3 \chi_6 - 30 \chi_2 F_4 + 1800 \chi_2^3 - 180 \chi_2 \chi_4, \end{align} $$

and

$$ \begin{align*}\beta_i = 12 \cdot \mathrm{G}(f_i).\end{align*} $$

The coefficients of $B(t)$ can then be computed as the symmetric polynomials in $\beta _i$ . We find

$$ \begin{align*} b_2 &= -12 \chi_2; \\ b_4 &= -b_2 \sum_{i=1}^5 \beta_i = 12 \chi_4 - 360 \chi_2^2, \\ b_6 &= b_2 \sum_{i < j} \beta_i \beta_j = -12 \chi_6 + 720 \chi_2 \chi_4 - 7200 \chi_2^3, \\ b_8 &= 12 \chi_8 - 432 \chi_4^2 - 14400 \chi_2^2 \chi_4 + 216000 \chi_2^4 - 4320 \chi_2^2 F_4; \\ b_{10} &= -12 \chi_{10} - 936 \chi_2 \chi_8 + 432 \chi_4 \chi_6 + 6480 \chi_2^2 E_6 + 32760 \chi_2^2 \chi_6 + 2376 \chi_2 \chi_4 F_4 \\ &\quad + 27936 \chi_2 \chi_4^2 + 362880 \chi_2^3 F_4 + 21600 \chi_2^3 \chi_4 - 11707200 \chi_2^5; \end{align*} $$

$$ \begin{align*} b_{12} &= 12 \chi_{12} - 216\chi_2 \chi_{10} + 360\chi_4\chi_8 - 216 \chi_6^2 - 1296 \chi_2 \chi_4 E_6 + 864\chi_2 \chi_6 F_4 - 648 \chi_4^2 F_4 \\ &\quad - 67968 \chi_2^2 \chi_8 + 28584 \chi_2 \chi_4 \chi_6 - 16992 \chi_4^3 + 83520\chi_2^3 E_6 - 5832 \chi_2^2 F_4^2 - 258336 \chi_2^2 \chi_4 F_4 \\ & \quad - 574560 \chi_2^3 \chi_6 + 871776 \chi_2^2 \chi_4^2 + 25574400 \chi_2^4 F_4 + 132096960 \chi_2^4 \chi_4 - 1523059200 \chi_2^6. \end{align*} $$

Remark 5.2. A priori, the coefficients $a_k$ and $b_k$ were allowed to have poles along the exceptional hyperplane $r^{\perp }$ (of order at most their weight). However, the computation of the coefficients shows that when we write

$$ \begin{align*}y^2 = x^3 + (t^3 + a_4 t + a_6) x^2 + (b_2 t^5 + \dots + b_{12}) x,\end{align*} $$

$a_4$ and $a_6$ are holomorphic along $r^{\perp }$ and all the $b_k$ have exactly a double pole there.

6.1 Modular forms and a Weierstrass equation for $S\hspace{-1.5pt} =\hspace{-1.5pt} H\hspace{-.5pt} \oplus\hspace{-.5pt} D_8(-1)\hspace{-.5pt} \oplus\hspace{-.5pt} D_4(-1).$

The graded algebra $M_*(\mathrm {O}^+(L))$ of modular forms was described in [Reference Wang and Williams56]. It is a free algebra that is generated in weights $2, 6, 8, 10, 12, 16, 20$ . The generators of weight $2$ and $6$ can be chosen to be the (unique) modular forms $E_2$ and $F_6$ whose leading Fourier–Jacobi coefficients are

$$ \begin{align*}1 + 24q + 24q^2 + 96q^3 + \dots \in M_2(\Gamma_0(2))\end{align*} $$

and

$$ \begin{align*}q + 32q^2 + 244q^3 + 1024q^4 + \dots \in M_6(\Gamma_0(2)),\end{align*} $$

and the generator $F_{10}$ of weight $10$ will be chosen to be the (unique) cusp form of that weight, normalized to have coprime Fourier coefficients. The other generators can be described as follows: there are modular forms $g_1,\dots ,g_5$ of weight four for the discriminant kernel $\mathrm {\widetilde {O}}(L)$ that are permuted under the action of $\mathrm {O}^+(L)$ and which satisfy

$$ \begin{align*}E_2^2 = \sum_{i=1}^5 g_i.\end{align*} $$

The generators of weights $8, 12, 16, 20$ can be taken as the power sums

$$ \begin{align*}p_{4k} = \sum_{i=1}^5 g_i^k, \quad k \in \{2,3,4,5\}.\end{align*} $$

