## 1 Introduction

### 1.1 Motivation: On the modular correspondences

Let $$j=j'=j(\tau )$$ be the elliptic modular j-function on the upper half plane. For $$m\ge 1$$ let $$\varphi _m\in {\mathbb {Z}}[j,j']$$ be the classical modular polynomial defined by

\begin{aligned} \varphi _m(j(\tau ),j(\tau '))=\prod _{A\in \mathrm {M}_2({\mathbb {Z}})\pmod {\mathrm {SL}_2({\mathbb {Z}})},\;\det A=m}(j(\tau )-j(A\tau ')). \end{aligned}

Put $$S=\mathrm {Spec}\,{\mathbb {Z}}[j,j']$$ and $$S_{\mathbb {C}}=\mathrm {Spec}\,{\mathbb {C}}[j,j']$$. Let $$T_m$$ and $$T_{m,{\mathbb {C}}}$$ be the arithmetic and geometric divisors defined by $$\varphi _m=0$$. We can view S as an arithmetic threefold $$\mathcal{{S}}=\mathcal{{M}}\times _{\mathrm {Spec}\,{\mathbb {Z}}}\mathcal{{M}}$$, where $$\mathcal{{M}}$$ is the moduli stack of elliptic curves over $${\mathbb {Z}}$$, and $$T_m$$ as the moduli stack $$\mathcal{{T}}_m$$ of isogenies of elliptic curves of degree m. In the 19th century Hurwitz has computed the intersection

\begin{aligned} (T_{m_1,{\mathbb {C}}}\cdot T_{m_2,{\mathbb {C}}}):=\dim _{\mathbb {C}}{\mathbb {C}}[j,j']/(\varphi _{m_1},\varphi _{m_2}) \end{aligned}

of complex curves. Gross and Keating [3] discovered that $$(T_{m_1,{\mathbb {C}}}\cdot T_{m_2,{\mathbb {C}}})$$ is related to the Fourier coefficients of the Siegel Eisenstein series of weight 2 for $$Sp_2({\mathbb {Z}})$$. Moreover, they gave an explicit expression for the intersection

\begin{aligned} (T_{m_1}\cdot T_{m_2}\cdot T_{m_3}):=\log \sharp {\mathbb {Z}}[j,j']/(\varphi _{m_1},\varphi _{m_2},\varphi _{m_3}) \end{aligned}

of 3 arithmetic modular correspondences. It is already mentioned in the introduction of [3] that computations of Kudla or Zagier strongly suggest that $$\deg {\mathscr {Z}}(B)$$ equals the B-th Fourier coefficient of the derivative of the Siegel Eisenstein series of weight 2 for $$Sp_3({\mathbb {Z}})$$, up to multiplication by a constant which is independent of B. A complete proof of this identity has been given in [20] (cf. [13]).

The purpose of this paper is to compute the Fourier coefficients of the derivative of the Siegel Eisenstein series of weight 2 for $$Sp_4({\mathbb {Z}})$$. One may expect that these coefficients are related to the intersection of 4 modular correspondences. Naively, the fiber product $${\mathcal {T}}_{m_1}\times _{{\mathcal {S}}} {\mathcal {T}}_{m_2}\times _{{\mathcal {S}}}{\mathcal {T}}_{m_3}\times _{{\mathcal {S}}} {\mathcal {T}}_{m_4}$$ comes up to our mind. However, it may be hard to relate it with the summation of local contribution as in (1.1) of [20] because four modular polynomials never make up a regular sequence in $$\mathbb {Z}[j,j']$$. In the face of this situation, each local contribution itself is an interesting object, and our result (see Theorem 1.2) suggests each local contribution should be described with three elements in $$W(\overline{\mathbb {F}}_p)[[j,j']]$$ which make up a regular sequence, where $$W(\overline{\mathbb {F}}_p)$$ denotes the Witt ring of $$\overline{{\mathbb {F}}}_p$$. What remains is how we properly subsume the local contributions into the (arithmetic) intersection of four cycles defined by four modular polynomials. This will be handled in a future work or left for the interested readers.

In the intervening years Kudla and others have gone a long way towards proving such relations in much greater generality. In [10], he introduced a certain family of Eisenstein series of genus g and weight $$\frac{g+1}{2}$$. They have an odd functional equation and hence have a natural zero at their center of symmetry. The central derivatives of such series, which he refers to as incoherent Eisenstein series, have a connection with arithmetic algebraic geometry of cycles on integral models of Shimura varieties attached to orthogonal groups of signature $$(2,g-1)$$, at least when $$g\le 4$$. We refer the reader to [16] for $$g=1$$, to [10, 14, 17] for $$g=2$$, to [13, 20, 28] for $$g=3$$, to [15] for $$g=4$$, and to [18] for an arbitrary positive integer g.

### 1.2 The Fourier coefficients of derivative of Eisenstein series

In this paper we compute the Fourier coefficients of derivatives of incoherent Eisenstein series of genus g and weight $$\frac{g}{2}$$. In this introductory section we will consider classical Eisenstein series of level 1. Let g be a positive integer that is divisible by 4. Let

\begin{aligned} E_g(Z,s)=\sum _{\{C,D\}}\det (CZ+D)^{-g/2}|\det (CZ+D)|^{-(2s+1)/2}(\det Y)^{(2s+1)/4} \end{aligned}

be the Siegel Eisenstein series of genus g, where $$\{C,D\}$$ runs over a complete set of representatives of the equivalence classes of coprime symmetric pairs of degree g, and Z is a complex symmetric matrix of degree g with positive definite imaginary part Y. This series converges absolutely for $$\mathfrak {R}s>\frac{g+1}{2}$$, admits a meromorphic continuation to the whole s-plane and satisfies a functional equation by the general theory of Langlands. It is worth noting that $$s=0$$ is the central point on the real axis for this functional equation.

If $$\frac{g}{4}$$ is even, then $$E_g(Z,s)$$ is holomorphic at $$s=-\frac{1}{2}$$ and the T-th Fourier coefficient of $$E_g\bigl (Z,-\frac{1}{2}\bigl )$$ is equal to

\begin{aligned} 2\biggl (\sum _i\frac{1}{N(L_i,L_i)}\biggl )^{-1}\sum _i\frac{N(L_i,T)}{N(L_i,L_i)} \end{aligned}
(1.1)

by the Siegel formula (see [12, 27, 31]), where $$\{L_i\}$$ is the set of isometry classes of positive definite even unimodular lattices of rank g. Here $$N(L,L')$$ denotes the number of isometries $$L'\rightarrow L$$ for two quadratic spaces $$L,L'$$ over $${\mathbb {Z}}$$. In particular, the nondegenerate Fourier coefficients are supported on a single rational equivalence class.

On the other hand, if $$\frac{g}{4}$$ is odd, then $$E_g(Z,s)$$ has a zero at $$s=-\frac{1}{2}$$ by Corollary 5.5 of [31] and Lemma 2.1. Our main object of study in this paper is the derivative

\begin{aligned} \frac{\partial }{\partial s}E_g(Z,s)|_{s=-1/2}=\sum _{T>0}C_g(T)e^{2\pi \sqrt{-1}\mathrm {tr}(TZ)}+\sum _{\text {other }T}C_g(T,Y)e^{2\pi \sqrt{-1}\mathrm {tr}(TZ)}. \end{aligned}

Fix a positive definite symmetric half-integral $$n\times n$$ matrix T and a rational prime p. Let $${\mathbb {Q}}^{(p)}$$ be a subring of $${\mathbb {Q}}$$, consisting of the numbers of the form $$\frac{a}{p^{n}}$$ with $$n\in {\mathbb {N}}$$ and $$a\in {\mathbb {Z}}$$. We define the additive character $${\mathbf {e}}_p$$ of $${\mathbb {Q}}_p$$ by setting $${\mathbf {e}}_p(x)=e^{-2\pi \sqrt{-1}y}$$ with $$y\in {\mathbb {Q}}^{(p)}$$ such that $$x-y\in {\mathbb {Z}}_p$$. The Siegel series attached to T and p is defined by

\begin{aligned}b_p(T,s)=\sum _{z\in \mathrm {Sym}_n({\mathbb {Q}}_p)/\mathrm {Sym}_n({\mathbb {Z}}_p)} {\mathbf {e}}_p(-\mathrm {tr}(Tz))\nu [z]^{-s}, \end{aligned}

where $$\nu [z]$$ is the product of denominators of elementary divisors of z. Put $$D_T=(-4)^{[n/2]}\det T$$. We denote the primitive Dirichlet character corresponding to $${\mathbb {Q}}(\sqrt{D_T})$$ by $$\chi _T$$ and its conductor by $${\mathfrak {d}}^T$$. Put $$\xi ^T_p=\chi _T(p)$$. Let $$e^T_p={\text {ord}}_pD_T$$ or $$e^T_p={\text {ord}}_pD_T-{\text {ord}}_p{\mathfrak {d}}^T$$ according as n is odd or even. There exists a polynomial $$F_p^T(X)\in {\mathbb {Z}}[X]$$ such that

\begin{aligned} b_p(T,s)=\gamma ^T_p(p^{-s})F^T_p(p^{-s}), \end{aligned}

where

\begin{aligned}\gamma ^T_p(X) =(1-X)\prod _{j=1}^{[n/2]}(1-p^{2j}X^2)\times {\left\{ \begin{array}{ll} 1 &{}\text { if }n \text { is odd, } \\ \frac{1}{1-\xi ^T_p p^{n/2}X} &{}\text { if }n \text { is even. } \end{array}\right. } \end{aligned}

The symbol $$\eta _p^T$$ stands for the normalized Hasse invariant of T over $${\mathbb {Q}}_p$$ (see Definition 2.1). We write $$\mathrm {Diff}(T)$$ for the finite set of prime numbers p such that $$\eta _p^T=-1$$. A direct calculation gives the following formula:

Assume that $$\frac{g}{4}$$ is odd. Let T be a positive definite symmetric half-integral matrix of size g.

1. (1)

If $$\chi _T=1$$, then $$C_g(T)=0$$ unless $$\mathrm {Diff}(T)$$ is a singleton.

2. (2)

If $$\chi _T=1$$ and $$\mathrm {Diff}(T)=\{p\}$$, then

\begin{aligned} C_g(T)=-\frac{2^{(g+2)/2}p^{-(g+e^T_p)/2}\log p}{\zeta \bigl (1-\frac{g}{2}\bigl )\prod _{i=1}^{(g-2)/2}\zeta (1-2i)}\frac{\partial F^T_p}{\partial X}(p^{-g/2})\prod _{p\ne \ell |D_T}\ell ^{-e^T_\ell /2}F_\ell ^T(\ell ^{-g/2}). \end{aligned}
3. (3)

If $$\chi _T\ne 1$$, then

\begin{aligned} C_g(T)=-\frac{2^{(g+2)/2}L(1,\chi _T)}{\zeta \bigl (1-\frac{g}{2}\bigl )\prod _{i=1}^{(g-2)/2}\zeta (1-2i)}\prod _{\ell |D_T}p^{-e^T_\ell /2}F_\ell ^T(\ell ^{-g/2}). \end{aligned}

### Remark 1.1

If $$\chi _T\ne 1$$, then $$L(1,\chi _T)=\frac{\sqrt{{\mathfrak {d}}^T}}{\log \epsilon }h$$ by Dirichlet’s class number formula, where h is the class number of the real quadratic field $${\mathbb {Q}}(\sqrt{\det T})$$ and $$\epsilon =\frac{t+u\sqrt{{\mathfrak {d}}^T}}{2}$$ ($$t>0$$, $$u>0$$) is the solution to the Pell equation $$t^2-{\mathfrak {d}}^Tu^{2}=4$$ for which u is smallest.

The following theorem is a special case of Theorem 4.1 and allows us to compute $$\frac{\partial F^T_p}{\partial X}(\xi _p^Tp^{-g/2})$$. For simplicity we here assume p to be odd.

### Theorem 1.1

Let p be an odd rational prime and $$T=\mathrm {diag}[t_1,\dots ,t_g]$$ with $$0\le {\text {ord}}_pt_1\le \cdots \le {\text {ord}}_pt_g$$. Put $$T'=\mathrm {diag}[t_1,\dots ,t_{g-1}]$$. Suppose that g is even and $$p\not \mid {\mathfrak {d}}^T$$. Then

\begin{aligned} F_p^T(\xi _p^Tp^{-g/2})=p^{e^T_p/2}F_p^{T'}(\xi _p^Tp^{-g/2}). \end{aligned}

If $$\eta ^T_p=-1$$, then

\begin{aligned}\frac{\xi ^T_p}{p^{g/2}}\frac{\partial F_p^T}{\partial X}\biggl (\frac{\xi ^T_p}{p^{g/2}}\biggl ) =\frac{F_p^{T'}(\xi ^T_p p^{(2-g)/2})}{p-1} -p^{e_p^T/2}\frac{\xi ^T_p}{p^{g/2}}\frac{\partial F_p^{T'}}{\partial X}\biggl (\frac{\xi ^T_p}{p^{g/2}}\biggl ). \end{aligned}

Our key ingredient is the explicit formula for $$F_p^T(X)$$, given by Ikeda and Katsurada in [6], which expresses the polynomial $$F_p^T$$ in terms of the (naive) extended Gross–Keating datum H of T over $${\mathbb {Z}}_p$$. The polynomial $$F_p^{T'}=F_p^{H'}$$ is defined in terms of a subset $$H'\subsetneq H$$ for any p in a uniform way. Actually, if $$g=4$$, then the values $$\frac{\partial F^{H'}_p}{\partial X}(p^{-2})$$ and $$F_p^{H'}(p^{-1})$$ depend only on $$(a_1,a_2,a_3)$$ if we write $$(a_1,a_2,a_3,a_4)$$ for the Gross–Keating invariant of T over $${\mathbb {Z}}_p$$.

