On some free algebras of orthogonal modular forms

Abstract

For 25 orthogonal groups of signature (2, n) related to the root lattices A1,

The classification of free algebras of orthogonal modular forms

“…It was proved in [HU14] that the algebra of modular forms on O + (2U ⊕ E 8 (−1)) is freely generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, and in [DKW19] that the generators can be chosen as additive lifts of Jacobi Eisenstein series. The other 25 cases were proved in a universal and elementary way in [WW20]. A general rule characterizing the weights of generators was also given.…”

“…After Igusa, more free algebras of O(2, n)modular forms were constructed in [AI05, DK03, DK06, Kri05, FH00, FS07, HU14, Vin10, Vin18]. Recently, the author proved joint with B. Williams that the spaces of orthogonal modular forms are free algebras for 25 groups in a universal way in [WW20].…”

On some free algebras of orthogonal modular forms II

“…The bigraded ring of weak Jacobi forms invariant under the orthogonal group O(2A 1 ) is freely generated over M * (SL 2 (Z)) by three forms of index one φ 0,2A 1 , φ −2,2A 1 , and φ −4,2A 1 . In fact, this type of Jacobi forms is the so-called Weyl invariant Jacobi forms associated to the root system B 2 (see [Wir92] and [WW20,§2]). We fix the model of 2A 1 :…”

“…The first example of such free algebras was determined by Igusa [Igu62], which is related to the orthogonal group of signature (2,3). By means of the theory of Weyl invariant Jacobi forms [Wir92], the author proved the freeness of 25 graded algebras of orthogonal modular forms in a universal method joint with B. Williams [WW20]. In [Wan20] we established a necessary and sufficient condition for M * (Γ) to be free, which is based on the existence of a particular modular form which vanishes exactly on the mirrors of reflections in Γ and equals the Jacobian determinant of n + 1 generators.…”

“…The bigraded ring of weak Jacobi forms invariant under the Weyl group W (D 4 ) is freely generated over M * (SL 2 (Z)) by five forms φ 0,D 4 φ −2,D 4 , φ −4,D 4 , ψ 0,D 4 , and φ −6,D 4 ,2 (see [Wir92] and [WW20,§2]). The five generators were constructed in [AG19] and we can choose the generators of index one with the following Fourier expansions:…”