Following Proposition 2.6, the van Geemen–Sarti equation for the general S-polarized K3 surface can be put in the form

$$ \begin{align*}y^2 = x^3 + 2 C(t) x^2 + D(t) x,\end{align*} $$

where

$$ \begin{align*}C(t) = c_2 t^2 + c_6 t + c_{10}\end{align*} $$

and

$$ \begin{align*}D(t) = t^5 + d_8 t^3 + d_{12} t^2 + d_{16} t + d_{20},\end{align*} $$

and where the $c_k$ and $d_k$ are modular forms for $\mathrm {O}^+(L)$ of weight k. If we factor

$$ \begin{align*}D(t) = \prod_{i=1}^5 (t - \gamma_i),\end{align*} $$

then each $\gamma _i$ is modular of weight $4$ on the discriminant kernel. Since they sum to zero, the ring structure implies that $\gamma _i$ is a constant multiple of $g_i - \frac {1}{5} E_2^2$ : without loss of generality (applying a substitution if necessary), we may assume the roots are

$$ \begin{align*}\gamma_i = g_i - \frac{1}{5} E_2^2, \quad i=1,2,3,4,5.\end{align*} $$

6.2 A limit process

Consider the equation

$$ \begin{align*}y^2 = x^3 + (t^3 + a_4 t + a_6) x^2 + b_2 x \cdot \prod_{t=1}^5 (t - \beta_i)\end{align*} $$

with $a_4, a_6, b_2$ and $\beta _i$ defined as in §5.6. The forms $a_4, a_6,$ and $\beta _i$ are holomorphic along $r^{\perp }$ and their restrictions can be determined using the “pullback trick”, namely,

$$ \begin{align*}\mathrm{res}(\beta_i) = E_2, \quad i=1,2,3,4,5\end{align*} $$

and

$$ \begin{align*}\mathrm{res}(a_4) = -3E_2^2, \quad \mathrm{res}(a_6) = 2 E_2^3.\end{align*} $$

Define the following (holomorphic) modular forms of weight 4 and level $\mathrm {\widetilde {O}}(L_5)$ :

$$ \begin{align*}f_j := b_2 \cdot \Big(-\beta_j + \frac{1}{5} \sum_{i=1}^5 \beta_i \Big) = \sum_{i \ne j} \frac{b_2 (\beta_i - \beta_j)}{5}, \quad j=1,\dots,5.\end{align*} $$

These are Gritsenko lifts; they can be written as the additive theta lifts of the unique vector-valued modular forms $\varphi _j$ of weight $3/2$ on the Weil representation attached to $L_5$ with constant Fourier coefficient

$$ \begin{align*}\frac{576}{5} \mathfrak{e}_{\gamma_i} - \frac{144}{5} \sum_{j \ne i} \mathfrak{e}_{\gamma_j},\end{align*} $$

where $\gamma _1,\dots ,\gamma _5$ are the nonzero cosets of $(L_5' / L_5)$ of norm zero, and $\mathfrak {e}_{\gamma }$ are the associated basis elements of the group ring $\mathbb {C}[(L_5' / L_5)]$ . (For the definition of the Weil representation we refer to [Reference Borcherds3].) This allows the restrictions of $f_j$ to $r^{\perp }$ (which are nonzero) to be computed using the pullback method as well:

$$ \begin{align*}\mathrm{res}(f_j) = -3 \cdot g_j.\end{align*} $$

In particular, the forms $h_j := \frac {C}{3} (a_4 / 5 - f_j)$ restrict to the zeros $\gamma _j$ of D.

This motivates the following substitution:

$$ \begin{align*}t = \frac{1}{b_2} \cdot T + \frac{1}{5} \Big( -\frac{a_4}{b_2} + \sum_{j=1}^5 \beta_j \Big) = \frac{1}{ b_2} \cdot T - \frac{a_4 + b_4}{5 b_2},\end{align*} $$

such that

$$ \begin{align*}t - \beta_j = \frac{1}{b_2} \cdot \Big( T - ( \tfrac{a_4}{5} - f_j) \Big).\end{align*} $$

Together with the substitutions $x = X / b_2^2$ and $y = Y / b_2^3$ , this transforms the equation

$$ \begin{align*}y^2 = x^3 + A(t) x^2 + B(t) x\end{align*} $$

into

$$ \begin{align*}Y^2 = X^3 + b_2^2 A \left( \frac{1}{b_2} \cdot T - \frac{a_4 + b_4}{5 b_2} \right) X^2 + \prod_{j=1}^5 (T - h_j) \cdot X.\end{align*} $$