### 1.3 Applications

#### 1.3.1 On the average of the representation numbers

Theorem 1.1 combined with the Siegel formula will identify (1.1) with four times the average of the representation numbers of a symmetric matrix of size $$g-1$$ (see Conjecture 5.1 and Proposition 5.2). The following result is a special case of Proposition 5.2.

If T is a positive definite symmetric half-integral matrix of size 4 which satisfies $$\chi ^T=1$$ and $$\eta ^T_\ell =1$$ for $$\ell \ne p$$, then there exists a positive definite symmetric half-integral matrix $$T'$$ of size 3 such that

\begin{aligned} \sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T)}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}=2\sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T')}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}, \end{aligned}

where $$(E,E')$$ extends over all pairs of isomorphism classes of supersingular elliptic curves over $${\bar{{\mathbb {F}}}}_p$$.

#### 1.3.2 On the Fourier coefficients and the modular correspondences

The factor $$\frac{\partial F_p^{H'}}{\partial X}(\xi ^T_pp^{-g/2})$$ appears in Fourier coefficients of central derivatives of incoherent Eisenstein series of genus $$g-1$$ and weight $$\frac{g}{2}$$, which have close connection with arithmetical geometry on Shimura varieties. We will be mostly interested in the case $$g=4$$. When $$T_{m_1}$$, $$T_{m_2}$$ and $$T_{m_3}$$ intersect properly, the formula of Gross and Keating in [3] can be stated as follows:

\begin{aligned} (T_{m_1}\cdot T_{m_2}\cdot T_{m_3})=\sum _B\deg {\mathscr {Z}}(B), \end{aligned}

where B extends over all positive definite symmetric half-integral matrices with diagonal entries $$m_1,m_2,m_3$$. Here $$\deg {\mathscr {Z}}(B)=0$$ unless $$\mathrm {Diff}(B)$$ consists of a single rational prime p, in which case

\begin{aligned} \deg {\mathscr {Z}}(B)=-\frac{(\log p)}{2p^2}\frac{\partial F^B_p}{\partial X}\left( \frac{1}{p^2}\right) \sum _{(E,E')}\frac{N(\mathrm {Hom}(E',E),B)}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}. \end{aligned}
(1.2)

The degree $$\deg {\mathscr {Z}}(B)$$ equals the B-th Fourier coefficient of the derivative of the Siegel Eisenstein series of weight 2 and genus 3 up to a negative constant (cf. Theorem 2.2 of [20]). We combine (1.2), Theorem 5.1 and Corollary 5.1 to obtain the following formula:

### Theorem 1.2

If T is a positive definite symmetric half-integral matrix of size 4, $$\chi _T=1$$ and $$\mathrm {Diff}(T)$$ consists of a single prime number p, then there exists a positive definite symmetric half-integral matrix $$T'$$ of size 3 such that

\begin{aligned} \frac{C_4(T)}{-2^8\cdot 3^2}=\deg {\mathscr {Z}}(T')+\frac{F_p^{T'}(p^{-1})}{2\sqrt{p}^{e^T_p}(p-1)}\log p\sum _{(E,E')}\frac{N(\mathrm {Hom}(E',E),T')}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}, \end{aligned}

where $$(E,E')$$ extends over all pairs of isomorphism classes of supersingular elliptic curves over $${\bar{{\mathbb {F}}}}_p$$.

Since $$\mathrm {Hom}(E',E)$$ is a quaternary quadratic space, if S has rank greater than 4, then $$N(\mathrm {Hom}(E,E'),S)=0$$. Therefore when $$g\ge 5$$, the nature of Fourier coefficients of the derivative of Eisenstein series of weight 2 and genus g should be much different. The case $$g=4$$ should be a boundary case. We will explicitly compute $$F_p^{T'}(p^{-1})$$ in Lemma 5.2 and show that

\begin{aligned} \biggl |\frac{C_4(T)}{-2^8\cdot 3^2\cdot \deg {\mathscr {Z}}(T')}-1\biggl |<\frac{20}{p\sqrt{p}}. \end{aligned}

Moreover, Corollary 5.2 says that for a fixed prime number p

\begin{aligned} \lim _{{\text {ord}}_p(\det T)\rightarrow \infty }\frac{C_4(T)}{-2^8\cdot 3^2\cdot \deg {\mathscr {Z}}(T')}=1. \end{aligned}

### 1.4 Organizations

We now explain the lay-out of this paper. Section 2 extends the notion of incoherent Eisenstein series to the case where the point at which the Eisenstein series is evaluated lies within the left half-plane. We calculate the Fourier coefficients of those Eisenstein series and their derivatives. In Sect. 3 we derive a general formula for Fourier coefficients of derivatives of incoherent Eisenstein series. Section 4 is devoted to a local study of the Siegel series. We give the inductive expression for the special value of the derivative of the Siegel series. Section 5 is devoted to proving Theorem 5.1.

## 2 Notations

For a finite set A, we denote by $$\sharp A$$ the number of elements in A. For a ring R we denote by $$\mathrm {M}_{i,j}(R)$$ the set of $$i\times j$$-matrices with entries in R and write $$\mathrm {M}_m(R)$$ in place of $$\mathrm {M}_{m,m}(R)$$. The group of all invertible elements of $$\mathrm {M}_m(R)$$ and the set of symmetric matrices of size m with entries in R are denoted by $$\mathrm {GL}_m(R)$$ and $$\mathrm {Sym}_m(R)$$, respectively. Let $$\mathcal{{E}}_m(R)$$ be the set of elements $$(a_{ij})\in \mathrm {Sym}_m(R)$$ such that $$a_{ii}\in 2R$$ for every i. For matrices $$B\in \mathrm {Sym}_m(R)$$ and $$G\in \mathrm {M}_{m,n}(R)$$ we use the abbreviation $$B[G]=\,^t\!GBG$$, where $$\,^t\!G$$ is the transpose of G. If $$A_1, \dots , A_r$$ are square matrices, then $$\mathrm {diag}[A_1, \dots , A_r]$$ denotes the matrix with $$A_1, \dots , A_r$$ in the diagonal blocks and 0 in all other blocks. Let $$\mathbf{1}_m$$ be the identity matrix of degree m. Put

\begin{aligned} Sp_g(R)&=\left\{ G\in \mathrm {GL}_{2g}(R)\;\biggl |\;G\begin{pmatrix} 0 &{} \mathbf{1}_g \\ -\mathbf{1}_g &{} 0\end{pmatrix}\,^t\!G=\begin{pmatrix} 0 &{} \mathbf{1}_g \\ -\mathbf{1}_g &{} 0\end{pmatrix}\right\} , \\ M_g(R)&=\left\{ {\mathbf {m}}(A)=\begin{pmatrix} A &{} 0 \\ 0 &{} \,^t\!A^{-1}\end{pmatrix}\;\biggl |\;A\in \mathrm {GL}_g(R)\right\} , \\ N_g(R)&=\left\{ {\mathbf {n}}(B)=\begin{pmatrix} \mathbf{1}_g &{} B \\ 0 &{} \mathbf{1}_g\end{pmatrix}\;\biggl |\;B\in \mathrm {Sym}_g(R)\right\} . \end{aligned}

Let $${\mathbb {Z}}$$ be the set of integers and $$\mu _n$$ the group of n-th roots of unity. If x is a real number, then we put $$[x]=\max \{m\in {\mathbb {Z}}\;|\;m\le x\}$$.

## 3 Eisenstein series

Let k be a totally real number field with integer ring $${\mathfrak {o}}$$. The set of real places of k is denoted by $${\mathfrak {S}}_\infty$$. The completion of k at a place v is denoted by $$k_v$$. Let $$(\;,\;)_{k_v}:k_v^\times \times k_v^\times \rightarrow \mu _2$$ denote the Hilbert symbol. We let $${\mathfrak {p}}$$ denote a finite place of k above a prime number p and do not use the letters $$p,q,{\mathfrak {p}},{\mathfrak {q}}$$ for a real place. Let $$q_{\mathfrak {p}}=\sharp {\mathfrak {o}}/{\mathfrak {p}}$$ be the order of the residue field. We define the character $${\mathbf {e}}_{\mathfrak {p}}$$ of $$k_{\mathfrak {p}}$$ by $${\mathbf {e}}_{\mathfrak {p}}(x)={\mathbf {e}}(-y)$$ with $$y\in {\mathbb {Q}}^{(p)}$$ such that $$\mathrm {Tr}_{k_{\mathfrak {p}}/{\mathbb {Q}}_p}(x)-y\in {\mathbb {Z}}_p$$ if p is the rational prime divisible by $${\mathfrak {p}}$$. Put $${\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z}$$ for $$z\in {\mathbb {C}}$$ and $${\mathbf {e}}_\infty (z)=\prod _{v\in {\mathfrak {S}}_\infty }{\mathbf {e}}(z_v)$$ for $$z\in \prod _{v\in {\mathfrak {S}}_\infty }{\mathbb {C}}$$.

Once and for all we fix a positive integer $$g\ge 2$$. Let $$(V,(\;,\;))$$ be a quadratic space of dimension m over $$k_v$$. Whenever we speak of a quadratic space, we always assume that $$(\;,\;)$$ is nondegenerate, i.e., $$(u,V)=0$$ implies that $$u=0$$. Put

\begin{aligned} s_0=\frac{1}{2}(m-g-1). \end{aligned}

Given $$u=(u_1,\dots ,u_g)\in V^g$$, we write (uu) for the $$g\times g$$ symmetric matrix with (ij) entry equal to $$(u_i,u_j)$$. We write $$\det V$$ for the element in $$k_v^\times /k_v^{\times 2}$$ represented by the determinant of the matrix representation of the bilinear form $$(\;,\;)$$ with respect to any basis for V over $$k_v$$. We define the character $$\chi ^V:k_v^\times \rightarrow \mu _2$$ by

\begin{aligned} \chi ^V(t)=(t,(-1)^{m(m-1)/2}\det V)_{k_v}. \end{aligned}
(2.1)

We normalize our Hasse invariant $$\eta ^V$$ so that it depends only on the isomorphism class of an anisotropic kernel of V (cf. [2, 26]).

### Definition 2.1

We associate to the quadratic space V over $$k_{\mathfrak {p}}$$ of dimension m an invariant $$\eta ^V\in \mu _2$$ according to the type of V as follows:

• If m is odd, then an anisotropic kernel of V has dimension $$2-\eta ^V$$.

• If m is even and $$\chi ^V\ne 1$$ and if we choose an element $$c\in k_{\mathfrak {p}}^\times$$ such that $$\chi ^V(c)=\eta ^V$$, then V is the orthogonal sum of a split form of dimension $$m-2$$ with the norm form scaled by the factor c on the quadratic extension of $$k_{\mathfrak {p}}$$ corresponding to $$\chi ^V$$.

• If m is even and $$\chi ^V=1$$, then V is split or the orthogonal sum of the norm form on the quaternion algebra over $$k_{\mathfrak {p}}$$ with a split form of dimension $$m-4$$ according as $$\eta ^V=1$$ or $$-1$$.

We denote the set of positive definite symmetric matrices over $${\mathbb R}$$ of rank g by $$\mathrm {Sym}_g({\mathbb R})^+$$. Let

\begin{aligned} {\mathfrak {H}}_g=\{X+\sqrt{-1}Y\in \mathrm {Sym}_g({\mathbb {C}})\;|\;Y\in \mathrm {Sym}_g({\mathbb R})^+\} \end{aligned}

be the Siegel upper half-space of genus g. The real symplectic group $$Sp_g({\mathbb R})$$ acts transitively on $${\mathfrak {H}}_g$$ by $$GZ=(AZ+B)(CZ+D)^{-1}$$ for $$Z\in {\mathfrak {H}}_g$$ and $$G=\begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix}\in Sp_g({\mathbb R})$$. We define the maximal compact subgroups by

\begin{aligned} K_{\mathfrak {p}}&=Sp_g({\mathfrak {o}}_{\mathfrak {p}}),&K_v&=\{G\in Sp_g(k_v)\;|\;G(\sqrt{-1}\mathbf{1}_g)=\sqrt{-1}\mathbf{1}_g\} \end{aligned}

for $$v\in {\mathfrak {S}}_\infty$$. We have the Iwasawa decomposition

\begin{aligned} Sp_g(k_v)=M_g(k_v)N_g(k_v)K_v. \end{aligned}

Denote the two-fold metaplectic cover of $$Sp_g(k_v)$$ by $$\mathrm {Mp}_v$$. There is a canonical splitting $$N_g(k_v)\rightarrow \mathrm {Mp}_v$$. When $${\mathfrak {p}}$$ does not divide 2, we have a canonical splitting $$K_{\mathfrak {p}}\rightarrow \mathrm {Mp}_{\mathfrak {p}}$$. We still use $$N_g(k_v)$$ and $$K_{\mathfrak {p}}$$ to denote the images of these splittings. Let $$\tilde{K}_v$$ denote the pull-back of $$K_v$$ in $$\mathrm {Mp}_v$$. Define the map $$|a(\cdot )|:\mathrm {Mp}_v\rightarrow {\mathbb R}^\times _+$$ by writing $$\tilde{G}={\mathbf {n}}(b)\tilde{m}\tilde{k}\in \mathrm {Mp}_v$$ with $$b\in \mathrm {Sym}_g(k_v)$$, $$a\in \mathrm {GL}_g(k_v)$$, $$\tilde{m}=({\mathbf {m}}(a),\zeta )$$ and $$\tilde{k}\in \tilde{K}_v$$ and setting $$|a(\tilde{G})|=|\det a|_v$$. We refer to Section 1.1 of [31] for additional explanation.

