--- title: Paramodular Siegel Eisenstein Series url: https://www.emergentmind.com/topics/paramodular-siegel-eisenstein-series type: topic --- # Paramodular Siegel Eisenstein Series Paramodular Siegel Eisenstein series are Siegel modular forms attached to paramodular groups, most explicitly in genus \(2\) as holomorphic Eisenstein series of weight \(k\ge 4\) and paramodular level \(N^2\) built from a primitive Dirichlet character \(\eta\) of conductor \(N\). In the genus-\(2\) construction developed in 2025, the series \(E_{k,\eta}(Z)\) is defined both classically and adelically for \(\mathrm{GSp}_4\), its Fourier expansion is computed, and it is shown to be a paramodular newform whose adelization generates an irreducible automorphic representation; a subsequent arithmetic study proves that all of its Fourier coefficients lie in an explicit number field [2509.04395] [2510.22762]. ## 1. Definition in genus \(2\) and the paramodular group For degree \(2\), the ambient Hermitian symmetric domain is the Siegel upper half space \[ \mathbb{H}_2=\{Z=X+iY\in M_2(\mathbb{C})\mid Z^t=Z,\; Y>0\}, \] with \[ Z=\begin{bmatrix}\tau & z\\ z & \tau'\end{bmatrix}. \] If \[ g'=\begin{bmatrix}A & B\\ C & D\end{bmatrix}\in \mathrm{Sp}_4(\mathbb{R}), \] then \[ gZ=(AZ+B)(CZ+D)^{-1},\qquad J(g,Z)=CZ+D. \] A scalar Siegel modular form of degree \(2\) and weight \(k\) for a discrete subgroup \(\Gamma\subset\mathrm{Sp}_4(\mathbb{Q})\) is a holomorphic function \(F:\mathbb{H}_2\to\mathbb{C}\) satisfying \[ (F|_k\gamma)(Z)=F(Z),\qquad \gamma\in\Gamma, \] with slash operator \[ (F|_k g)(Z)=\lambda(g)^k\det(J(g,Z))^{-k}F(gZ). \] The relevant level subgroup is the paramodular group \(K(M)\), in the genus-\(2\) setting. In the 2025 construction, the pertinent level is \(M=N^2\). These groups are described as arithmetic subgroups of \(\mathrm{GSp}_4(\mathbb{Q})\) or \(\mathrm{Sp}_4(\mathbb{Q})\), depending on the chosen model, and are natural from the viewpoint of abelian surfaces with polarization of type \((1,N)\). They also feature prominently in the paramodular conjecture [2509.04395]. Fix an even integer \(k\ge 4\) and a primitive Dirichlet character \(\eta\) of conductor \(N\). For \[ \mathcal{C}_0(x)= \begin{bmatrix} 1&0&x&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}, \] the cusp Eisenstein series attached to \(\mathcal{C}_0(bN)\) is \[ E_k(Z,K(N^2),\mathcal{C}_0(bN)) = \sum_{\gamma\in (K(N^2)\cap \mathcal{C}_0(bN)^{-1}P^1(\mathbb{Q})\mathcal{C}_0(bN))\backslash K(N^2)} \det(J(\mathcal{C}_0(bN)\gamma,Z))^{-k}. \] The paramodular Siegel Eisenstein series attached to \(\eta\) is then \[ E_{k,\eta}(Z) = \frac12\sum_{b\in(\mathbb{Z}/N\mathbb{Z})^\times}\eta(b)\,E_k(Z,K(N^2),\mathcal{C}_0(bN)). \] In this sense, a paramodular Siegel Eisenstein series of level \(N^2\) is a holomorphic genus-\(2\) Siegel modular form obtained as a specific linear combination of Eisenstein series attached to \(0\)-dimensional cusps. ## 2. Adelic construction and the origin of the level \(N^2\) The construction begins with the primitive Dirichlet character \(\eta\bmod N\) and its associated idele class character \[ \chi=\bigotimes_{p\le\infty}\chi_p. \] The parity condition is \[ \chi_\infty(-1)=\eta(-1)=(-1)^k. \] From \(\chi\) one forms the normalized induced representation \[ J_\chi(s)=\chi|\cdot|^{s-3/2}\,1_{\mathrm{GL}_2}\rtimes \chi^{-1}|\cdot|^{-s+3/2}. \] In the flat model, for \[ b=\begin{bmatrix}A & *\\ 0 & u\hat A\end{bmatrix}\in P(\mathbb{A}), \] one has \[ f(bg)=\chi(u^{-1}\det A)\,|u^{-1}\det A|^s\,f(g). \] At the archimedean place, \(J_{\chi_\infty}(s)\) contains a unique scalar \(K_\infty\)-type \((k,k)\), denoted \(\alpha_k\), and one chooses a section \(f^{(k)}_{s,\infty}\) spanning that type, normalized by \(f^{(k)}_{s,\infty}(1)=1\). For \(s=k\), this vector lies in the holomorphic discrete series \(\mathcal{D}(k,k)\). At finite places, one chooses local paramodular newforms and assembles them into the global section \[ f_s^{(k)}=f^{(k)}_{s,\infty}\otimes\bigotimes_{p<\infty}f_{s,p}. \] The