{"paper":{"title":"A note on the growth of nearly holomorphic vector-valued Siegel modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abhishek Saha, Ameya Pitale, Ralf Schmidt","submitted_at":"2016-12-14T19:27:31Z","abstract_excerpt":"Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\\rho$ with respect to some congruence subgroup of $\\mathrm{Sp}_{2n}(\\mathbb Q)$. In this note, we prove that the function on $\\mathrm{Sp}_{2n}(\\mathbb R)$ obtained by lifting $F$ has the moderate growth (or \"slowly increasing\") property. This is a consequence of the following bound that we prove: $\\|\\rho(Y^{1/2})F(Z) \\| \\ll \\prod_{i=1}^n (\\mu_i(Y)^{\\lambda_1/2} + \\mu_i(Y)^{-\\lambda_1/2})$ where $ \\lambda_1 \\ge \\ldots \\ge \\lambda_n$ is the highest weight of $\\rho$ and $\\mu_i(Y)$ are the eigenvalues of the matrix $Y$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04778","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}