Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

Abstract

We prove the explicit structure theorems of modules k M det k ⊗Sym(10) (Sp 2 (Z 2 )) of vector valued Siegel modular forms of degree 2, where k runs over the set of even integers or odd integers. We also check the conjecture given by Ibukiyama [7] for modules of vector valued Siegel modular forms of degree 2 of weights det * ⊗Sym(8) and det * ⊗Sym(10).

Covariants of binary sextics and modular forms of degree 2 with character

“…We are interested in the structure of the R-modules M ev j (Γ 2 , ǫ) = ⊕ k even M j,k (Γ 2 , ǫ) and M odd j (Γ 2 , ǫ) = ⊕ k odd M j,k (Γ 2 , ǫ) . The structure of the analogous modules for modular forms without character M ev j (Γ 2 ) = ⊕ k even M j,k (Γ 2 ) and M odd j (Γ 2 ) = ⊕ k odd M j,k (Γ 2 ) is known for some values of j by work of Satoh, Ibukiyama, van Dorp, Kiyuna, and Takemori, see [17,12,8,15,19]. The next table summarizes the results.…”

“…) is known for some values of j by work of Satoh, Ibukiyama, van Dorp, Kiyuna, and Takemori, see [17,12,8,15,19]. The next table summarizes the results.…”

“…We get a form F 4,18 of weight (4, 18) by putting F 4,18 = (16875/8)ν(C 5,4 )χ 3 5 ; it is holomorphic and its Fourier expansion starts with Finally, the covariant ξ 5 = C 2 2,0 C 5,4 − 10 C 2,0 C 7,4 + 1000 C 9,4 yields the form F 4,22 = (3189375/32)ν(ξ 5 )χ 3 5 with Fourier expansion Theorem 11. The R-module M odd 6 (Γ 2 , ǫ) is free with generators of weight (6, 3), (6, 5), (6,11), (6,13), (6,17), (6,19) and (6,21).…”

“…Theorem 11. The R-module M odd 6 (Γ 2 , ǫ) is free with generators of weight (6, 3), (6, 5), (6,11), (6,13), (6,17), (6,19) and (6, 21).…”

“…The next table summarizes the results. Satoh [17] Ibukiyama [12] Ibukiyama [12] Kiyuna [15] Takemori [19] odd Ibukiyama [12] Ibukiyama [12] van Dorp [8] Kiyuna [15] Takemori [19] The difficult part is the construction of the generators and the authors just mentioned used an array of methods to construct generators. For example, Satoh used generalized Rankin-Cohen brackets, Ibukiyama used theta series for even unimodular lattices, van Dorp used differential operators, and so on.…”