adelic Eisenstein series is \[ \mathbf{E}_\chi(g,s,f^{(k)})= \sum_{\gamma\in P(\mathbb{Q})\backslash \mathrm{GSp}_4(\mathbb{Q})} f_s^{(k)}(\gamma g). \] It is right-invariant under \(\prod_{p<\infty}K(p^{2n_p})\), and the resulting global level is exactly \(N^2\). The explanation given in the construction is representation-theoretic: the conductor of the induced representation is \(N^2\), and this is the smallest paramodular level at which a non-zero paramodular vector exists [2509.04395]. To descend from the adelic setting to \(\mathbb{H}_2\), one uses the standard point \(I=\begin{bmatrix}i&0\\0&i\end{bmatrix}\) and, for \(Z\in\mathbb{H}_2\), the unique upper triangular element \(b_Z\in\mathrm{GSp}_4(\mathbb{R})^+\) with positive diagonal such that \(b_Z\cdot I=Z\). If \(\Phi\) satisfies the \(K_\infty\)-equivariance determined by \(\alpha_k\), then \[ F(Z)=\det(J(b_Z,I))^k\,\Phi(b_Z)=\det(Y)^{-k/2}\Phi(b_Z). \] Applied to \(\Phi(g)=\mathbf{E}_\chi(g,k,f^{(k)})\), this yields the classical form \(E_{k,\eta}(Z)\). The identification relies on a double coset decomposition of \(\mathrm{GSp}_4(\mathbb{Q})\) relative to \(P^1(\mathbb{Q})\) and \(K(N^2)\), using Poor–Yuen’s classification of \(0\)-dimensional cusps. ## 3. Fourier expansion and local–global coefficient formulas The Fourier expansion of the genus-\(2\) paramodular Eisenstein series has the form \[ E_{k,\eta}(Z)=\delta_{\eta=1}+\sum_{T\neq 0} a(T)e^{2\pi i\operatorname{tr}(TZ)}, \] where \[ T=\begin{bmatrix}n&r/2\\ r/2&m\end{bmatrix},\qquad n,r,m\in\mathbb{Z},\qquad T\ge 0, \] and the support condition \[ N^2\mid m \] is built into the expansion. The constant term is \(1\) when \(\eta\) is trivial and \(0\) otherwise [2509.04395]. On the adelic side, the coefficient attached to \(T\) is obtained from the Fourier integral \[ c_T(g,s,f)= \int_{\mathrm{Sym}_2(\mathbb{Q})\backslash \mathrm{Sym}_2(\mathbb{A})} \mathbf{E}_\chi\!\left(\begin{bmatrix}1&X\\0&1\end{bmatrix}g,s,f\right) \psi(-\operatorname{tr}(TX))\,dX, \] and factors as \[ c_T(g,s,f)=\prod_p I_p(T,g_p,s). \] For rank \(2\), the Bruhat decomposition yields four terms \(c_{T,i}\), but only \(c_{T,4}\) is non-zero. At the real place, for \(s=k\ge 2\), \[ I_\infty(T,1)= \begin{cases} \dfrac{(4\pi)^{2k-1}\det(T)^{k-\frac32}e^{-2\pi\operatorname{tr}(T)}}{2(2k-2)!} & \text{if }T>0,\\[1ex] 0 & \text{otherwise.} \end{cases} \] For \(g=b_Z\), this becomes \[ I_\infty(T,b_Z)= \det(Y)^{k/2}\, \dfrac{(4\pi)^{2k-1}\det(T)^{k-\frac32}e^{2\pi i\operatorname{tr}(TZ)}}{2(2k-2)!} \qquad (T>0). \] This is the point at which the Koecher principle appears in the calculation: the archimedean integral vanishes unless \(T\) is positive semidefinite. For non-archimedean places not dividing \(N\), the coefficients are expressed in terms of the discriminant factorization \[ -\Delta=Df^2,\qquad \Delta=4nm-r^2, \] with \(D\) a fundamental discriminant. The multiplicative arithmetic factor is packaged into \[ \tilde H_{D,s,\eta}(e,f) = \sum_{d\mid e}\eta(d)^{-1}d^{s-1} \sum_{g\mid \frac{f}{d}}\mu(g)\chi_D(g)\eta(g)^{-1}g^{s-2} \sum_{h\mid \frac{f}{dg}}\eta(h)^{-2}h^{2s-3}. \] For primes \(p\mid N\), the ramified local integral has the form \[ I_p= \delta_{v_p(m)\ge 2n_p}\,p^{n_p(5/2-2s)}\varepsilon(1/2,\chi_p,\psi_p) \Bigl( \delta_{v_p(r)=0}\chi_p(-r) + \delta_{v_p(r)>0}p^{n_p(2-s)}\chi_p(-p^{n_p})K(s,T,\chi_p) \Bigr), \] where \(K(s,T,\chi_p)\) is a local integral. In certain cases, specifically for odd \(p\) and quadratic \(\chi_p\), this local factor can be expressed in terms of point counts on elliptic curves over \(\mathbb{F}_p\) [2509.04395]. The final coefficient formulas separate rank \(1\) and rank \(2\). In rank \(1\), the coefficient is piecewise: the full-level term \(\sigma_{k-1}(n)/\zeta(k)\) occurs only when \(N=1\) and \[ T=\begin{bmatrix}n&0\\0&0\end{bmatrix},\qquad n>0, \] while the paramodular case involves \(\sigma_{k-1,\eta}(e_{\hat N})/L(k,\eta)\), the condition \(m>0\), and \[ r_N=\frac{(2m)_N}{N}; \] otherwise the coefficient is zero. In rank \(2\), one obtains \[ a(T) = \frac{(4\pi)^{2k-1}\det(T)^{k-\frac32}}{2(2k-2)!