Let V be a quadratic space over $$k_v$$ and $$\omega _v$$ the Weil representation of $$\mathrm {Mp}_v$$ with respect to $${\mathbf {e}}_v$$ on the space $$\mathcal{{S}}(V^g)$$ of the Schwartz functions on $$V^g$$. We associate to $$\varphi \in \mathcal{{S}}(V^g)$$ the function on $$\mathrm {Mp}_v\times {\mathbb {C}}$$ by

\begin{aligned} f_\varphi ^{(s)}(\tilde{G})=(\omega _v(\tilde{G})\varphi )(0)|a(\tilde{G})|^{s-s_0}. \end{aligned}

The real metaplectic group acts on the half-space $${\mathfrak {H}}_g$$ through $$Sp_g({\mathbb R})$$. There is a unique factor of automorphy $$\jmath _v:\mathrm {Mp}_v\times {\mathfrak {H}}_g\rightarrow {\mathbb {C}}^\times$$ whose square descends to the automorphy factor on $$Sp(k_v)\times {\mathfrak {H}}_g$$ given by $$\jmath _v(G_v,Z_v)^2=\det (C_vZ_v+D_v)$$ for $$G_v=\begin{pmatrix} * &{} * \\ C_v &{} D_v \end{pmatrix}\in Sp(k_v)$$. We define an automorphy factor $$\jmath :\prod _{v\in {\mathfrak {S}}_\infty }(\mathrm {Mp}_v\times {\mathfrak {H}}_g)\rightarrow {\mathbb {C}}^\times$$ by $$\jmath (\tilde{G},Z)=\prod _v\jmath _v(\tilde{G}_v,Z_v)$$.

Let $${\mathbb {A}}$$ be the adele ring of k and $${{\mathbb {A}}_{\mathbf{f}}}$$ the finite part of the adele ring. We arbitrarily fix a quadratic character $$\chi$$ of $${\mathbb {A}}^\times /k^\times$$ such that $$\chi _v=\mathrm {sgn}^{m(m-1)/2}$$ for $$v\in {\mathfrak {S}}_\infty$$.

### Definition 2.2

Let $$\mathcal{{C}}=\{\mathcal{{C}}_v\}$$ be a collection of local quadratic spaces of dimension m such that $$\chi ^{\mathcal{{C}}_v}=\chi _v$$ for all v, such that $$\mathcal{{C}}_v$$ is positive definite for $$v\in {\mathfrak {S}}_\infty$$ and such that $$\eta ^{\mathcal{{C}}_{\mathfrak {p}}}=1$$ for almost all $${\mathfrak {p}}$$. We say that $$\mathcal{{C}}$$ is coherent if it is the set of localizations of a global quadratic space. Otherwise we call $$\mathcal{{C}}$$ incoherent.

One can derive the following criterion from the theorem of Minkowski-Hasse (see Theorem 4.4 of [25]).

### Lemma 2.1

Put $$d=[k:{\mathbb {Q}}]$$. When m is odd, $$\mathcal{{C}}$$ is coherent if and only if

\begin{aligned} (-1)^{d(m^2-1)/8}\prod _{\mathfrak {p}}\eta ^{\mathcal{{C}}_{\mathfrak {p}}}=1. \end{aligned}

When m is even, $$\mathcal{{C}}$$ is coherent if and only if $$(-1)^{dm(m-2)/8}\prod _{\mathfrak {p}}\eta ^{\mathcal{{C}}_{\mathfrak {p}}}=1$$.

Recall from the beginning of Sect. 2 that v stands for an arbitrary place of k and $${\mathfrak {p}}$$ stands for a finite place of k. There is a unique splitting $$Sp_g(k)\hookrightarrow \mathrm {Mp}_g$$ by which we regard $$Sp_g(k)$$ as the subgroup of the two-fold metaplectic cover $$\mathrm {Mp}_g$$ of $$Sp_g({\mathbb {A}})$$. Let $$P_g=M_gN_g$$ be the Siegel parabolic subgroup of $$Sp_g$$. Given any pure tensor $$\varphi =\otimes _{\mathfrak {p}}\varphi _{\mathfrak {p}}\in \otimes _{\mathfrak {p}}'\mathcal{{S}}(\mathcal{{C}}_{\mathfrak {p}}^g)$$, we consider the function

\begin{aligned} f_\varphi ^{(s)}(\tilde{G})&=\prod _{\mathfrak {p}}f_{\varphi _{\mathfrak {p}}}^{(s)}(\tilde{G}_{\mathfrak {p}}),&f_{\varphi _{\mathfrak {p}}}^{(s)}(\tilde{G}_{\mathfrak {p}})&=(\omega _{\mathfrak {p}}(\tilde{G}_{\mathfrak {p}})\varphi _{\mathfrak {p}})(0)|a(\tilde{G}_{\mathfrak {p}})|^{s-s_0} \end{aligned}

on $$\mathrm {Mp}_g\times {\mathbb {C}}$$ and the Eisenstein series on $$\prod _{v\in {\mathfrak {S}}_\infty }{\mathfrak {H}}_g$$

\begin{aligned} E(Z,f_\varphi ^{(s)})=(\det Y)^{(s-s_0)/2}\sum _{\gamma \in P_g(k)\backslash Sp_g(k)}|\jmath (\gamma ,Z)|^{s_0-s}\jmath (\gamma ,Z)^{-g}f_\varphi ^{(s)}(\gamma ), \end{aligned}

where Y is the imaginary part of Z. The series is absolutely convergent for $$\mathfrak {R}s>\frac{g+1}{2}$$. It admits a meromorphic continuation to the whole plane and its Laurent coefficients define automorphic forms. Moreover, it is holomorphic at $$s=s_0$$, and if $$\mathcal{{C}}$$ is coherent, then the Siegel–Weil formula holds by [12].

From now on we require that $$m\le g+1$$. Let V be a totally positive definite quadratic space of dimension m over k. We normalize the invariant measure $$\mathrm {d}h$$ on $$\mathrm {O}(V,k)\backslash \mathrm {O}(V,{\mathbb {A}})$$ to have total volume 1 and define the integral

\begin{aligned} I(Z,\varphi )=\int _{\mathrm {O}(V,k)\backslash \mathrm {O}(V,{\mathbb {A}})}\varTheta (Z,h;\varphi )\,\mathrm {d}h \end{aligned}

of the theta function

\begin{aligned} \varTheta (Z,h;\varphi )=\sum _{u\in V(k)^g}\varphi (h^{-1}u){\mathbf {e}}_\infty (\mathrm {tr}((u,u)Z)). \end{aligned}

Under coherent situation, the Siegel–Weil formula can now be stated as follows:

\begin{aligned} E(Z,f_\varphi ^{(s)})|_{s=s_0}=2I(Z,\varphi ). \end{aligned}
(2.2)

The reader who is interested in this identity can consult Theorem 2.2(i) of [31]. On the other hand, if $$\mathcal{{C}}$$ is incoherent, then the series $$E(Z,f_\varphi ^{(s)})$$ has a zero at $$s=s_0$$ by Corollary 5.5 of [31].

Under incoherent situation, consider the Fourier expansions

\begin{aligned} E(Z,f_\varphi ^{(s)})&=\sum _{T\in \mathrm {Sym}_g(k)}A(T,Y,\varphi ,s){\mathbf {e}}_\infty (\mathrm {tr}(TZ)), \\ \frac{\partial }{\partial s}E(Z,f_\varphi ^{(s)})|_{s=s_0}&=\sum _{T\in \mathrm {Sym}_g(k)}C(T,Y,\varphi ){\mathbf {e}}_\infty (\mathrm {tr}(TZ)), \end{aligned}

where

\begin{aligned} Z&=X+\sqrt{-1}Y,&C(T,Y,\varphi )&=\frac{\partial }{\partial s}A(T,Y,\varphi ,s)|_{s=s_0}. \end{aligned}

Put $$\mathrm {Sym}_g^\mathrm {nd}=\mathrm {Sym}_g(k)\cap \mathrm {GL}_g(k)$$. When $$T\in \mathrm {Sym}_g^\mathrm {nd}$$, by Lemma 2.4 of [12] the Fourier coefficient has an explicit expression as an infinite product

\begin{aligned} A(T,Y,\varphi ,s)=a(T,Y,s)\prod _{\mathfrak {p}}W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big ) \end{aligned}
(2.3)

for $$\mathfrak {R}s\gg 0$$, where

\begin{aligned} W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big )=\int _{\mathrm {Sym}_g(k_{\mathfrak {p}})}f_{\varphi _{\mathfrak {p}}}^{(s)}\left( \begin{pmatrix} 0 &{} \mathbf{1}_g \\ -\mathbf{1}_g &{} 0\end{pmatrix}{\mathbf {n}}(z_{\mathfrak {p}})\right) \overline{{\mathbf {e}}_{\mathfrak {p}}(\mathrm {tr}(Tz_{\mathfrak {p}}))}\,\mathrm {d}z_{\mathfrak {p}}\end{aligned}

and $$a(T,Y,s){\mathbf {e}}_\infty (\sqrt{-1}\mathrm {tr}(TY))$$ is a product of the confluent hypergeometric functions investigated in [21]. Given $$T\in \mathrm {Sym}_g^\mathrm {nd}$$, we define the quadratic form on $$V^T=k^g$$ by $$u\mapsto T[u]$$ and define the Hecke character $$\chi ^T=\prod _v\chi ^T_v$$ and the Hasse invariants $$\eta ^T_{\mathfrak {p}}$$, where $$\chi ^T_v$$ is defined in (2.1). Let $$\mathrm {Diff}(T,\mathcal{{C}})$$ denote the set of places v of k such that T is not represented by $$\mathcal{{C}}_v$$. Let $$\mathrm {Sym}_g^+$$ denote the set of totally positive definite symmetric $$g\times g$$ matrices over k.

### Lemma 2.2

Let $$\varphi _{\mathfrak {p}}\in \mathcal{{S}}(\mathcal{{C}}_{\mathfrak {p}}^g)$$ and $$T\in \mathrm {Sym}_g^\mathrm {nd}$$.

1. (1)

a(TYs) and $$W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big )$$ are entire functions in s.

2. (2)

$$\displaystyle \lim _{s\rightarrow s_0}W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big )=0$$ unless T is represented by $$\mathcal{{C}}_{\mathfrak {p}}$$.

3. (3)

If $$m=g$$, $$T\in \mathrm {Sym}_g^+$$, $$\chi ^T=\chi$$ and $$\mathcal{{C}}$$ is incoherent, then $$\mathrm {Diff}(T,\mathcal{{C}})$$ is a finite set of odd cardinality. Here $$\chi$$ is the quadratic character associated to $$\mathcal{{C}}$$ as explained in Definition 2.2.

### Proof

The first part is well-known (see [7, 21]). Lemma on p. 73 of [19] implies (2). By assumption $$\mathrm {Diff}(T,\mathcal{{C}})=\{{\mathfrak {p}}\;|\;\eta ^{\mathcal{{C}}_{\mathfrak {p}}}=-\eta ^T_{\mathfrak {p}}\}$$. Since $$\mathcal{{C}}$$ is incoherent, Lemma 2.1 implies $$\prod _{\mathfrak {p}}\eta ^{\mathcal{{C}}_{\mathfrak {p}}}=-\prod _{\mathfrak {p}}\eta ^T_{\mathfrak {p}}$$, which proves (3). $$\square$$

Let $$T\in \mathrm {Sym}_g^+$$. Then $$a(T,Y,s_0)$$ is independent of Y, and so by Lemma 2.2(2), (3) and the Leibniz rule, $$C(T,Y,\varphi )$$ is also independent of Y. Put

\begin{aligned} c_m(T)&=a(T,Y,s_0),&C(T,\varphi )&=C(T,Y,\varphi ),&D_T&=\mathrm {N}_{k/{\mathbb {Q}}}(\det (2T)). \end{aligned}

Let $${\mathfrak {d}}_k$$ denote the absolute value of the discriminant of k. Note that

\begin{aligned} c_g(T)&=c_g\cdot D_T^{-1/2},&c_g&={\mathfrak {d}}_k^{-g(g+1)/4}\biggl ({\mathbf {e}}\biggl (\frac{g^2}{8}\biggl )\frac{2^g\pi ^{g^2/2}}{\Gamma _g\bigl (\frac{g}{2}\bigl )}\biggl )^d \end{aligned}
(2.4)

by applying (1.21K), (3.15) and (4.34K) of [21] with $$\alpha =\frac{g}{2}$$ and $$\beta =0$$, where

\begin{aligned}\Gamma _g(s)=\pi ^{g(g-1)/4}\prod _{i=0}^{g-1}\Gamma \Big (s-\frac{i}{2}\Big ). \end{aligned}