Here,

(6.1) $$ \begin{align} \begin{aligned} b_2^2 A \left( \frac{1}{b_2} \cdot T - \frac{a_4 + b_4}{5b_2} \right) & = \frac{1}{b_2} T^3 - \frac{3}{5} \frac{a_4 + b_4}{b_2} T^2 + \left( b_2 a_4 + \frac{3}{25} \frac{a_4^2}{b_2} + \frac{6}{25} \frac{a_4 b_4}{b_2} + \frac{3}{25} \frac{b_4^2}{b_2} \right) T \\ &- \frac{1}{5} b_2 a_4^2 - \frac{1}{5} b_2 a_4 b_4 + b_2^2 a_6 - \frac{1}{125} \frac{a_4^3}{b_2} - \frac{3}{125} \frac{a_4^2 b_4}{b_2} - \frac{3}{125} \frac{a_4 b_4^2}{b_2} - \frac{b_4^3}{125 b_2}. \end{aligned} \end{align} $$

The asymptotic behavior of the coefficients $b_2, b_4$ in the limit is governed by Theorem 3.2 and Equation (3.1): letting $r^{\perp }$ be the exceptional hyperplane we obtain, asymptotically as $\langle r, Z \rangle \rightarrow 0,$

$$ \begin{align*}b_2(Z) \sim -12 \cdot -\frac{1}{4\pi^2 \langle r, Z \rangle^2} + E_2 + O(\langle r, Z \rangle)\end{align*} $$

and

$$ \begin{align*}b_4(Z) = -b_2(Z) \sum_{i=1}^5 \beta_i(Z) \sim 12 \cdot -\frac{5 E_2}{4\pi^2 \langle r, Z \rangle^2}.\end{align*} $$

This shows that

$$ \begin{align*}3 b_2 a_4 + \frac{9}{25} \frac{b_4^2}{b_2} \sim -\frac{9}{\pi^2 \langle r, Z \rangle^2} E_2^2 + \frac{9}{\pi^2 \langle r, Z \rangle^2} E_2^2,\end{align*} $$

and similarly that

$$ \begin{align*}-\frac{1}{5}b_2 a_4^2 - \frac{1}{5} b_2 a_4 b_4 + b_2^2 a_6 - \frac{3}{125} \frac{a_4 b_4^2}{b_2} - \frac{b_4^3}{125 b_2}\end{align*} $$

remains holomorphic in the limit $1/b_2 \rightarrow 0$ .

6.3 Determining the coefficients

The coefficients $a_k, b_k$ of the van Geemen–Sarti equation

$$ \begin{align*}y^2 = (t^3 + a_4 t + a_6) x^2 + (b_2 t^5 + \dots + b_{12}) x\end{align*} $$

have “level two” Fourier–Jacobi expansions in addition to level one Fourier–Jacobi expansions, due to the isometry

$$ \begin{align*}L_5 = H \oplus H \oplus A_1(-1)^{\oplus 5} = H \oplus H(2) \oplus D_4(-1) \oplus A_1(-1).\end{align*} $$

More precisely, we can represent modular forms on $\mathrm {O}^+(L_5)$ as series

$$ \begin{align*}F(Z) = \sum_{n=0}^{\infty} \phi_n(\tau, z) e^{2\pi i n w},\end{align*} $$

where each $\phi _n$ is a meromorphic Jacobi form of lattice index $D_4 \oplus A_1$ with respect to the congruence subgroup $\Gamma _0(2) \le \mathrm {SL}_2(\mathbb {Z})$ . The two Fourier–Jacobi expansions can be quite different, and it is not generally clear how to compute the level two expansions of a modular form that is given as a level one Fourier–Jacobi series, but the expansions of a Gritsenko lift or Borcherds product at either cusp can be computed using the formulas of [Reference Borcherds3].

In the level two expansions,

$$ \begin{align*}\chi_k = \sum_{n=0}^{\infty} \chi_{k, n}(\tau, \mathfrak{z}, z) e^{2\pi i n w}, \quad \mathfrak{z} \in D_4 \otimes \mathbb{C}, \; z \in A_1 \otimes \mathbb{C},\end{align*} $$

the terms $\chi _{k, 0}$ are independent of $\mathfrak {z}$ and define elliptic functions (in the variable z) on the lattice $\mathbb {Z} \oplus 2\tau \mathbb {Z}$ which are holomorphic away from the lattice points. For example, we have

$$ \begin{align*} \chi_{2, 0} &= \frac{1}{(2\pi i)^2} \wp(2\tau, z) + \frac{1}{12} E_2(\tau) - \frac{1}{6} E_2(2\tau) \\ &= \frac{1}{\zeta^{-1} - 2 + \zeta} - 2q + (\zeta^{-1} - 4 + \zeta) q^2 - 8q^3 + (2 \zeta^{-2} + \zeta^{-1} - 8 + \zeta + 2 \zeta^2) q^4 + O(q^5),\end{align*} $$

and

$$ \begin{align*} \chi_{4, 0} &= \frac{5}{(2\pi i)^4} \wp"(2\tau, z) + 2\frac{ \eta(2\tau)^{16}}{\eta(\tau)^8} \\ &= \frac{5 \zeta^{-1} + 20 + 5 \zeta}{(\zeta^{-1} - 2 + \zeta)^2} + 2q + (5 \zeta^{-1} + 16 + 5\zeta) q^2 + 56q^3 \\ & \qquad + (40 \zeta^{-2} + 5 \zeta^{-1} + 128 + 5 \zeta + 40 \zeta^2) q^4 + O(q^5), \end{align*} $$

and the higher terms $\chi _{k, n}, n> 0$ are weak Jacobi forms of lattice index.