}\, N^{2-2k}\,f_{\hat N}^{3-2k}\,\eta(f_{\hat N}^2)\, \tilde H_{D,k,\eta}(e_{\hat N},f_{\hat N}) \frac{L(k-1,\chi_D\eta)}{L(k,\eta)L(2k-2,\eta^2)}\,G(\eta)\,\Xi(T), \] with \[ \Xi(T) = \Bigg(\prod_{p\mid N,\;p\nmid r}\chi_p(r)\Bigg) \Bigg(\prod_{p\mid N,\;p\mid r}p^{n_p(2-k)}\chi_p(p^{n_p})K(k,T,\chi_p)\Bigg). \] For \(\eta=1\), these formulas reduce to the classical Maass and Eichler–Zagier formulas for the degree-\(2\) Eisenstein series on \(\mathrm{Sp}_4(\mathbb{Z})\). ## 4. Paramodular newforms and automorphic representation theory The genus-\(2\) paramodular Eisenstein series \(E_{k,\eta}\) is not merely an Eisenstein series with a prescribed level. It is shown to be a paramodular newform in the sense of the local newform theory for \(\mathrm{GSp}_4\) developed by Roberts–Schmidt. For each finite prime \(p\), if \[ \pi_p=J_{\chi_p}(k), \] then \(\pi_p\) is irreducible of type IIb, and its chosen \(K(p^{2n_p})\)-fixed vector is the local paramodular newform [2509.04395]. Globally, the automorphic representation generated by the adelization of \(E_{k,\eta}\) is \[ \pi=\bigotimes_p \pi_p, \] with \[ \pi_\infty=\mathcal{D}(k,k),\qquad \pi_p= \chi_p|\cdot|^{k-3/2}\,1_{\mathrm{GL}_2}\rtimes \chi_p^{-1}|\cdot|^{-k+3/2}. \] The representation \(\pi\) is irreducible, and the adelization of the classical form \(E_{k,\eta}(Z)\) is exactly the adelic Eisenstein series \(\mathbf{E}_\chi(g,k,f^{(k)})\). The conductor statement is equally precise: each local factor \(\pi_p\) has minimal paramodular level \(p^{2n_p}\), hence the global paramodular conductor is \(N^2\). Consequently, \(E_{k,\eta}\) is new at level \(K(N^2)\): any trace, or oldform projection, from level \(K(N^2)\) down to a paramodular subgroup \(K(M)\) with \[ M\mid N^2,\qquad Mn+1\) the Eisenstein space is one-dimensional. In this setting the normalized Eisenstein series is identified, via the Siegel–Weil formula, with the genus theta series attached to the unique genus of paramodular lattice chains. This higher-degree theory is part of a larger program involving Hecke algebras, boundary components, Garrett’s doubling method, and the basis problem for paramodular cusp forms [2011.09597]. Comparison with the full-level genus-\(2\) series clarifies what is genuinely paramodular in the 2025 construction. The differences are the level \(K(N^2)\) rather than \(\mathrm{Sp}_4(\mathbb{Z})\), the twist by a primitive Dirichlet character \(\eta\), the support condition \(N^2\mid m\) in the Fourier expansion, and the ramified local factors \(K(k,T,\chi_p)\) at primes dividing \(N\). For \(\eta=1\) and \(N=1\), the formulas reduce exactly to the classical full-level formulas [2509.04395]. A further analytic perspective comes from the study of real-analytic Siegel Eisenstein series at full level. Nagaoka gives an explicit residue formula at \(s=m/2\) in terms of a constant term built from Koecher–Maass zeta data and a rank-\(1\) Fourier expansion with confluent hypergeometric factors. The accompanying exposition states that many of these structures are local or independent of level. This suggests a template for future paramodular residue formulas: the archimedean hypergeometric factors should remain unchanged, while the finite local factors should be modified at primes dividing the paramodular level [2105.05437]. Two misconceptions are thereby avoided. First, not every “paramodular Eisenstein series” is of the same parabolic type: Siegel-parabolic constructions such as \(E_{k,\eta}\) and Klingen-type constructions such as \(\widehat E_k(Z,f,N)\) are distinct objects. Second, in the constructions presently available, the natural paramodular level is often \(N^2\), not \(N\). This is not an artifact of normalization, but a reflection of the conductor of the underlying induced representation and of the local paramodular invariants. Source: https://www.emergentmind.com/topics/paramodular-siegel-eisenstein-series