### Proposition 2.1

Let $$m=g$$ and $$T\in \mathrm {Sym}_g^+$$. Suppose that $$\mathcal{{C}}$$ is incoherent. If $$\chi ^T=\chi$$, then $$C(T,\varphi )=0$$ unless $$\mathrm {Diff}(T,\mathcal{{C}})$$ is a singleton. Moreover, if $$\mathrm {Diff}(T,\mathcal{{C}})=\{{\mathfrak {p}}\}$$, then

\begin{aligned}C(T,\varphi )=c_gD_T^{-1/2}\lim _{s\rightarrow -1/2}\beta ^T(s)\cdot \beta ^T_{\mathfrak {p}}(s)\frac{\partial W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big )}{\partial s}\prod _{{\mathfrak l}\ne {\mathfrak {p}}}\beta ^T_{\mathfrak l}(s)W_T\Big (f_{\varphi _{\mathfrak l}}^{(s)}\Big ), \end{aligned}

where

\begin{aligned} \beta ^T(s)&=\frac{L\bigl (s+\frac{1}{2},\chi ^T\chi \bigl )}{\prod ^{[(g+1)/2]}_{j=1}\zeta (2s+2j-1)}\times {\left\{ \begin{array}{ll} 1 &{}\text { if }2\not \mid g, \\ L\bigl (s+\frac{g+1}{2},\chi \bigl )^{-1} &{}\text { if }2|g, \end{array}\right. }\\ \beta ^T_{\mathfrak {q}}(s)&=\frac{\prod ^{[(g+1)/2]}_{j=1}\zeta _{\mathfrak {q}}(2s+2j-1)}{L\bigl (s+\frac{1}{2},\chi ^T_{\mathfrak {q}}\chi _{\mathfrak {q}}\bigl )}\times {\left\{ \begin{array}{ll} 1 &{}\text { if }2\not \mid g, \\ L\bigl (s+\frac{g+1}{2},\chi _{\mathfrak {q}}\bigl ) &{}\text { if }2|g. \end{array}\right. } \end{aligned}

Note that

\begin{aligned} \beta ^T(s)=\prod _{\mathfrak {q}}\beta ^T_{\mathfrak {q}}(s)^{-1}. \end{aligned}

### Proof

For given $$\varphi$$ and T, let $${\mathfrak {S}}$$ be a finite set of rational primes of k which contains $$\mathrm {Diff}(T,\mathcal{{C}})$$ and such that if $${\mathfrak {q}}\notin {\mathfrak {S}}$$, then $${\mathfrak {q}}$$ does not divide 2, $$\chi _{\mathfrak {q}}$$ is unramified, $${\mathbf {e}}_{\mathfrak {q}}$$ is of order 0, $$T\in \mathrm {GL}_g({\mathfrak {o}}_{\mathfrak {q}})$$, and the restriction of $$f^{(s)}_{\varphi _{\mathfrak {q}}}$$ to $$K_{\mathfrak {q}}$$ is 1. If $${\mathfrak {q}}\notin {\mathfrak {S}}$$, then

\begin{aligned} W_T\bigl (f_{\varphi _{\mathfrak {q}}}^{(s)}\bigl )=\beta ^T_{\mathfrak {q}}(s)^{-1} \end{aligned}
(2.5)

by [22, Proposition 14.9] and [24, Section A1] (cf. Proposition 3.1 and (4.1)). By (2.3) the T-th Fourier coefficient of $$E(Z,f_\varphi ^{(s)})$$ is given by

\begin{aligned} A(T,Y,\varphi ,s)=\beta ^T(s)a(T,Y,s)\prod _{{\mathfrak {q}}\in {\mathfrak {S}}}\beta ^T_{\mathfrak {q}}(s)W_T\Big (f_{\varphi _{\mathfrak {q}}}^{(s)}\Big ). \end{aligned}
(2.6)

Notice that the product $$\beta ^T_{\mathfrak {q}}(s)W_T\Big (f_{\varphi _{\mathfrak {q}}}^{(s)}\Big )$$ is holomorphic at $$s=-\frac{1}{2}$$. Indeed, if $$\chi ^T_{\mathfrak {q}}=\chi _{\mathfrak {q}}$$, then $$\beta ^T_{\mathfrak {q}}(s)$$ is holomorphic at $$s=-\frac{1}{2}$$ while if $$\chi ^T_{\mathfrak {q}}\ne \chi _{\mathfrak {q}}$$, then $$\beta ^T_{\mathfrak {q}}(s)$$ has a simple pole at $$s=-\frac{1}{2}$$, but $$W_T\Big (f_{\varphi _{\mathfrak {q}}}^{(s)}\Big )$$ has a zero at $$s=-\frac{1}{2}$$ by Lemma 2.2(2).

Assume that $$\chi ^T=\chi$$. Then $$\beta ^T(s)$$ is holomorphic and has no zero at $$s=-\frac{1}{2}$$. If $${\mathfrak {q}}\in \mathrm {Diff}(T,\mathcal{{C}})$$, then $$\beta ^T_{\mathfrak {q}}(s)W_T\Big (f_{\varphi _{\mathfrak {q}}}^{(s)}\Big )$$ has a zero at $$s=-\frac{1}{2}$$ by Lemma 2.2(2), which combined with (2.6) proves the first statement. We obtain the desired formula by differentiating (2.6) at $$s=-\frac{1}{2}$$. $$\square$$

### Corollary 2.1

If $$m=g$$, $$\mathcal{{C}}$$ is incoherent and $$T\in \mathrm {Sym}_g^+$$ with $$\chi ^T\ne \chi$$, then

\begin{aligned} C(T,\varphi )=c_gD_T^{-1/2}\lim _{s\rightarrow -1/2}\frac{\partial \beta ^T}{\partial s}(s)\prod _{\mathfrak {p}}\beta ^T_{\mathfrak {p}}(s)W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big ). \end{aligned}

### Proof

Since $$\chi _v^{}=\chi ^T_v=\mathrm {sgn}^{g(g-1)/2}$$ for every $$v\in {\mathfrak {S}}_\infty$$, the factor $$\frac{L(s+\frac{1}{2},\chi ^T\chi )}{\zeta (2s+1)}$$ of $$\beta ^T(s)$$ has a zero at $$s=-\frac{1}{2}$$ unless $$\chi =\chi ^T$$. Therefore $$\beta ^T(s)$$ has a zero at $$s=-\frac{1}{2}$$ if $$\chi \ne \chi ^T$$. We can deduce Corollary 2.1 from (2.6) and the fact that $$W_T\Big (f_{\varphi _{\mathfrak {q}}}^{(s)}\Big )=\beta ^T_{\mathfrak {q}}(s)^{-1}$$ for $${\mathfrak {q}}\notin {\mathfrak {S}}$$. $$\square$$

## 4 Fourier coefficients of derivatives of Eisenstein series

For $$t\in k^\times _v$$ there is an 8th root of unity $$\gamma _v(t)$$ such that for all Schwartz functions $$\phi$$ on $$k_v$$

\begin{aligned} \int _{k_v}\phi (x_v){\mathbf {e}}_v(tx_v^2)\,\mathrm {d}x_v=\gamma _v(t)|2t|_v^{-1/2}\int _{k_v}\mathcal{{F}}\phi (x_v){\mathbf {e}}_v\left( -\frac{x_v^2}{4t}\right) \,\mathrm {d}x_v, \end{aligned}

where $$\mathrm {d}x_v$$ is the self-dual Haar measure on $$k_v$$ with respect to the Fourier transform

\begin{aligned} \mathcal{{F}}\phi (y)=\int _{k_v}\phi (x_v){\mathbf {e}}_v(x_vy)\,\mathrm {d}x_v. \end{aligned}

Put

\begin{aligned} \gamma (\mathcal{{C}}_v)=\varepsilon _v(\mathcal{{C}}_v)\gamma _v\left( \frac{1}{2}\right) ^{m-1}\gamma _v\left( \frac{1}{2}\det \mathcal{{C}}_v\right) . \end{aligned}

Let $$L_{\mathfrak {p}}$$ be an integral lattice of $$\mathcal{{C}}_{\mathfrak {p}}$$, i.e., a finitely generated $${\mathfrak {o}}_{\mathfrak {p}}$$-submodule of $$\mathcal{{C}}_{\mathfrak {p}}$$ which spans $$\mathcal{{C}}_{\mathfrak {p}}$$ over $$k_{\mathfrak {p}}$$ and such that $$(u,u)\in {\mathfrak {o}}_{\mathfrak {p}}$$ for every $$u\in L_{\mathfrak {p}}$$. Note that $$2(u,w)\in {\mathfrak {o}}_{\mathfrak {p}}$$ for all $$u,w\in L_{\mathfrak {p}}$$. Let

\begin{aligned} L_{\mathfrak {p}}^*=\{u\in \mathcal{{C}}_{\mathfrak {p}}\;|\;2(u,w)\in {\mathfrak {o}}_{\mathfrak {p}}\text { for every }w\in L_{\mathfrak {p}}\} \end{aligned}

be its dual lattice. Let $$\mathrm {ch}\langle L_{\mathfrak {p}}^g\rangle \in \mathcal{{S}}(\mathcal{{C}}_{\mathfrak {p}}^g)$$ be the characteristic function of $$L_{\mathfrak {p}}^g$$. We write $$S_{\mathfrak {p}}$$ for the matrix for the quadratic form on $$\mathcal{{C}}_{\mathfrak {p}}$$ with respect to a fixed basis of $$L_{\mathfrak {p}}$$. For nondegenerate symmetric matrices $$T\in \frac{1}{2}\mathcal{{E}}_g({\mathfrak {o}}_{\mathfrak {p}})$$ and $$S\in \frac{1}{2}\mathcal{{E}}_m({\mathfrak {o}}_{\mathfrak {p}})$$ the local density of representing T by S is defined by

\begin{aligned} \alpha _{\mathfrak {p}}(S,T)=\lim _{i\rightarrow \infty }q_{\mathfrak {p}}^{ig((g+1)-2m)/2}A_i(S,T), \end{aligned}

where

\begin{aligned} A_i(S,T)=\sharp \{X\in \mathrm {M}_{m,g}({\mathfrak {o}}/{\mathfrak {p}}^i)\;|\;S[X]-T\in {\mathfrak {p}}^i\mathrm {Sym}_g({\mathfrak {o}}_{\mathfrak {p}})\}. \end{aligned}

Let $${\mathfrak {d}}_{k_{\mathfrak {p}}}$$ denote the norm of the different of $$k_{\mathfrak {p}}$$ over $${\mathbb {Q}}_p$$.

### Proposition 3.1

(cf. [10]) For every non-negative integer r we have

\begin{aligned} \lim _{s\rightarrow r+s_0}W_T\Big (f_{\mathrm {ch}\langle L_{\mathfrak {p}}^g\rangle }^{(s)}\Big )=\frac{\alpha _{\mathfrak {p}}\left( S_{\mathfrak {p}}\perp \frac{1}{2}\begin{pmatrix} &{} \mathbf{1}_r \\ \mathbf{1}_r &{} \end{pmatrix} ,T\right) }{\gamma (\mathcal{{C}}_{\mathfrak {p}})^g{\mathfrak {d}}_{k_{\mathfrak {p}}}^{-g/2}[L_{\mathfrak {p}}^*:L_{\mathfrak {p}}]^{g/2}}. \end{aligned}

Here, $$s_0=\frac{1}{2}(m-g-1)$$ is associated to $$\mathcal{{C}}_{\mathfrak {p}}$$.

### Proof

Since

\begin{aligned} \lim _{s\rightarrow r+s_0}W_T\Big (f_{\mathrm {ch}\langle L_{\mathfrak {p}}^g\rangle }^{(s)}\Big )=W_T\Big (f_{\mathrm {ch}\langle L_{\mathfrak {p}}^g\oplus \mathrm {M}_{2r,g}({\mathfrak {o}}_{\mathfrak {p}})\rangle }^{(s_0)}\Big ), \end{aligned}

this result can be deduced from the proof of [32, Lemma 8.3(2)]. $$\square$$

Let $$\mathcal{{V}}$$ be a totally positive definite quadratic space of dimension g over k. Fix an integral lattice L in $$\mathcal{{V}}$$. Put

\begin{aligned} L_{\mathfrak {p}}&=L\otimes _{\mathfrak {o}}{\mathfrak {o}}_{\mathfrak {p}},&\mathrm {ch}\langle L^g\rangle&=\otimes _{\mathfrak {p}}\mathrm {ch}\langle L^g_{\mathfrak {p}}\rangle . \end{aligned}

For $$h\in \mathrm {O}(\mathcal{{V}},{\mathbb {A}})$$ we write hL for the lattice defined by $$(hL)_{\mathfrak {p}}=h_{\mathfrak {p}}L_{\mathfrak {p}}$$. Put

\begin{aligned} K_L&=\{h\in \mathrm {SO}(\mathcal{{V}},{\mathbb {A}})\;|\; hL=L\},&\mathrm {SO}(L)&=\{h\in \mathrm {SO}(\mathcal{{V}},k)\;|\; hL=L\}. \end{aligned}

### Definition 3.1

We mean by the genus (resp. class) of L the set of all lattices of the form hL with $$h\in \mathrm {O}(\mathcal{{V}},{\mathbb {A}})$$ (resp. $$h\in \mathrm {O}(\mathcal{{V}},k)$$). The proper class of L consists of all lattices of the form hL with $$h\in \mathrm {SO}(\mathcal{{V}},k)$$.

We write $$\Xi '(L)$$ and $$\Xi (L)$$ for the sets of classes and proper classes in the genus of L, respectively. Define the mass of the genus of L by

\begin{aligned} {\mathfrak {m}}'(L)&=\sum _{{\mathscr {L}}\in \Xi '(L)}\frac{1}{\sharp \mathrm {O}({\mathscr {L}})},&{\mathfrak {m}}(L)&=\sum _{{\mathscr {L}}\in \Xi (L)}\frac{1}{\sharp \mathrm {SO}({\mathscr {L}})}. \end{aligned}

### Remark 3.1

For each finite prime $${\mathfrak {p}}$$ there is $$h\in \mathrm {O}(\mathcal{{V}},k_{\mathfrak {p}})$$ with $$\det h=-1$$ such that $$hL_{\mathfrak {p}}=L_{\mathfrak {p}}$$. The genus of L therefore consists of lattices hL with $$h\in \mathrm {SO}(\mathcal{{V}},{\mathbb {A}})$$. We identify $$\Xi (L)$$ with double cosets for $$\mathrm {SO}(\mathcal{{V}},k)\backslash \mathrm {SO}(\mathcal{{V}},{\mathbb {A}})/K_L$$ via the map $$h\mapsto hL$$.

