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Abstract:Let $f$ be a primitive form of weight $2k+j-2$ for $\SL_2(\ZZ)$, and let $\frkp$ be a prime ideal of the Hecke field of $f$. We denote by $\SP_m(\ZZ)$ the Siegel modular group of degree $m$. Suppose that $k \equiv 0 \mod 2, \ j \equiv 0 \mod 4$ and that $\frkp$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. 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 $\SP_2(\ZZ)$ such that $[\scri_2(f)]^{\bf k}$ is congruent to $\scra^{(I)}_4(F)$ modulo $\frkp$. Here, $[\scri_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $\scri_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $\SP_4(\ZZ)$, and $\scra^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $\SP_4(\ZZ)$. 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 of the paper arXiv:2306.07582v2.

Submission history

From: Hidenori Katsurada [view email]
[v1] Tue, 13 Jun 2023 07:11:38 UTC (47 KB)
[v2] Tue, 8 Aug 2023 06:12:02 UTC (49 KB)
[v3] Tue, 16 Jun 2026 00:19:27 UTC (51 KB)