Starting with a primitive Dirichlet character of conductor N, we construct a paramodular Siegel Eisenstein series of level N^2 and weight k≥4. We calculate the Fourier expansion of the holomorphic Siegel modular form thus constructed. The function is a paramodular newform, and its adelization generates an irreducible automorphic representation.

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to L-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for L-functions. Significantly stronger results are available when the L-function is known to be automorphic.

We compute the local integrals appearing in the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods of Sp_4 in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of our identity for the growth of Petersson norms, the size of Fourier coefficients, and non-vanishing of central L-values.

We carry out "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight $2$ with level, and Eisenstein series with character.

We consider simple supercuspidal representations of $\mathrm{GSp}_4$ over a $p$-adic field and show that they have conductor exponent 5. We study (paramodular) newvectors and minimal vectors in these representations, obtain formulas for their matrix coefficients, and compute key local integrals involving these as test vectors. Our local computations lead to several explicit global period formulas involving automorphic representations $\pi$ of $\mathrm{GSp}_4(\mathbb{A})$ whose local components (at ramified primes) are simple supercuspidal representations, and where the global test vectors are chosen to be (diagonal shifts of) newforms or automorphic forms of minimal type. As an analytic application of our work to the sup-norm problem, we show the existence of paramodular newforms on $\mathrm{GSp}_4(\mathbb{A})$ of conductor $p^5$ that take ``large values" on a fixed compact set as $p\rightarrow \infty$.

For a p-adic field F, we determine the Bessel functionals of the irreducible, admissible, supercuspidal representations of GSp(4,F) of depth zero.

Let F be a non-archimedean local field of characteristic zero. We calculate the Fourier-Jacobi modules of the irreducible, admissible, non-generic, supercuspidal representations of GSp(4,F) of depth zero.

Let F be a nonarchimedean local field of characteristic zero. In this chapter we prove a number of important results concerning stable Klingen vectors in representations of $$\mathrm {GSp}(4,F)$$ induced from the Siegel parabolic subgroup P of $$\mathrm {GSp}(4,F)$$ . These results are key ingredients for the dimension formulas proved in the next chapter.

In this chapter we translate the local upper block level changing operators from the first part of this work to operators on Siegel modular forms defined with respect to the stable Klingen congruence subgroups. We give a slash formula for each such operator; since this formula involves only upper block matrices, we are able to calculate the Fourier and Fourier-Jacobi expansions of the resulting Siegel modular forms.

We introduce the family of stable Klingen congruence subgroups of GSp(4). We use these subgroups to study both local paramodular vectors and Siegel modular forms of degree 2 with paramodular level. In the first part, when F is a nonarchimedean local field of characteristic zero and (π,V) is an irreducible, admissible representation of GSp(4,F) with trivial central character, we establish a basic connection between the subspaces V_s(n) of V fixed by the stable Klingen congruence subgroups and the spaces of paramodular vectors in V and derive a fundamental partition of the set of paramodular representations into two classes. We determine the spaces V_s(n) for all (π,V) and n. We relate the stable Klingen vectors in V to the two paramodular Hecke eigenvalues of π by introducing two stable Klingen Hecke operators and one level lowering operator. In contrast to the paramodular case, these three new operators are given by simple upper block formulas. We prove further results about stable Klingen vectors in V especially when π is generic. In the second part we apply these local results to a Siegel modular newform F of degree 2 with paramodular level N that is an eigenform of the two paramodular Hecke operators at all primes p. We present new formulas relating the Hecke eigenvalues of F at p to the Fourier coefficients a(S) of F for p^2 | N. We verify that these formulas hold for a large family of examples and indicate how to use our formulas to generally compute Hecke eigenvalues at p from Fourier coefficients of F for p^2 | N. Finally, for p^2 | N we express the formal power series in p^-s with coefficients given by the radial Fourier coefficients a(p^t S), t≥ 0, as an explicit rational function in p^-s with denominator L_p(s,F)^-1, where L_p(s,F) is the spin L-factor of F at p.

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 $$S_k(\mathrm {K}(N))_{\mathrm {new}}$$ of degree two with paramodular level N. Assuming that F is an eigenform for the Hecke operators $$T(1,1,p,p)$$ and $$T(p,1,p,p^2)$$ 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 $$p^2 \mid N$$ . 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 $$\varGamma _0(N)$$ on the vector space of complex valued functions on the set of positive semi-definite $$2 \times 2$$ 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 $$a(p^t S)$$ for $$t \geq 0$$ and $$p^2 \mid N$$ satisfy a recurrence relation determined by the spin L-factor of F at p. This extends results known in other cases.

Let F be a nonarchimedean local field of characteristic zero, and let $$(\pi ,V)$$ be an irreducible admissible representation of $$\mathrm {GSp}(4,F)$$ with trivial central character. We will say that $$\pi $$ is Iwahori-spherical if the space $$V^I$$ of vectors in V fixed by I is non-zero, where I is the Iwahori subgroup of $$\mathrm {GSp}(4,F)$$ . The goal of this chapter is to describe the actions of the stable Hecke operators on $$V_s(1)$$ when $$\pi $$ is Iwahori-spherical. Since $$\mathrm {K}_s(\mathfrak {p})=\mathrm {Kl}(\mathfrak {p})$$ , $$V_s(1)$$ is simply the space $$V^{\mathrm {Kl}(\mathfrak {p})}$$ of vectors fixed by the Klingen congruence subgroup $$\mathrm {Kl}(\mathfrak {p})$$ of level $$\mathfrak {p}$$ .