Lemma 5.6(1) of [23] says that

\begin{aligned} {\mathfrak {m}}(L)=2{\mathfrak {m}}'(L). \end{aligned}
(3.1)

We consider the following sums of representation numbers of $$T\in \mathrm {Sym}_g(k)$$:

\begin{aligned} R'(L,T)&=\sum _{{\mathscr {L}}\in \Xi '(L)}\frac{N({\mathscr {L}},T)}{\sharp \mathrm {O}({\mathscr {L}})},&R(L,T)&=\sum _{{\mathscr {L}}\in \Xi (L)}\frac{N({\mathscr {L}},T)}{\sharp \mathrm {SO}({\mathscr {L}})}, \end{aligned}

where $$N(L,T)=\sharp \{u\in L^g\;|\;(u,u)=T\}$$.

### Proposition 3.2

Notation being as above, we have

\begin{aligned} 2\frac{R(L,T)}{{\mathfrak {m}}(L)}=c_gD_T^{-1/2}\lim _{s\rightarrow -1/2}\beta ^T(s)\prod _{\mathfrak {p}}\beta ^T_{\mathfrak {p}}(s)W_T\Big (f_{\mathrm {ch}\langle L_{\mathfrak {p}}^g\rangle }^{(s)}\Big ). \end{aligned}

### Proof

This equality is nothing but the Siegel formula (cf. [27, Satz 2 on p. 555]). Nevertheless we reproduce its proof here because of its importance for us. Since both sides are zero unless $$V^T\simeq \mathcal{{V}}$$ by Lemma 2.2(2), we may identify $$V^T$$ with $$\mathcal{{V}}$$. As is well-known, there exists $$h\in \mathrm {O}(V^T,k_{\mathfrak {p}})$$ such that $$hL_{\mathfrak {p}}=L_{\mathfrak {p}}$$ and $$\det h=-1$$. Since $$\mathrm {SO}(V^T,{\mathbb {A}})\backslash \mathrm {O}(V^T,{\mathbb {A}})=\mu _2({\mathbb {A}})$$, we have

\begin{aligned} I(Z,\mathrm {ch}\langle L^g\rangle )&=\frac{1}{2}\int _{\mathrm {SO}(V^T,k)\backslash \mathrm {SO}(V^T,{\mathbb {A}})}\varTheta (Z,h;\mathrm {ch}\langle L^g\rangle )\,\mathrm {d}h. \end{aligned}

Choose a finite set of double coset representatives $$h_i\in \mathrm {SO}(V^T,{{\mathbb {A}}_{\mathbf{f}}})$$ so that

\begin{aligned} \mathrm {SO}(V^T,{\mathbb {A}})=\bigsqcup _i\mathrm {SO}(V^T,k)h_iK_L. \end{aligned}

Then

\begin{aligned} I(Z,\mathrm {ch}\langle L^g\rangle )=\frac{1}{2}\mathrm {vol}(K_L)\sum _i\frac{\varTheta (Z,h_i;\mathrm {ch}\langle L^g\rangle )}{\sharp \mathrm {SO}(h_iL)}. \end{aligned}

Since $$\{h_iL\}$$ is a complete set of representatives for $$\Xi (L)$$ and since the left coset $$\mathrm {SO}(V^T,k)\backslash \mathrm {SO}(V^T,{\mathbb {A}})$$ can be identified with a disjoint union $$\bigsqcup _i\mathrm {SO}(h_iL)\backslash h_iK_L$$ in view of Remark 3.1, we have

\begin{aligned} 2=\int _{\mathrm {SO}(V^T,k)\backslash \mathrm {SO}(V^T,{\mathbb {A}})}\mathrm {d}h=\mathrm {vol}(K_L)\sum _i\frac{1}{\sharp \mathrm {SO}(h_iL)}=\mathrm {vol}(K_L){\mathfrak {m}}(L). \end{aligned}

The T-th Fourier coefficient of $$I(Z,\mathrm {ch}\langle L^g\rangle )$$ is equal to $$\frac{R(L,T)}{{\mathfrak {m}}(L)}$$. The Siegel–Weil formula (2.2) and (2.4), (2.6) prove the declared identity. $$\square$$

An examination of the proof of Proposition 3.2 confirms that

\begin{aligned} \frac{R(L,T)}{{\mathfrak {m}}(L)}=\frac{R'(L,T)}{{\mathfrak {m}}'(L)}. \end{aligned}
(3.2)

We can prove the following result by combining Propositions 2.1 and 3.2.

### Proposition 3.3

We assume that $$\mathrm {Diff}(T,\mathcal{{C}})=\{{\mathfrak {p}}\}$$, notation and assumption being as in Proposition 2.1. Take an integral lattice L in $$V^T$$ such that

\begin{aligned} \lim _{s=-1/2}W_T\Big (f_{\mathrm {ch}\langle L^g_{\mathfrak {p}}\rangle }^{(s)}\Big )\ne 0. \end{aligned}

If $$\varphi ^{}_{\mathfrak l}=\mathrm {ch}\langle L_{\mathfrak l}^g\rangle$$ for every prime ideal $${\mathfrak l}$$ distinct from $${\mathfrak {p}}$$, then

\begin{aligned} C(T,\varphi )=2\frac{R(L,T)}{{\mathfrak {m}}(L)}\lim _{s\rightarrow -1/2}W_T\Big (f_{\mathrm {ch}\langle L^g_{\mathfrak {p}}\rangle }^{(s)}\Big )^{-1}\frac{\partial W_T\Big (f_{\varphi _{\mathfrak {p}}}^{(s)}\Big )}{\partial s}. \end{aligned}

## 5 Siegel series

In this section we drop the subscript $$_{\mathfrak {p}}$$. Thus k is a nonarchimedean local field of characteristic zero with integer ring $${\mathfrak {o}}$$. We denote the maximal ideal of $${\mathfrak {o}}$$ by $${\mathfrak {p}}$$ and the order of the residue field $${\mathfrak {o}}/{\mathfrak {p}}$$ by q. Fix a prime element $$\varpi$$ of $${\mathfrak {o}}$$. We define the additive order $${\text {ord}}:k^\times \rightarrow {\mathbb {Z}}$$ by $${\text {ord}}(\varpi ^i{\mathfrak {o}}^\times )=i$$.

Let $$T\in \frac{1}{2}\mathcal{{E}}_g({\mathfrak {o}})$$ with $$\det T\ne 0$$. Denote the conductor of $$\chi ^T$$ by $${\mathfrak {d}}^T$$. Put

\begin{aligned} D_T&=(-4)^{[g/2]}\det T, \\ e^T&={\left\{ \begin{array}{ll} {\text {ord}}\,D_T &{}\text { if }g \text { is odd, }\\ {\text {ord}}\,D_T-{\text {ord}}\,{\mathfrak {d}}^T &{}\text { if }g \text { is even, } \end{array}\right. }\\ \xi ^T&={\left\{ \begin{array}{ll} 1 &{}\text { if }D_T\in k^{\times 2}, \\ -1 &{}\text { if }D_T\notin k^{\times 2} \text { and }{\mathfrak {d}}^T={\mathfrak {o}}, \\ 0 &{}\text { if }D_T\notin k^{\times 2} \text { and }{\mathfrak {d}}^T\ne {\mathfrak {o}}. \end{array}\right. } \end{aligned}

Note that $$\chi ^T(\varpi )=\xi ^T$$ if $${\mathfrak {d}}^T={\mathfrak {o}}$$.

The Siegel series associated to T is defined by

\begin{aligned} b(T,s)=\sum _{z\in \mathrm {Sym}_g(k)/\mathrm {Sym}_g({\mathfrak {o}})} \psi (-\mathrm {tr}(T z))\nu [z]^{-s}, \end{aligned}

where $$\nu [z]=[z{\mathfrak {o}}^g+{\mathfrak {o}}^g:{\mathfrak {o}}^g]$$ and $$\psi$$ is an arbitrarily fixed additive character on k which is trivial on $${\mathfrak {o}}$$ but nontrivial on $${\mathfrak {p}}^{-1}$$. By Proposition 14.9 of [22] there exists a polynomial $$A_T(X)\in {\mathbb {Z}}[X]$$ such that $$A_T(q^{-s})=b(T,s)$$. Moreover, this polynomial $$A_T(X)$$ is divisible by the following polynomial

\begin{aligned}\gamma ^T(X) =(1-X)\prod _{j=1}^{[g/2]}(1-q^{2j}X^2)\times {\left\{ \begin{array}{ll} 1 &{}\text { if }g \text { is odd, } \\ \frac{1}{1-\xi ^T q^{g/2}X} &{}\text { if }g \text { is even. } \end{array}\right. } \end{aligned}

Put

\begin{aligned} A_T(X)&=\gamma ^T(X)F^T(X),&{\widetilde{\mathcal{{F}}}}^T(X)&=X^{-e^T/2}F^T(q^{-(g+1)/2}X). \end{aligned}

It is proved in [4, 8] that if g is even, then $${\widetilde{\mathcal{{F}}}}^T\in {\mathbb {Q}}[\sqrt{q}][X, X^{-1}]$$ satisfies the functional equation $${\widetilde{\mathcal{{F}}}}^T(X)={\widetilde{\mathcal{{F}}}}^T(X^{-1})$$.

Let $$\mathcal{{C}}$$ be a g-dimensional quadratic space over k. Recall that S is the matrix for the quadratic form on $$\mathcal{{C}}$$ with respect to a fixed basis of L, where L is an integral lattice of $$\mathcal{{C}}$$ as explained at the beginning of Sect. 3. If g is even, $$\chi =\chi ^\mathcal{{C}}$$ is unramified, $$\det (2S)\in {\mathfrak {o}}^\times$$ and $$\eta ^\mathcal{{C}}=1$$, then Lemma 14.8 combined with Proposition 14.3 of [22] gives

\begin{aligned} \alpha \left( S\perp \frac{1}{2}\begin{pmatrix} &{} \mathbf{1}_r \\ \mathbf{1}_r &{} \end{pmatrix} ,T\right) =A_T(\chi (\varpi )q^{-(g+2r)/2}). \end{aligned}
(4.1)

For the rest of this paper we require g to be even.

### Proposition 4.1

If g is even, $$\chi$$ is unramified, $$\chi ^T=\chi$$, $$\eta ^T=-1$$, $$\eta ^\mathcal{{C}}=1$$ and L is a self-dual lattice of $$\mathcal{{C}}$$, then

\begin{aligned}\frac{\partial }{\partial s}W_T\Big (f_{\mathrm {ch}\langle L^g\rangle }^{(s)}\Big )\Big |_{s=-1/2} =-\frac{\sqrt{{\mathfrak {d}}_k}^g\log q}{\gamma (\mathcal{{C}})^g}\frac{\xi ^T}{\sqrt{q}^g}\gamma ^T\biggl (\frac{\xi ^T}{\sqrt{q}^g}\biggl )\frac{\partial F^T}{\partial X}\biggl (\frac{\xi ^T}{\sqrt{q}^g}\biggl ). \end{aligned}

### Proof

The formula is analogous to Proposition A.6 of [10], which deals with the case $$\dim \mathcal{{C}}=g+1$$. Recall that we use the additive character $${\mathbf {e}}_p\circ \mathrm {Tr}^k_{{\mathbb {Q}}_p}$$ on k, where p is the residual characteristic of k. Let $$\delta _k$$ be the different of k over $${\mathbb {Q}}_p$$. The factor

\begin{aligned} {\mathfrak {d}}_k^{g/2}=[L:\delta _k L]^{-g/2}[\mathcal{{E}}_g({\mathfrak {o}}):\delta _k\mathcal{{E}}_g({\mathfrak {o}})] \end{aligned}

intervenes the formula. Though the proof is similar, we will produce it here. Let $$\varphi =\mathrm {ch}\langle L^g\rangle$$. Since $$\chi (\varpi )=\xi ^T$$ and $$s_0=-\frac{1}{2}$$ by assumption, we combine Proposition 3.1 and (4.1) with Lemmas A.2-A.3 of [10] to see that

\begin{aligned} W_T\Big (f_\varphi ^{(s)}\Big )&=\gamma (\mathcal{{C}})^{-g}\sqrt{{\mathfrak {d}}_k}^g A_T\Big (\xi ^T q^{-(g+1+2s)/2}\Big ) \\&=\gamma (\mathcal{{C}})^{-g}\sqrt{{\mathfrak {d}}_k}^g\gamma ^T\Big (\xi ^T q^{-(g+1+2s)/2}\Big )F^T\Big (\xi ^T q^{-(g+1+2s)/2}\Big ). \end{aligned}

By assumption $$\displaystyle \lim _{s\rightarrow -1/2}W_T(f_\varphi ^{(s)})=0$$ in view of Lemma 2.2(2). Since $$\chi ^T=\chi$$, we have $$\gamma ^T\bigl (\xi ^Tq^{-g/2}\bigl )\ne 0$$ and hence we see that $$F^T(\xi ^T q^{-g/2})=0$$. We can obtain the stated identity by differentiating this equality at $$s=-\frac{1}{2}$$. $$\square$$

### Definition 4.1

Let $$T=(t_{ij})\in \frac{1}{2}\mathcal{{E}}_g({\mathfrak {o}})\cap \mathrm {GL}_g(k)$$. We denote by S(T) the set of all nondecreasing sequences $$(a_1,\dots ,a_g)$$ of nonnegative integers such that $${\text {ord}}t_{ii}\ge a_i$$ and $${\text {ord}}(2t_{ij})\ge \frac{a_i+a_j}{2}$$ for $$1\le i,j\le g$$. The Gross–Keating invariant $$\mathrm {GK}(T)$$ of T is the greatest element of $$\bigcup _{U\in \mathrm {GL}_g({\mathfrak {o}})}S(T[U])$$ with respect to the lexicographic order.

