On some free algebras of orthogonal modular forms. II (Q2045736)

The author identifies the structure of algebras of orthogonal modular forms for lattices~\(2 U \oplus L(-1)\), where~\(U\) is the hyperbolic plane with Gram matrix~\(\left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)\) and~\(L\) runs through a set of~\(8\) rescaled root lattices: \begin{gather*} A_1(2) \text{,}\; A_1(3) \text{,}\; A_1 (4) \text{,}\; 2 A_1 (2) \text{,}\; A_2(2) \text{,}\; A_2 (3) \text{,}\; A_3 (2) \text{,}\; D_4 (2) \text{.} \end{gather*} He shows that they are free polynomials algebras and gives concrete generators in terms of additive and multiplicative Borcherds lifts. The structure of algebras of modular forms is related to the geometry of modular varieties via the Bailey-Borel compactification. As such the case of free algebras corresponds to modular varieties whose Bailey-Borel compactifications are weighted projective spaces. In previous work [Part I, Compos. Math. 157, No. 9, 2026--2045 (2021; Zbl 1482.11074)], the author gave a further characterization: Algebras of orthogonal modular forms for signature~\((2,n)\) are free if and only if there is a modular form that is the Jacobian of~\(n+1\) modular forms and whose divisor is the sum of the reflexive divisors associated with the corresponding modular group, which is necessarily a reflection group. In this case, these~\(n+1\) modular forms are generators of the algebra. The author proceeds by constructing such~\(n+1\) modular forms in each of his eight cases.