Setting $z=0$ , or equivalently $\zeta = 1$ in the Fourier expansions of $\chi _k$ and using the formulas from before yields

$$ \begin{align*}\mathrm{res} \Big(\!\!-\frac{3}{5} \frac{a_4 + b_4}{b_2} \Big) = 3 E_2;\end{align*} $$

$$ \begin{align*}\mathrm{res} \Big( b_2 a_4 + \frac{3}{25} \frac{a_4^2}{b_2} + \frac{6}{25} \frac{a_4 b_4}{b_2} + \frac{3}{25} \frac{b_4^2}{b_2} \Big) = \frac{18}{5} E_2^3 - \frac{864}{5} F_6;\end{align*} $$

and

$$ \begin{align*} &\quad \mathrm{res} \Big(\!\!-\frac{1}{5} b_2 a_4^2 - \frac{1}{5} b_2 a_4 b_4 + b_2^2 a_6 - \frac{1}{125} \frac{a_4^3}{b_2} - \frac{3}{125} \frac{a_4^2 b_4}{b_2} - \frac{3}{125} \frac{a_4 b_4^2}{b_2} - \frac{b_4^3}{125 b_2}\Big) \\ &= \frac{81}{100} E_2^5 - \frac{2592}{25} E_2^2 F_6 + \frac{27}{20} E_2 p_8 + \frac{31104}{5} F_{10}. \end{align*} $$

So in the limit $b_2 \rightarrow \infty $ , Equation (6.1) becomes

$$ \begin{align*}Y^2 = X^3 + 2 C(T) X^2 + D(T) X\end{align*} $$

with

$$ \begin{align*}C(T) = \frac{3}{2} E_2 \cdot T^2 + \Big( \frac{9}{5} E_2^3 - \frac{432}{5} F_6 \Big) T + \frac{81}{200} E_2^5 - \frac{1296}{25} E_2^2 F_6 + \frac{27}{40} E_2 p_8 + \frac{15552}{5} F_{10}.\end{align*} $$

The coefficients of

$$ \begin{align*}D(T) = \prod_{i=1}^5 (T - \gamma_i) = T^5 + d_8 T^3 + d_{12} T^2 + d_{16} T + d_{20}\end{align*} $$

are easily determined in terms of $E_2$ and $p_{4k} = \sum _i g_i^k$ by expanding the product and substituting $\gamma _i = g_i - \frac {1}{5} E_2^2 = g_i - \frac {1}{5} \sum _{j=1}^5 g_j$ . We obtain

$$ \begin{align*}d_8 = -\frac{1}{2} \sum_{i=1}^5 \gamma_i^2 = -\frac{1}{2} p_8 + \frac{1}{10} E_2^4; \qquad \qquad\qquad\qquad\end{align*} $$

$$ \begin{align*}d_{12} = -\frac{1}{3} \sum_{i=1}^5 \gamma_i^3 = -\frac{1}{3} p_{12} + \frac{1}{5} E_2^2 p_8 - \frac{2}{75} E_2^6;\qquad \qquad\qquad\end{align*} $$

$$ \begin{align*}d_{16} = -\frac{1}{4} \sum_{i=1}^5 \gamma_i^4 + \frac{1}{8} \Big( \sum_{i=1}^5 \gamma_i^2 \Big)^2 = -\frac{1}{4} p_{16} + \frac{1}{5} E_2^2 p_{12} - \frac{11}{100} E_2^4 p_8 + \frac{11}{1000} E_2^8 + \frac{1}{8} p_8^2;\end{align*} $$

$$ \begin{align*} d_{20} &= -\frac{1}{5} \sum_{i=1}^5 \gamma_i^5 + \frac{1}{6} \Big( \sum_{i=1}^5 \gamma_i^3 \Big) \Big( \sum_{i=1}^5 \gamma_i^2 \Big) \\ &= -\frac{1}{5} p_{20} + \frac{1}{5} E_2^2 p_{16} - \frac{17}{150} E_2^4 p_{12} + \frac{37}{750} E_2^6 p_8 - \frac{37}{9375} E_2^{10} + \frac{1}{6} p_{12} p_8 - \frac{1}{10} E_2^2 p_8^2 \end{align*} $$