Harder's conjecture II

Let f be a primitive form of weight 2 ⁢ 𝑘 + 𝑗 − 2 for S L 2 ⁢ ( ℤ ) , and let 𝔭 be a prime ideal of the Hecke field of f . We denote by S p 𝑚 ⁢ ( ℤ ) the Siegel modular group of degree m . Suppose that 𝑘 ≡ 0 m o d ⁡ 2 , 𝑗 ≡ 0 m o d ⁡ 4 and that 𝔭 divides the algebraic part of 𝐿 ⁡ ( 𝑘 + 𝑗 , 𝑓 ) . Put 𝐤 = ( 𝑘 + 𝑗 / 2 , 𝑘 + 𝑗 / 2 , 𝑗 / 2 + 4 , 𝑗 / 2 + 4 ) . Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform F in the space of modular forms of weight ( 𝑘 + 𝑗 , 𝑘 ) for S p 2 ⁢ ( ℤ ) such that [ ℐ 2 ⁡ ( 𝑓 ) ] 𝐤 is congruent to 𝒜 ( 𝐼 ) 4 ⁡ ( 𝐹 ) modulo 𝔭 . Here, [ ℐ 2 ⁡ ( 𝑓 ) ] 𝐤 is the Klingen-Eisenstein lift of the Saito-Kurokawa lift ℐ 2 ⁡ ( 𝑓 ) of f to the space of modular forms of weight k for S p 4 ⁢ ( ℤ ) , and 𝒜 ( 𝐼 ) 4 ⁡ ( 𝐹 ) is a certain lift of F to the space of cusp forms of weight k for S p 4 ⁢ ( ℤ ) . As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of F and some quantities related to the Hecke eigenvalues of f .