Abstract
In this final chapter we present some applications of the local theory of the first part of this work to the Hecke eigenvalues and Fourier coefficients of Siegel modular newforms F in Sk(K(N))new\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_k(\mathrm {K}(N))_{\mathrm {new}}$$\end{document} of degree two with paramodular level N. Assuming that F is an eigenform for the Hecke operators T(1,1,p,p)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T(1,1,p,p)$$\end{document} and T(p,1,p,p2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T(p,1,p,p^2)$$\end{document} for all primes p, we begin by proving that the local results from the first part of this text imply identities involving F and its images under the upper block operators from the previous chapter at p for p2∣N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2 \mid N$$\end{document}. We prove that these identities yield relations between Fourier coefficients and paramodular Hecke eigenvalues as well as conditions which determine properties of the attached local representations at p. We then show that these formulas can be rewritten in terms of the action of the Hecke ring of Γ0(N)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varGamma _0(N)$$\end{document} on the vector space of complex valued functions on the set of positive semi-definite 2×2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2 \times 2$$\end{document} matrices with rational entries. We conclude this chapter with two applications. First, we show that our formulas do indeed hold for known examples, and we indicate how our equations could be used to calculate paramodular Hecke eigenvalues from Fourier coefficients in other instances. Finally, we prove that the radial Fourier coefficients a(ptS)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a(p^t S)$$\end{document} for t≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t \geq 0$$\end{document} and p2∣N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2 \mid N$$\end{document} satisfy a recurrence relation determined by the spin L-factor of F at p. This extends results known in other cases.