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

“…Since is contained in , the non-zero modular forms for have even weight. It is easy to prove that using the argument in [WW20, § 3]. Indeed, we take in [WW20, Inequality (3.3) in § 3].…”

“…It is easy to prove that using the argument in [WW20, § 3]. Indeed, we take in [WW20, Inequality (3.3) in § 3]. Notice that and .…”

“…Since is a sublattice of , is a finite index subgroup of . We know from [WW20] that is a free algebra. Thus the above conjecture implies the freeness of the algebra .…”

“…After Igusa, more free algebras of -modular forms were constructed in [AI05, DK03, DK04, Kri05, FH00, FS07, HU14, Vin10, Vin18]. Recently, the author proved jointly with B. Williams that the spaces of orthogonal modular forms are free algebras for 25 groups in a universal way in [WW20].…”

“…It was proved in [HU14] that the algebra of modular forms on 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.…”