Let F be a nonarchimedean local field of characteristic zero, let $$(\pi ,V)$$ be an irreducible, admissible representation of $$\mathrm {GSp}(4,F)$$ with trivial central character, and let n be a non-negative integer. Assume that $$\pi $$ is paramodular. In this chapter we investigate the relationship between $$V_s(n)$$ and its subspace $$V(n-1)+V(n)$$ of paramodular vectors. Except for a few non-generic $$\pi $$ , the investigation of the relation between $$V_s(n)$$ and its subspace $$V(n-1)+V(n)$$ is reduced to considering $$\pi $$ that are generic. In a previous chapter we proved that if $$\pi $$ is generic and a category 1 representation, then $$V_s(n)=V(n-1) \oplus V(n) \oplus \bigoplus _{\substack {i,j \geq 0\\ i+j=n-N_\pi +1}} \mathbb {C} \tau ^i \theta ^j W_s$$ where $$W_s$$ is the shadow of the newform $$W_{\mathrm {new}}$$ in $$V(N_\pi )$$ . Most of this chapter is devoted to generalizing this result to all generic $$\pi $$ , though we also give a complete account for all non-generic $$\pi $$ .

We prove several dimension formulas for spaces of scalar‐valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of GSp(4,A)${\rm GSp}(4,{\mathbb {A}})$ whose local component at p=2$p=2$ admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at p=2$p=2$ , we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

Let F be a nonarchimedean local field of characteristic zero, and let $$(\pi ,V)$$ be a generic, irreducible, admissible representation of $$\mathrm {GSp}(4,F)$$ with trivial central character. In this chapter we prove several additional results about stable Klingen vectors in $$\pi $$ . Our first result generalizes a fundamental theorem from the paramodular theory. Assume that V is the Whittaker model of $$\mathcal {W}(\pi ,\psi _{c_1,c_2})$$ of $$\pi $$ , and let n be a non-negative integer. We prove that if W is in $$V_s (n)$$ , then W is non-zero if and only if W does not vanish on the diagonal subgroup of $$\mathrm {GSp}(4,F)$$ . To describe our second result, assume that $$n \geq \max (N_{\pi ,s},2)$$ , and recall the surjective level lowering operator $$\sigma _{n-1}:V_s(n) \to V_s(n-1)$$ . We prove that $$V_s(n) = (V(n-1) \oplus V(n)) + \mathrm {ker}\, (\sigma _{n-1})$$ for integers $$n \geq \max (N_{\pi ,s},2)$$ . Consequently, $$\sigma _{n-1}$$ induces an isomorphism $$(V(n-1) \oplus V(n))/\mathcal {K}_{n} \stackrel {\sim }{\longrightarrow } V_s(n-1)$$ for integers $$n \geq \max (N_{\pi ,s},2)$$ , where $$\mathcal {K}_n$$ is the intersection of $$V(n-1) \oplus V(n)$$ with $$\ker (\sigma _{n-1})$$ . We characterize $$\mathcal {K}_n$$ for $$n \geq \mathrm {max}(N_{\pi ,s},2)$$ ; in particular, this subspace is at most two-dimensional. To prove these results we use the just mentioned nonvanishing statement. Finally, we consider $$\pi $$ such that $${L(s,\pi )=1}$$ ; this includes all $$\pi $$ that are supercuspidal. Assuming that $$L(s,\pi )=1$$ , we are able to define another, graphical, model for $$V_s(n)$$ for integers $$n \geq N_{\pi ,s}$$ . The existence of this alternative model also uses the result about the non-vanishing of stable Klingen vectors on the diagonal of $$\mathrm {GSp}(4,F)$$ . In this model, our level changing operators have simple and visual interpretations. Finally, still under the hypothesis that $$L(s,\pi )=1$$ , we prove that $$N_\pi \geq 4$$ .

Let F be a nonarchimedean local field of characteristic zero, and let $$\pi $$ be a paramodular, irreducible, admissible representation of $$\mathrm {GSp}(4,F)$$ with trivial central character such that $$N_\pi \geq 2$$ . In this chapter we consider the action of the stable Klingen Hecke operators on the vector spaces $$V_s(N_{\pi ,s})$$ and $$V_s(N_\pi )$$ . We will show that the actions of these operators can be described in terms of the paramodular Hecke eigenvalues of $$\pi $$ . We will also explain how these results can be used to compute these paramodular Hecke eigenvalues and determine whether $$\pi $$ is non-generic. As outlined in the introduction of this work, this has useful applications to Siegel paramodular newforms.

The remainder of this text explores applications of the local theory developed in the first part of this work to Siegel modular forms of degree two. In this chapter, we recall some essential definitions. In the next chapter we translate the operators on stable Klingen vectors defined in previous chapters to modular forms.

We prove the expected algebraicity property for the critical values of character twists of the standard L L -function associated to vector-valued holomorphic Siegel cusp forms of archimedean type ( k 1 (k_1 , k 2 k_2 , …, k n ) k_n) , where k n ≥ n + 1 k_n \ge n+1 and all k i k_i are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.