Here, the lexicographic order is defined as follows: $$(y_1,\dots ,y_g)$$ is greater than $$(z_1,\dots ,z_g)$$ if there is an integer $$1\le j\le g$$ such that $$y_i=z_i$$ for $$i<j$$ and $$y_j>z_j$$. If q is odd, then it is easy to compute $$\mathrm {EGK}(T)$$; the formula with odd q will be given in the paragraph just before Theorem 4.1.

Ikeda and Katsurada [6] define a set $$\mathrm {EGK}(T)$$ of invariants of T attached to $$\mathrm {GK}(T)$$, which they call the extended Gross–Keating datum of T. They associated to an extended Gross–Keating datum H a polynomial

\begin{aligned} {\widetilde{\mathcal{{F}}}}(H;Y,X)\in {\mathbb {Z}}[Y,Y^{-1},X^{1/2},X^{-1/2}] \end{aligned}

and show that

\begin{aligned} {\widetilde{\mathcal{{F}}}}(\mathrm {EGK}(T);\sqrt{q},X)={\widetilde{\mathcal{{F}}}}^T(X). \end{aligned}

When g is even and $${\mathfrak {d}}^T={\mathfrak {o}}$$, one can associate to $$\mathrm {EGK}(T)$$ truncated extended Gross–Keating datum $$\mathrm {EGK}(T)'$$ of length $$g-1$$ by Proposition 4.4 of [6]. By Definitions 4.2-4.4 of [6]

\begin{aligned} {\widetilde{\mathcal{{F}}}}(\mathrm {EGK}(T);Y,X)=&Y^{{\mathfrak {e}}'/2}X^{-({\mathfrak {e}}-{\mathfrak {e}}'+2)/2}\frac{1-\xi ^T Y^{-1}X}{X^{-1}-X}{\widetilde{\mathcal{{F}}}}(\mathrm {EGK}(T)';Y,YX)\\&+Y^{{\mathfrak {e}}'/2}X^{({\mathfrak {e}}-{\mathfrak {e}}'+2)/2}\frac{1-\xi ^T Y^{-1}X^{-1}}{X-X^{-1}}{\widetilde{\mathcal{{F}}}}(\mathrm {EGK}(T)';Y,YX^{-1}), \end{aligned}

where $$\mathrm {GK}(T)=(a_1,\cdots ,a_g)$$, $${\mathfrak {e}}=2\left[ \frac{a_1+\cdots +a_g}{2}\right]$$ and $${\mathfrak {e}}'=a_1+\cdots +a_{g-1}$$. It is worth noting that since $${\mathfrak {d}}^T={\mathfrak {o}}$$, we have $${\mathfrak {e}}=a_1+\cdots +a_g=e^T$$. We put

\begin{aligned} F^H(X)=(q^{(g+1)/2}X)^{{\mathfrak {e}}/2}{\widetilde{\mathcal{{F}}}}(H;\sqrt{q}, q^{(g+1)/2}X). \end{aligned}

Then

\begin{aligned} F^{\mathrm {EGK}(T)}(X)=(q^{(g+1)/2}X)^{e^T/2}{\widetilde{\mathcal{{F}}}}^T(q^{(g+1)/2}X)=F^T(X). \end{aligned}
(4.2)

If q is odd, then T is equivalent to a diagonal matrix $$\mathrm {diag}[t_1,\cdots ,t_g]$$ with $${\text {ord}}\,t_1\le \cdots \le {\text {ord}}t_g$$ and the (naive) extended Gross–Keating datum $$\mathrm {EGK}(T)=(a_1,\cdots ,a_g;\varepsilon _1,\dots ,\varepsilon _g)$$ is given by

\begin{aligned} a_i&={\text {ord}}\,t_i,&T^{(i)}&=\mathrm {diag}[t_1,\cdots ,t_i],&\varepsilon _i&={\left\{ \begin{array}{ll} \eta ^{T^{(i)}} &{}\text { if }i \text { is odd, }\\ \xi ^{T^{(i)}} &{}\text { if }i \text { is even } \end{array}\right. } \end{aligned}

and $$\mathrm {EGK}(T)'=(a_1,\cdots ,a_{g-1};\varepsilon _1,\dots ,\varepsilon _{g-1})$$.

### Theorem 4.1

Assume that g is even and that $${\mathfrak {d}}^T={\mathfrak {o}}$$. Then

\begin{aligned} F^H(\xi ^Tq^{-g/2})=q^{e^T/2}F^{H'}(\xi ^Tq^{-g/2}), \end{aligned}

where we put $$H=\mathrm {EGK}(T)$$ and $$H'=\mathrm {EGK}(T)'$$. If $$\eta ^T=-1$$, then

\begin{aligned}\frac{\xi ^T}{\sqrt{q}^g}\frac{\partial F^H}{\partial X}\biggl (\frac{\xi ^T}{\sqrt{q}^g}\biggl ) =\frac{F^{H'}(\xi ^T q^{(2-g)/2})}{q-1} -\sqrt{q}^{e^T}\frac{\xi ^T}{\sqrt{q}^g}\frac{\partial F^{H'}}{\partial X}\biggl (\frac{\xi ^T}{\sqrt{q}^g}\biggl ). \end{aligned}

### Proof

Substituting $$Y=\sqrt{q}$$ into $${\widetilde{\mathcal{{F}}}}(H;Y,X)$$, we get

\begin{aligned} {\widetilde{\mathcal{{F}}}}(H;\sqrt{q},X) =&X^{-({\mathfrak {e}}+2)/2}\frac{1-\xi ^Tq^{-1/2}X}{X^{-1}-X}(\sqrt{q}X)^{{\mathfrak {e}}'/2}{\widetilde{\mathcal{{F}}}}(H';\sqrt{q},\sqrt{q}X)\\&+X^{({\mathfrak {e}}+2)/2}\frac{1-\xi ^Tq^{-1/2}X^{-1}}{X-X^{-1}}(\sqrt{q}X^{-1})^{{\mathfrak {e}}'/2}{\widetilde{\mathcal{{F}}}}(H';\sqrt{q},\sqrt{q}X^{-1})\\ =&X^{-(e^T+2)/2}\frac{1-\xi ^Tq^{-1/2}X}{X^{-1}-X}F^{H'}(q^{(1-g)/2}X)\\&+X^{(e^T+2)/2}\frac{1-\xi ^Tq^{-1/2}X^{-1}}{X-X^{-1}}F^{H'}(q^{(1-g)/2}X^{-1}). \end{aligned}

By letting $$X=\xi ^T\sqrt{q}$$, we get

\begin{aligned} (\xi ^T\sqrt{q})^{-e^T/2}F^H(\xi ^Tq^{-g/2})={\widetilde{\mathcal{{F}}}}(H;\sqrt{q},\xi ^T\sqrt{q})=(\xi ^T\sqrt{q})^{e^T/2}F^{H'}(\xi ^Tq^{-g/2}). \end{aligned}

In the proof of Proposition 4.1 we have seen by (4.2) that if $$\eta ^T=-1$$, then

\begin{aligned} F^H(\xi ^Tq^{-g/2})=F^T(\xi ^Tq^{-g/2})=0 \end{aligned}

and hence $$F^{H'}(\xi ^Tq^{-g/2})=0$$. We can prove the stated identity by differentiating the equality above at $$X=\xi ^T\sqrt{q}$$. $$\square$$

We will use the following result in the next section.

### Lemma 4.1

If T is a split symmetric half-integral matrix of size 4 over $${\mathbb {Z}}_p$$, namely, $$\chi ^T=1$$ and $$\eta ^T=1$$, then there exists a nondegenerate isotropic symmetric half-integral matrix B of size 3 over $${\mathbb {Z}}_p$$ such that $$F^B(X)=F^{\mathrm {EGK}(T)'}(X)$$.

### Proof

If $$p=2$$, then the existence of B with $$\mathrm {EGK}(B)=\mathrm {EGK}(T)'$$ follows from Proposition 6.4 of [5]. If p is odd, then T is equivalent to a diagonal matrix $$\mathrm {diag}[t_1,t_2,t_3,t_4]$$ with $${\text {ord}}t_1\le {\text {ord}}t_2\le {\text {ord}}t_3\le {\text {ord}}t_4$$. Then we may choose B as $$\mathrm {diag}[t_1,t_2,t_3]$$ by using the argument explained in the paragraph just before Theorem 4.1 so that $$\mathrm {EGK}(B)=\mathrm {EGK}(T)'$$. Now we have $$F^B(X)=F^{\mathrm {EGK}(B)}(X)=F^{\mathrm {EGK}(T)'}(X)$$ by (4.2). $$\square$$

## 6 Proofs of the main results

We discuss the classical Eisenstein series of Siegel. For this it is simplest to work over $$k={\mathbb {Q}}$$. For the moment we let g be a multiple of 4. Consider the series

\begin{aligned} E_g(Z,s)=\sum _{\{C,D\}}\det (CZ+D)^{-g/2}|\det (CZ+D)|^{-(2s+1)/2}(\det Y)^{(2s+1)/4}. \end{aligned}

Here the sum extends over all symmetric coprime pairs modulo $$\mathrm {GL}_g({\mathbb {Z}})$$. Let $$\mathcal{{C}}_p=\mathcal{{H}}({\mathbb {Q}}_p)^{g/2}$$ be the split quadratic space of dimension g over $${\mathbb {Q}}_p$$. Define $$\varphi =\otimes _p\varphi _p$$ by taking $$\varphi _p=\mathrm {ch}\langle \mathrm {M}_{g,g}({\mathbb {Z}}_p)\rangle \in \mathcal{{S}}(\mathcal{{C}}_p^g)$$. It is known that

\begin{aligned} E_g(Z,s)=E(Z,f_\varphi ^{(s)}) \end{aligned}

(see §IV.2 of [11]). We say that the series is incoherent if the collection $${\mathcal {C}}$$ of local quadratic spaces defining the Eisenstein series is incoherent. The series is incoherent if and only if $$\frac{g}{4}$$ is odd due to Lemma 2.1.

Fix a positive definite symmetric half-integral matrix T of size g. Recall that $$\chi _T$$ stands for the primitive Dirichlet character corresponding to $$\chi ^T$$. Since the series $$E_g(Z,s)$$ has level 1, the character $$\chi$$ is trivial and $$\beta ^T_p(s)=\gamma ^T(p^{-(g+1+2s)/2})^{-1}$$. We see that

\begin{aligned} \beta ^T_p(s)W_T(f_\varphi ^{(s)}) =\gamma (\mathcal{{C}})^{-g}F^T_p\Big (q^{-(g+1+2s)/2}\Big ) \end{aligned}
(5.1)

as in the proof of Proposition 4.1. Recall that $$F^T_p(X)=1$$ if p and $$D_T$$ are coprime. By (2.6) the T-th Fourier coefficient of $$E_g(Z,s)$$ is given by

\begin{aligned} A(T,Y,s)=\frac{a(T,Y,s)L\bigl (s+\frac{1}{2},\chi _T\bigl )}{\zeta \bigl (s+\frac{g+1}{2}\bigl )\prod _{i=1}^{g/2}\zeta (2s+2i-1)}\prod _{p|D_T}F_p^T(p^{-(2s+g+1)/2}). \end{aligned}

The T-th Fourier coefficient of $$\frac{\partial }{\partial s}E_g(Z,s)|_{s=-1/2}$$ is given by

\begin{aligned} C_g(T)=\frac{\partial }{\partial s}A(T,Y,s)|_{s=-1/2}. \end{aligned}

Recall that $$\mathrm {Diff}(T)=\{p\;|\;\eta ^T_p=-1\}$$.

### Proposition 5.1

Assume that $$\frac{g}{4}$$ is odd. Let $$T\in \frac{1}{2}\mathcal{{E}}_g({\mathbb {Z}})\cap \mathrm {Sym}_g^+$$.

1. (1)

If $$\chi _T=1$$, then $$C_g(T)=0$$ unless $$\mathrm {Diff}(T)$$ is a singleton.

