Elliptic modular forms are analytic functions on the upper half plane that are invariant under the action of the congruence subgroup \(\Gamma_0(N)\) of \(\mathrm{SL}(2,\mathbb Z)\). The present book deals with automorphic forms on the general symplectic group \(\mathrm{GSp}(4)\), also known as Siegel modular forms of degree \(2\). Of particular interest are the paramodular forms since they appear in modularity theorems for abelian surfaces.
The Introduction presents the classical theory of automorphic forms on \(\mathrm{GL}(1)\) (Dirichlet characters, ideles, \(L\)-functions and strong multiplicity one) and on \(\mathrm{GL}(2)\) (modular forms, Hecke operators, \(L\)-functions, local factors etc.), and also explains what is and what is not known in the case of Siegel modular forms of degree \(2\).
Part I begins with the second chapter, which presents the “local theory”: the representation theory of the group \(\mathrm{GSp}(4)\) over a nonarchimedean local field of characteristic \(0\), and a brief survey of paramodular vectors as introduced by B. Roberts and R. Schmidt [Local newforms for \(\mathrm{GSp}(4)\). Berlin: Springer (2007; Zbl 1126.11027)].
In the next few chapters, Klingen subgroups of \(\mathrm{GSp}(4,F)\) and Klingen vectors are introduced, together with level raising and level lowering operators and Hecke operators between subspaces defined by smooth representations.
Part II is dedicated to Siegel modular forms. After providing some background, the authors define level raising and level lowering operators as well as Hecke operators, and explain how to compute Fourier and Fourier-Jacobi expansions of the resulting Siegel modular forms. The last chapter provides examples, and this is followed by an appendix containing tables, a detailed bibliography and an index.