Harder's conjecture II

Abstract

Let $f$ be a primitive form of weight $2k+j-2$ 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 $k$ is congruent to $0$ modulo $4$, $j$ is congruent to $0$ modulo $4$, and that $P$ divides the algebraic part of $L(k+j,f)$. Put $\mathbf{k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. 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 $[I_2(f){]}^{\mathbf{k}}$ is congruent to $A_4^{(I)}(F)$ modulo $P$. Here, $[I_2(f){]}^{\mathbf{k}}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $I_2(f)$ of $f$ to the space of modular forms of weight $\mathbf{k}$ for $S{p}_4(Z)$, and $A_4^{(I)}(F)$ is a certain lift of $F$ to the space of cusp forms of weight $\mathbf{k}$ 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 $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.

arXiv admin note: text overlap with arXiv:2109.10551