2. (2)

If $$\chi _T=1$$ and $$\mathrm {Diff}(T)=\{p\}$$, then

\begin{aligned}C_g(T)=-\frac{2^{(g+2)/2}p^{-(g+e^T_p)/2}\log p}{\zeta \bigl (1-\frac{g}{2}\bigl )\prod _{i=1}^{(g-2)/2}\zeta (1-2i)}\frac{\partial F^T_p}{\partial X}(p^{-g/2})\prod _{p\ne \ell |D_T}\ell ^{-e^T_\ell /2}F_\ell ^T(\ell ^{-g/2}). \end{aligned}
3. (3)

If $$\chi _T\ne 1$$, then

\begin{aligned}C_g(T)=-\frac{2^{(g+2)/2}L(1,\chi _T)}{\zeta \bigl (1-\frac{g}{2}\bigl )\prod _{i=1}^{(g-2)/2}\zeta (1-2i)}\prod _{p|D_T}p^{-e^T_p/2}F_p^T(p^{-g/2}). \end{aligned}

### Proof

We have already proved (1) in Proposition 2.1. Taking

\begin{aligned} \zeta (2i)=(-1)^i\frac{(2\pi )^{2i}}{2(2i-1)!}\zeta (1-2i) \end{aligned}

into account, we have

\begin{aligned} \zeta \biggl (\frac{g}{2}\biggl )\prod _{i=1}^{(g-2)/2}\zeta (2i)=\frac{(2\pi )^{g^2/4}\zeta \bigl (1-\frac{g}{2}\bigl )}{2^{g/2}\bigl (\frac{g}{2}-1\bigl )!}\prod _{i=1}^{(g-2)/2}\frac{\zeta (1-2i)}{(2i-1)!} \end{aligned}

Recall that $$a\bigl (T,Y,-\frac{1}{2}\bigl )=\frac{2^g\pi ^{g^2/2}}{\Gamma _g(\frac{g}{2})D_T^{1/2}}$$ by (2.4). Since

\begin{aligned} \varGamma _g\biggl (\frac{g}{2}\biggl )&=\frac{\pi ^{g^2/4}}{2^{(g^2-2g)/4}}\prod _{i=1}^{(g-2)/2}(2i)!,&\zeta (0)&=-\frac{1}{2},&L'(0,\chi _T)&=\frac{\sqrt{{\mathfrak {d}}^T}}{2}L(1,\chi _T), \end{aligned}

we get (2) and (3) from Proposition 2.1, Corollary 2.1 and (5.1). Here $$L(0,\chi _T)=0$$ since the positivity of $$D_T$$ yields $$\chi _T(-1)=1$$. $$\square$$

Now we let $$g=4$$. By a quaternion algebra over a field k we mean a central simple algebra over k of dimension 4. Let $${\mathbb {B}}_p$$ denote the definite quaternion algebra over $$k={\mathbb {Q}}$$ that ramifies only at a prime number p. The reduced norm $$\mathrm {Nrd}$$ on $${\mathbb {B}}_p$$ defines a positive definite quadratic space $$\mathcal{{V}}_p$$. Fix a maximal order $$\mathcal{{O}}_p$$ of $${\mathbb {B}}_p$$. Let $$\varphi _\ell \in \mathcal{{S}}(\mathcal{{C}}_\ell ^4)$$ be the characteristic function of $$\mathrm {M}_2({\mathbb {Z}}_\ell )^4$$ and $$\varphi _p'\in \mathcal{{S}}(\mathcal{{V}}^4_p({\mathbb {Q}}_p))$$ the characteristic function of $$\mathcal{{O}}_p^4\otimes {\mathbb {Z}}_p$$. We regard $$\varphi '=\varphi '_p\otimes (\otimes _{\ell \ne p}\varphi _\ell )$$ as the characteristic function of $$\mathcal{{O}}_p^4\otimes {\hat{{\mathbb {Z}}}}$$. We write $$S_p$$ for the matrix representation of $$\mathcal{{V}}_p$$ with respect to a $${\mathbb {Z}}$$-basis of $$\mathcal{{O}}_p$$. Put

\begin{aligned} S_0=\mathrm {diag}\biggl [\begin{pmatrix} 0 &{} \frac{1}{2} \\ \frac{1}{2} &{} 0\end{pmatrix}, \begin{pmatrix} 0 &{} \frac{1}{2} \\ \frac{1}{2} &{} 0\end{pmatrix}\biggl ]. \end{aligned}

### Lemma 5.1

Let $$T\in \mathrm {Sym}_4({\mathbb {Q}}_p)$$.

1. (1)

If $$T\notin \frac{1}{2}\mathcal{{E}}_4({\mathbb {Z}}_p)$$, then $$W_T\Big (f_{\varphi _p}^{(s)}\Big )$$ is identically zero.

2. (2)

If $$T\in \frac{1}{2}\mathcal{{E}}_4({\mathbb {Z}}_p)$$ with $$\det T\ne 0$$, $$\chi ^T=1$$ and $$\eta ^T_p=-1$$, then

\begin{aligned}\lim _{s\rightarrow -1/2}\frac{W_{S_p}\Big (f_{\varphi _p'}^{(s)}\Big )}{W_T\Big (f_{\varphi _p'}^{(s)}\Big )}\frac{\frac{\partial }{\partial s}W_T\Big (f_{\varphi _p}^{(s)}\Big )}{pW_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )}=\biggl (p^{-2}\frac{\partial F^{H'}_p}{\partial X}(p^{-2})-\frac{p^{-e^T_p/2}}{p-1}F_p^{H'}(p^{-1})\biggl )\log p, \end{aligned}

where we put $$H'=\mathrm {EGK}_p(T)'$$.

### Proof

The first part is trivial. Since

\begin{aligned} \alpha _p(S_p,T)=p^{(e^T_p-2)/2}\alpha _p(S_p,S_p) \end{aligned}

by Hilfssatz 17 of [27], it follows from Proposition 3.1 that

\begin{aligned} \lim _{s\rightarrow -1/2}\frac{W_{S_p}\Big (f_{\varphi _p'}^{(s)}\Big )}{W_T\Big (f_{\varphi _p'}^{(s)}\Big )}=p^{-(e^T_p-2)/2}. \end{aligned}

On the other hand, since $$\xi _p^{S_0}=1$$ and $$F_p^{S_0}(X)=1$$, we have

\begin{aligned} W_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )=\gamma (\mathcal{{H}}({\mathbb {Q}}_p)^2)^{-4}\gamma ^{S_0}_p(p^{-(5+2s)/2}) \end{aligned}

by Proposition 3.1 and (4.1), where $$\mathcal{{H}}({\mathbb {Q}}_p)^2$$ is the split quaternary quadratic space over $${\mathbb {Q}}_p$$. It is a special case of Proposition 4.1 that

\begin{aligned}\frac{\partial }{\partial s}W_T\Big (f_{\varphi _p}^{(s)}\Big )\Big |_{s=-1/2} =-\frac{\log p}{\gamma (\mathcal{{C}}_p)^4}p^{-2}\gamma ^T_p(p^{-2})\frac{\partial F_p^T}{\partial X}(p^{-2}). \end{aligned}

Since $$\gamma ^{S_0}_p(X)=\gamma ^T_p(X)$$ and $$\gamma (\mathcal{{C}}_p)^2=\gamma (V_p)^2$$ by definition, we get

\begin{aligned} \lim _{s\rightarrow -1/2}\frac{\frac{\partial }{\partial s}W_T\Big (f_{\varphi _p}^{(s)}\Big )}{W_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )} =-p^{-2}\frac{\partial F_p^T}{\partial X}(p^{-2})\log p. \end{aligned}

Theorem 4.1 now gives

\begin{aligned}\lim _{s\rightarrow -1/2}\frac{\frac{\partial }{\partial s}W_T\Big (f_{\varphi _p}^{(s)}\Big )}{W_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )} =\biggl (p^{(e^T_p-4)/2}\frac{\partial F^{H'}_p}{\partial X}(p^{-2})-\frac{F_p^{H'}(p^{-1})}{p-1}\biggl )\log p. \end{aligned}

These complete our proof. $$\square$$

Let $${\bar{{\mathbb {F}}}}_p$$ be an algebraic closure of a finite field $${\mathbb {F}}_p$$ with p elements. For two supersingular elliptic curves $$E,E'$$ over $${\bar{{\mathbb {F}}}}_p$$ we consider the free $${\mathbb {Z}}$$-module $$\mathrm {Hom}(E',E)$$ of homomorphisms $$E'\rightarrow E$$ over $${\bar{{\mathbb {F}}}}_p$$ together with the quadratic form given by the degree. As E and $$E'$$ are supersingular, $$\mathrm {Hom}(E',E)$$ has rank 4 as a $${\mathbb {Z}}$$-module. For two quadratic spaces over $${\mathbb {Z}}$$ we write $$N(L,L')$$ for the number of isometries $$L'\rightarrow L$$.

We are now ready to prove our main result.

### Theorem 5.1

If $$T\in \frac{1}{2}\mathcal{{E}}_4({\mathbb {Z}})$$ is positive definite, $$\chi _T=1$$ and $$\mathrm {Diff}(T)$$ consists of a single prime p, then

\begin{aligned} C_4(T)=2^6\cdot 3^2\biggl (p^{-2}\frac{\partial F^{H'}_p}{\partial X}(p^{-2})-\frac{F_p^{H'}(p^{-1})}{\sqrt{p}^{e^T_p}(p-1)}\biggl )\log p\sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T)}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}, \end{aligned}

where we put $$H'=\mathrm {EGK}_p(T)'$$ and where $$(E',E)$$ extends over all pairs of isomorphism classes of supersingular elliptic curves over $${\bar{{\mathbb {F}}}}_p$$.

### Proof

Proposition 3.3 and (3.2) applied to $$L=\mathcal{{O}}_p$$ gives

\begin{aligned} C_4(T)&=R'(\mathcal{{O}}_p,T)c\lim _{s\rightarrow -1/2}\frac{W_{S_p}\Big (f_{\varphi _p'}^{(s)}\Big )}{W_T\Big (f_{\varphi _p'}^{(s)}\Big )}\frac{\frac{\partial }{\partial s}W_T\Big (f_{\varphi _p}^{(s)}\Big )}{pW_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )}, \end{aligned}

where

\begin{aligned} c=\frac{2p}{{\mathfrak {m}}'(\mathcal{{O}}_p)}\lim _{s\rightarrow -1/2}\frac{W_{S_0}\Big (f_{\varphi _p}^{(s)}\Big )}{W_{S_p}\Big (f_{\varphi _p'}^{(s)}\Big )}. \end{aligned}

If $$T=S_p$$, then we claim that $$R'(\mathcal{{O}}_p,S_p)=1$$. To prove this, it suffices to show that $$N({\mathscr {L}},S_p)=0$$ if $${\mathscr {L}}$$ is not isometric to $$\mathcal{{O}}_p$$ and $$N(\mathcal{{O}}_p,S_p)=\sharp \mathrm {O}(\mathcal{{O}}_p)$$, where $${\mathscr {L}}\in \Xi '(\mathcal{{O}}_p)$$. If $$N({\mathscr {L}}, S_p)\ne 0$$, then there is an injection $$f:\mathcal{{O}}_p \rightarrow {\mathscr {L}}$$ as a lattice preserving the associated quadratic forms. Thus we only need to show that f is surjective. If it is not surjective, then $${\mathscr {L}}$$ and $$\mathcal{{O}}_p$$ have different discriminant, which is a contradiction to the assumption that $${\mathscr {L}}$$ and $$\mathcal{{O}}_p$$ are in the same genus.

Note that $$R(\mathcal{{O}}_p,T)=2R'(\mathcal{{O}}_p,T)$$ by (3.1) and (3.2), and

\begin{aligned} R(\mathcal{{O}}_p,T)=\sum _{{\mathscr {L}}\in \Xi (\mathcal{{O}}_p)}\frac{N({\mathscr {L}},T)}{\sharp \mathrm {SO}({\mathscr {L}})}=\sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T)}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')} \end{aligned}
(5.2)

by Proposition 4.1 of [29]. Our statement follows from Lemma 5.1(2) and the fact that $$c=2^7\cdot 3^2$$. Applying the Siegel formula stated in Proposition 3.2, (2.5) and (3.2) to $$T=S_p$$, we get the formula

\begin{aligned} \frac{2}{{\mathfrak {m}}'(\mathcal{{O}}_p)}=c_4D_{S_p}^{-1/2}\lim _{s\rightarrow -1/2}\beta ^T(s)\cdot \beta ^T_p(s)W_{S_p}\Big (f_{\varphi _p'}^{(s)}\Big ) \end{aligned}

It follows that

\begin{aligned} c&=pc_4D_{S_p}^{-1/2}\lim _{s\rightarrow -1/2}\prod _\ell W_{S_0}\Big (f_{\varphi _\ell }^{(s)}\Big )\\&=c_4\lim _{s\rightarrow -1/2}\prod _\ell \gamma ^S_\ell (\ell ^{-(5+2s)/2})=\frac{c_4}{\zeta (2)^2}\lim _{s\rightarrow -1/2}\frac{\zeta \bigl (s+\frac{1}{2}\bigl )}{\zeta (2s+1)}=2^7\cdot 3^2 \end{aligned}

as claimed. $$\square$$

We temporarily let g be an arbitrary multiple of 4 in the following conjecture:

### Conjecture 5.1

Let $$\mathcal{{V}}$$ be a totally positive definite quadratic space over a totally real number field k of dimension g. Fix a maximal integral lattice L of $$\mathcal{{V}}$$. Let $$T\in \frac{1}{2}\mathcal{{E}}_g({\mathfrak {o}})$$ be totally positive definite. If g is even and $$\chi ^\mathcal{{V}}=1$$, then there is a totally positive definite matrix $$T'\in \frac{1}{2}\mathcal{{E}}_{g-1}({\mathfrak {o}})$$ such that

\begin{aligned} R(L,T)=2R(L,T'). \end{aligned}

### Proposition 5.2

If $$k={\mathbb {Q}}$$ and $$g=4$$, then Conjecture 5.1 is true.

### Proof

Since $$R(L,T)=0$$ unless $$\chi ^T=1$$ and $$\mathrm {Diff}(T)=\mathrm {Diff}(\mathcal{{V}})$$, we assume that

\begin{aligned} \chi ^T&=1,&\mathrm {Diff}(T)&=\mathrm {Diff}(\mathcal{{V}}). \end{aligned}

Lemma 4.1 gives $$T'_p\in \frac{1}{2}\mathcal{{E}}_3({\mathbb {Z}}_p)$$ such that $$F^{T'_p}_p=F^{\mathrm {EGK}_p(T)'}_p$$ for every rational prime p. In addition, the proof of Lemma 4.1 yields that $$T'_p$$ is unimodular for almost all primes p. Thus we can find a positive rational number $$0<\delta \in {\mathbb {Q}}^\times$$ such that $$\delta ^{-1}\det T'_p\in {\mathbb {Z}}_p^\times$$ for every $$p\notin \mathrm {Diff}(\mathcal{{V}})$$. For $$p\in \mathrm {Diff}(\mathcal{{V}})$$ we fix an arbitrary anisotropic ternary quadratic form $$T_p'$$ over $${\mathbb {Z}}_p$$. Recall that $$\alpha _p(S_p,T_p')$$ is independent of the choice of $$T'_p$$.

