Harder's conjecture II

Mathematics > Number Theory [Submitted on 13 Jun 2023 (v1), last revised 16 Jun 2026 (this version, v3)] Title:Harder's conjecture II View PDF HTML (experimental)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) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.