Computations of Spaces of Paramodular Forms of General Level

Abstract

Abstract. This article gives upper bounds on the number of FourierJacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.

Antisymmetric paramodular forms of weight 3

“…, 12, 14, 15, 16, 18, 20, 24, 30, 36. In particular, H 3 (Γ t , C) is not trivial for all non exceptional polarisations. We note that dim S 3 (Γ t ) = 0 for these twenty t (see [5]). Due to the existence of canonical differential forms, the moduli space of (1, t)-polarised abelian surfaces might have trivial geometric genus only for the twenty exceptional polarisations.…”

“…In particular, it is positive for t = 167, 173, 223, 227, 251, 257, 269, 271, 283, 293. Moreover, we have It is known that dim S 3 (Γ * t ) = 0 for t ≤ 40 (see [5]). According to the calculation made by Jerry Shurman, the minimal level t with dim S 3 (Γ * t ) = 0 is 152 or 167.…”

“…(5) ) is equal to the Kac-Weyl denominator function of the affine Lie algebraĝ(2A 4 ). Similar to §5, we obtain the following theorem.…”

Graded rings of paramodular forms of levels 5 and 7

“…(The Koecher principle states that F extends holomorphically to the boundary of K(N )\H 2 so we omit this from the definition.) The invariance of F under the translations 3 ∈ Z implies that F is given by a Fourier series, which we write in the form The paramodular group is normalized by an additional map called the Fricke involution:…”

“…(See also the discussion in [15].) The space of real antisymmetric (4 × 4)-matrices admits a nondegenerate quadratic form (the Pfaffian) of signature (3,3) with pf(J) < 0 then the symplectic group Sp 4 (R) consists exactly of those matrices which preserve the orthogonal complement J ⊥ under conjugation, and the Klein correspondence identifies Sp 4 (R) with Spin(pf| J ⊥ ) = Spin 3,2 (R). Under this identification K(N ) + embeds into the spin group of the even lattice (Λ, N · pf) where Λ = {M ∈ J ⊥ : σ N M σ N ∈ Z 4×4 }, σ N = diag(1, 1, 1, N ), which is isomorphic to A 1 (N )⊕II 2,2 .…”