Since $$F^{uT'_p}_p=F^{T'_p}_p$$ for $$u\in {\mathbb {Z}}_p^\times$$, there is no harm in assuming that $$\delta =\det T'_p$$. Since $$\eta ^{T'_p}_p=1$$ for $$p\notin \mathrm {Diff}(\mathcal{{V}})$$, the Minkowski-Hasse theorem (cf. Lemma 2.1) gives $$z\in \mathrm {Sym}_3({\mathbb {Q}})$$ which is positive definite and such that $$z\in T'_p[\mathrm {GL}_3({\mathbb {Q}}_p)]$$ for every p. Take $$A\in \mathrm {GL}_3({{\mathbb {A}}_{\mathbf{f}}})$$ so that $$z=T'_p[A_p]$$ for every p. We can take $$D\in \mathrm {GL}_3({\mathbb {Q}})$$ in such a way that $$AD^{-1}\in \mathrm {GL}_3({\mathbb {Z}}_p)$$ for every p. Put $$T'=z[D^{-1}]$$. Then $$T'\in T'_p[\mathrm {GL}_3({\mathbb {Z}}_p)]$$ for every p. In particular, $$T'\in \frac{1}{2}\mathcal{{E}}_3({\mathbb {Z}})$$.

In view of (3.2) it suffices to show that

\begin{aligned} \frac{R'(L,T)}{{\mathfrak {m}}'(L)}=2\frac{R'(L,T')}{{\mathfrak {m}}'(L)}. \end{aligned}

Since the product $$\prod _p\zeta _p(s)$$ is absolutely convergent for $$\mathfrak {R}s>1$$, we can use $$\frac{\zeta _p(2s+1)}{\zeta _p\bigl (s+\frac{1}{2}\bigl )}$$ instead of $$\beta ^T_p(s)$$ as a convergence factor, and Proposition 3.2 gives

\begin{aligned} 2\frac{R(L,T)}{{\mathfrak {m}}(L)}=c_4D_T^{-1/2}\lim _{s\rightarrow -1/2}\frac{\zeta \bigl (s+\frac{1}{2}\bigl )}{\zeta (2s+1)}\prod _p \frac{\zeta _p(2s+1)}{\zeta _p\bigl (s+\frac{1}{2}\bigl )}W_T\Big (f_{\mathrm {ch}\langle L_p^g\rangle }^{(s)}\Big ). \end{aligned}

Since $$\frac{R'(L,T)}{{\mathfrak {m}}'(L)}=\frac{R(L,T)}{{\mathfrak {m}}(L)}$$ by (3.2),

\begin{aligned} \frac{R'(L,T)}{{\mathfrak {m}}'(L)}&=2^{-1}c_4D_T^{-1/2}\prod _p\lim _{s\rightarrow -1/2} \frac{W_T\bigl (f_{\mathrm {ch}\langle L_p^g\rangle }^{(s)}\bigl )}{2}\\&=2^4\pi ^4D_T^{-1/2}\prod _{p\in \mathrm {Diff}(\mathcal{{V}})} \frac{\alpha _p(S_p,T)}{2[L_p^*:L_p]^2}\prod _{q\notin \mathrm {Diff}(\mathcal{{V}})}(1-q^{-2})^2F_q^T(q^{-2}) \end{aligned}

by (2.4) and Proposition 3.1.

Recall that the nonarchimedean densities for $$p\in \mathrm {Diff}(\mathcal{{V}})$$ are given by

\begin{aligned} \alpha _p(S_p,T')&=2(p+1)(1+p^{-1}),&\alpha _p(S_p,T)=4p^{e^T_p/2}(p+1)^2 \end{aligned}

by [30, Theorem 1.1] and Proposition 5.7 of [1]. Since $$[L^*:L]=\prod _{p\in \mathrm {Diff}(\mathcal{{V}})}p^2$$ by assumption and since $$F_q^{T}(q^{-2})=q^{e^T/2}F_q^{T'}(q^{-2})$$ by Theorem 4.1, we obtain

\begin{aligned}\frac{R'(L,T)}{{\mathfrak {m}}'(L)} =d_\infty (L,T')2^3\prod _{p\in \mathrm {Diff}(\mathcal{{V}})}\alpha _p(S_p,T')\prod _{q\notin \mathrm {Diff}(\mathcal{{V}})}(1-q^{-2})^2F_q^{T'}(q^{-2}), \end{aligned}

where

\begin{aligned} d_\infty (L,T')=\frac{\prod _{i=2}^4\frac{\pi ^{i/2}}{\Gamma \left( \frac{i}{2}\right) }}{[L^*:L]^{3/2}} \end{aligned}

denotes the archimedean density. The final expression equals $$2\frac{R'(L,T')}{{\mathfrak {m}}'(L)}$$ by the Siegel formula for L and $$T'$$ (cf. [27, Satz 2 on p. 555]). $$\square$$

### Corollary 5.1

If T is a positive definite symmetric half-integral matrix of size 4 which satisfies $$\chi ^T=1$$ and $$\eta ^T_\ell =1$$ for $$\ell \ne p$$, then there exists a positive definite symmetric half-integral matrix $$T'$$ of size 3 such that

\begin{aligned} \sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T)}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}=2\sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T')}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}, \end{aligned}

where $$(E,E')$$ extends over all pairs of isomorphism classes of supersingular elliptic curves over $${\bar{{\mathbb {F}}}}_p$$.

### Proof

Proposition 4.1 of [29] gives

\begin{aligned}R(\mathcal{{O}}_p,T')=\sum _{L\in \Xi (\mathcal{{O}}_p)}\frac{N(L,T')}{\sharp \mathrm {SO}(L)}=\sum _{(E',E)}\frac{N(\mathrm {Hom}(E',E),T')}{\sharp \mathrm {Aut}(E)\sharp \mathrm {Aut}(E')}. \end{aligned}

We can derive Corollary 5.1 from (5.2) and Proposition 5.2. $$\square$$

Let $$T\in \frac{1}{2}\mathcal{{E}}_4({\mathbb {Z}}_p)$$ be an anisotropic symmetric matrix with (naive) extended Gross-Keating invariant $$(a_1,a_2,a_3,a_4;\varepsilon _1,\varepsilon _2,\varepsilon _3,\varepsilon _4)$$. Note that $$\varepsilon _1=\varepsilon _4=1$$ by definition. One can easily see that $$\varepsilon _2\ne 1$$ and $$\varepsilon _3=-1$$. Proposition 5.3 of [1] gives a partition $$\{1,2,3,4\}=\{i,j\}\cup \{k,l\}$$ such that

\begin{aligned} a_i\equiv a_j\not \equiv a_k\equiv a_l\pmod 2. \end{aligned}

### Lemma 5.2

1. (1)

If $$a_1\not \equiv a_2\pmod 2$$, then

\begin{aligned} F_p^{T'}(p^{-1}) =&\frac{p^{a_1+1}-1}{(p-1)(p^3-1)}\biggl (p^{\{a_1+3(a_2+1)\}/2}-\frac{p^{a_1+1}+1}{p+1}\biggl )\\&-\frac{p^{(a_1+a_2+2a_3+1)/2}}{p-1}\biggl \{(a_1+1)p^{(a_1+a_2+1)/2}-\frac{p^{a_1+1}-1}{p-1}\biggl \}. \end{aligned}
2. (2)

If $$a_1\equiv a_2\pmod 2$$, then

\begin{aligned} F_p^{T'}(p^{-1}) =&\frac{p^{a_1+1}-1}{(p-1)(p^3-1)}\biggl (p^{(a_1+3a_2)/2}-\frac{p^{a_1+1}+1}{p+1}\biggl )\\&-\frac{p^{(a_1+a_2+2a_3+2)/2}}{p-1}\biggl \{(a_1+1)p^{(a_1+a_2)/2}-\frac{p^{a_1+1}-1}{p-1}\biggl \}\\&+p^{(a_1+3a_2)/2}\frac{p^{a_1+1}-1}{p^2-1}(p^{a_1-a_2+1}+1). \end{aligned}

### Proof

We write the naive extended Gross-Keating invariant of T as

\begin{aligned} \mathrm {EGK}_p(T)=(a_1,a_2,a_3,a_4;1,\varepsilon _2,\varepsilon _3,1). \end{aligned}

Let $$\sigma$$ be either 1 or 2 according as $$a_1-a_2$$ is odd or even. Section 8 of [6] expresses $$F^{\mathrm {EGK}_p(T)'}_p(X)$$ in terms of $$\mathrm {EGK}_p(T)'=(a_1,a_2,a_3;1,\varepsilon _2,\varepsilon _3)$$:

\begin{aligned} F_p^{\mathrm {EGK}_p(T)'}(p^{-2}X)&=\sum _{i=0}^{a_1}\sum _{j=0}^{(a_1+a_2-\sigma )/2-i}p^{i+j}X^{i+2j}\\ & \varepsilon _3\sum _{i=0}^{a_1}\sum _{j=0}^{(a_1+a_2-\sigma )/2-i}p^{(a_1+a_2-\sigma )/2-j}X^{a_3+\sigma +i+2j}\\ & +\varepsilon _2^2p^{(a_1+a_2-\sigma +2)/2}\sum _{i=0}^{a_1}\sum _{j=0}^{a_3-a_2+2\sigma -4}\varepsilon _2^jX^{a_2-\sigma +2+i+j}. \end{aligned}

We now specialize the formula to $$X=p$$ and $$\varepsilon _3=-1$$. Then

\begin{aligned} F_p^{T'}(p^{-1}) =&\frac{p^{a_1+1}-1}{(p-1)(p^3-1)}\biggl (p^{\{a_1+3(a_2-\sigma +2)\}/2}-\frac{p^{a_1+1}+1}{p+1}\biggl )\\&-\frac{p^{(a_1+a_2+2a_3+\sigma )/2}}{p-1}\biggl ((a_1+1)p^{(a_1+a_2-\sigma +2)/2}-\frac{p^{a_1+1}-1}{p-1}\biggl )\\&+\varepsilon _2^2p^{\{a_1+3(a_2-\sigma +2)\}/2}\frac{(p^{a_1+1}-1)(1-(\varepsilon _2p)^{a_1-a_2+2\sigma -3})}{(p-1)(1-\varepsilon _2p)}. \end{aligned}

Since $$\varepsilon _2=0$$ or $$-1$$ according as $$a_1-a_2$$ is odd or even by Proposition 2.2 of [5] and Proposition 4.8 of [1], we obtain the stated formulas. $$\square$$

The degree $$\deg {\mathscr {Z}}(B)$$ is defined in (1.2) for positive definite symmetric half-integral $$3\times 3$$ matrices B such that $$\mathrm {Diff}(B)$$ is a singleton.

### Corollary 5.2

Let T be a positive definite symmetric half-integral $$4\times 4$$ matrix such that $$\chi _T=1$$ and $$\mathrm {Diff}(T)=\{p\}$$. Let $$\sigma$$ be either 1 or 2 according as $$a_1-a_2$$ is odd or even. If $$\deg {\mathscr {Z}}(T')\ne 0$$, then

\begin{aligned} \biggl |\frac{C_4(T)}{-2^8\cdot 3^2\cdot \deg {\mathscr {Z}}(T')}-1\biggl |<\frac{4}{p\sqrt{p}}\biggl (p^{-(a_4-3+\sigma )/2}+\frac{4p^{-(a_4-a_1)/2}}{a_1+1}\biggl ), \end{aligned}

where $$\mathrm {GK}_p(T)=(a_1,a_2,a_3,a_4)$$. In particular,

\begin{aligned} \biggl |\frac{C_4(T)}{-2^8\cdot 3^2\cdot \deg {\mathscr {Z}}(T')}-1\biggl |&<\frac{20}{p\sqrt{p}},&\lim _{e_p^T\rightarrow \infty }\frac{C_4(T)}{-2^9\cdot 3^2\cdot \deg {\mathscr {Z}}(T')}&=1. \end{aligned}

### Proof

By (2.12) and (2.13) of [30]

\begin{aligned} -p^{-2}\frac{\partial F^{H'}_p}{\partial X}(p^{-2})&\ge (a_1+1)p^{(a_1+a_2)/2}\biggl (\frac{a_3-a_2+2\sigma }{\sqrt{p}^\sigma }+\varepsilon _2^2\frac{a_3-a_2+1}{2}\biggl )\\&\ge (a_1+1)p^{(a_1+a_2-(2-\sigma ))/2}. \end{aligned}

Recall that if $$\sigma =1$$, then $$a_1<a_2\le a_3\le a_4$$ while if $$\sigma =2$$, then $$a_1\le a_2<a_3\le a_4$$. An examination of the proof of Lemma 5.2 confirms that

\begin{aligned} & \biggl |\frac{F_p^{H'}(p^{-1})}{\sqrt{p}^{e^T_p}(p-1)}\biggl |\le \frac{a_1+1}{(p-1)^2}p^{(a_1+a_2-a_4+2)/2}+\frac{p^{a_1+a_2-(a_3+a_4+3\sigma )/2+4}}{(p-1)^2(p^3-1)}\\ & \quad +\varepsilon _2^2\frac{p^{2a_1+2-(a_3+a_4)/2}}{(p-1)^2(p+1)}+\varepsilon _2^2\frac{p^{a_1+a_2+1-(a_3+a_4)/2}}{(p-1)^2(p+1)}\\ & \quad <4p^{(a_1+a_2)/2-1}\{(a_1+1)p^{-a_4/2}+2p^{-(a_4-a_1+3\sigma )/2}+2\varepsilon _2^2p^{-(a_4-a_1+1)/2}\}. \end{aligned}

Now our proof is completed by Theorem 1.2. $$\square$$