Twisting of Siegel paramodular forms

Abstract

Let $S_k(\Gamma^{\mathrm{para}}(N))$ be the space of Siegel paramodular forms of level $N$ and weight $k$. Let $p\nmid N$ and let $\chi$ be a nontrivial quadratic Dirichlet character mod $p$. Based on our previous work, we define a linear twisting map $\mathcal{T}_\chi:S_k(\Gamma^{\mathrm{para}}(N))\rightarrow S_k(\Gamma^{\mathrm{para}}(Np^4))$. We calculate an explicit expression for this twist and give the commutation relations of this map with the Hecke operators and Atkin-Lehner involution for primes $\ell\neq p$.Comment: 64 pages. In version 2, the paper has been shortened significantly and lengthy, technical proofs given in a separate appendix. In version 3, two typos are corrected. In version 4, we have included the full details of the proof of the local twisting theorem in the paper and improved the results with the L-function theore