Sturm bounds for Siegel modular forms of degree 2 and odd weights

Abstract

We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree 2 and of even weights modulo a prime number dividing 2 · 3. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.2. As mentioned in Introduction, in the case where p | 2 · 3 and k is even, the first named author stated the same property in [7]. However, the proof has some gaps and its method can give only more larger bounds. We give a new proof in subsection 5.1.3. We note that M k (Γ 2 ) = {0} if k is odd and k < 35.4. Other type bounds also were given in [8].By the result of [1] and a similar argument to them, we can prove the following.Corollary 2.3. Let Γ ⊂ Γ 2 be a congruence subgroup with level N, k ∈ Z ≥0 and f ∈ M k (Γ) Op . We put i = [Γ 2 : Γ]. For ν ∈ Z ≥1 , assume that a f (m, r, n) ≡ 0 mod p ν for all m, r, n ∈ 1 N Z with 0 ≤ m, n ≤ b ki . and 4mn − r 2 ≥ 0, then we have f ≡ 0 mod p ν .

Definite orthogonal modular forms: computations, excursions, and discoveries

“…It is not easy to be certain that an eigenvector is in the kernel of , since there is no effective Sturm bound to tell us when we can conclude that a linear combination is 0 from the first terms being all 0 (see e.g. Kikuta–Takemori [ 28 , Corollary 2.3], where in this case ). However, it seems reasonable to expect that if the kernel on the coefficients of the lattices of rank 2 of smallest discriminant is stable and nontrivial over a substantial range, then this is genuinely the kernel of .…”

A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

“…This is due to the explicit form of the Fourier expansion of H 8 (the same reason as in [7] Lemma 5.1);…”

“…Theorem 2.5 (Choi-Choie-Kikuta [1], Kikuta-Takemori [7]). Let k be a positive integer and p an any prime.…”

Theta operator on Hermitian modular forms over the Eisenstein field

“…By assumption and Sturm bound for the Siegel modular case (cf. [9]), we have F | S2 ≡ 0 (mod p). By Lemma 4.3, there exists The second main result can be stated as follows:…”