Correction to: Poles and residues of standard L-functions attached to Siegel modular forms

1 Correction to: Math. Ann. 289:589–612 (1991) https://doi.org/10.1007/BF01446591

Abstract

Correction to my paper on the poles of standard L-functions attached to Siegel modular forms.

Mathematics Subject Classification 11F46 $\cdot$ 11F66

In the second corollary of Theorem 3 of the original paper [5, p. 601], I made a statement (without proof) on the orders of possible poles of the completed standard L-function $\Lambda (s,f,{\underline{\mathrm {St}}})$ attached to a Siegel modular form f of weight k and degree n in case $k<n$. It asserts in particular that the orders of poles are at most two, but this is not the case.

This error was pointed out by a mail from Professor Chenevier on October 27, 2018 (for details, see the remark below). After that I checked the paper again and noticed that I had made some mistakes in deducing the claim.

To state the corrected version the following notation will be used. For $n, k\in {\mathbf{Z}}_{> 0}$ let $S_k^n$ be the space of holomorphic cusp forms of weight k for

$$\begin{aligned} {\Gamma }^{(n)}:={Sp}(n,\mathbf{Z})={Sp}_{2n}(\mathbf{Z})\, . \end{aligned}$$

Let $f\in S_k^n$ be a Hecke eigenform and $L(s,f,{\underline{\mathrm {St}}})$ be the standard L-function attached to f. Take $\varepsilon , \nu \in \{ 0,1\}$ such that $n\equiv \varepsilon \, \pmod {2}$ and $k\equiv \nu \, \pmod {2}$. Let

$$\begin{aligned} {\Gamma }_{\mathbf{R}}(s):={\pi }^{-\frac{s}{2}}{\Gamma }\left( \frac{s}{2}\right) \, , \quad {\Gamma }_{\mathbf{C}}(s):=2{(2\pi )}^{-s}{\Gamma }(s) \end{aligned}$$

and

$$\begin{aligned} \Lambda (s,f,{\underline{\mathrm {St}}}):={\Gamma }_{\mathbf{R}}(s+\varepsilon ) \prod _{j=1}^n {\Gamma }_{\mathbf{C}}(s+k-j)L(s,f,{\underline{\mathrm {St}}})\, . \end{aligned}$$

By Böcherer [1], $\Lambda (s,f,{\underline{\mathrm {St}}})$ has a meromorphic continuation to the whole s-plane and is invariant under the substitution $s\mapsto 1-s$. The symbol ${\mathrm {ord}}_{s=c}$ stands for the order of zero of a meromorphic function at $s=c$. The largest integer $\le x$ is denoted by [x].

Now we state the correction. The second corollary of Theorem 3 in [5] should be replaced by the following

Proposition 1

Suppose $k<n$. Then the possible poles of $\Lambda (s,f,{\underline{\mathrm {St}}})$ are contained in $[-n+k-\nu ,n-k+\nu +1]\cap \mathbf{Z}$ and for every $m\in [1,n-k+\nu +1]\cap \mathbf{Z}$ we have

$$\begin{aligned} {\mathrm {ord}}_{s=m}\, \Lambda (s,f,{\underline{\mathrm {St}}}) \ge \left[ \frac{m-n}{2}\right] +\frac{k-\nu }{2}-1\, . \end{aligned}$$

Proof

Put

$$\begin{aligned} {\Gamma }_n(s):=\prod _{j=1}^n {\Gamma }\left( s-\frac{j-1}{2}\right) , \quad C_n(s):=\prod _{j=1}^n\left( s+\frac{j-1}{2}\right) =\frac{{\Gamma }_n\left( s+\frac{n+1}{2}\right) }{{\Gamma }_n\left( s+\frac{n-1}{2}\right) }\, , \end{aligned}$$

and

$$\begin{aligned} \mu (n,k,s):=(-1)^{\frac{nk}{2}}2^{{\frac{n^2+3n}{2}}-2ns-nk+1} {\pi }^{\frac{n(n+1)}{2}} \frac{{\Gamma }_n\left( s+k-\frac{n+1}{2}\right) }{{\Gamma }_n\left( s+k\right) }\, . \end{aligned}$$

For $m\in {\mathbf{Z}}_{> 0}$ let ${\mathfrak H}_{m}$ be the Siegel upper half space of degree m. For $w\in 2{\mathbf{Z}}_{\ge 0}~, z\in {\mathfrak H}_{m}$ and $s\in \mathbf{C}$ with $\mathrm {Re}(w+2s)>m+1$ let

$$\begin{aligned} E_w^{(m)}(z,s):=\det (\mathrm {Im}(z))^s\sum _{\{ c,d\} } \det (cz+d)^{-w}|\det (cz+d)|^{-2s} \end{aligned}$$

be the nonholomorphic Eisenstein series of weight w for ${\Gamma }^{(m)}$. Here $\begin{pmatrix} *&{}*\\ c&{}d\end{pmatrix}$ runs over a complete set of representatives of . The Eisenstein series $E_w^{(m)}(z,s)$ has a meromorphic continuation to the whole s-plane. By Böcherer [1, p. 157],

$$\begin{aligned}&L(s,f,{\underline{\mathrm {St}}})f(Z)\nonumber \\&\quad ={\left( 2^nC_n\left( 1-\frac{s+k+\nu +n}{2} \right) \right) }^{-\delta (\frac{\nu -1}{2})} {\mu \left( n,k,\frac{s-k+\nu +n}{2} \right) }^{-1}\nonumber \\&\qquad \cdot \zeta (s+n)\prod _{j=0}^{n-1}\zeta (2s+2j)\nonumber \\&\qquad \cdot \left( f(W),\left( \left( D^{\nu }E_{k-\nu }^{(2n)}\right) \left( *,\frac{\bar{s}-k+\nu +n}{2} \right) \right) \begin{pmatrix}-\bar{Z}&{}0\\ 0&{}W\end{pmatrix}\right) \end{aligned}$$

(a)

for every $Z\in {\mathfrak H}_{n}.$ Here $\delta (x)=1$ or 0 according as $x\in \mathbf{Z}$ or not; for

$$\begin{aligned} {\mathcal {Z}}=\begin{pmatrix}Z&{}\quad U\\ {}^tU&{}\quad W\end{pmatrix}\in {\mathfrak {H}}_{2n} \quad {\text {with}}\ Z,\, W\in {\mathfrak {H}}_n \ {\text {and}}\ U=(u_{ij}), \end{aligned}$$

the differential operator D is defined by

$$\begin{aligned} D:=\det {\left( \frac{\partial }{\partial u_{ij}}\right) }_{1\le i,j\le n}. \end{aligned}$$

By the functional equation of $\Lambda (s,f,{\underline{\mathrm {St}}})$ we restrict our attention to the poles of the functions in (a) in the right half plane $\mathrm {Re}(s)>0$ . In the following let $g_j(s)\, (j=1,2,\ldots )$ be some functions which are holomorphic and nonzero in $\mathrm {Re}(s)>0$. Note that

$$\begin{aligned} C_n\left( 1-\frac{s+k+\nu +n}{2}\right) =(-1)^nC_n\left( \frac{s+k+\nu -1}{2} \right) =g_1(s) \end{aligned}$$

and

$$\begin{aligned} {\mu \left( n,k,\frac{s-k+\nu +n}{2} \right) }^{-1}=g_2(s)\cdot {{\Gamma }_n\left( \frac{s+k+\nu -1}{2} \right) }^{-1}\, . \end{aligned}$$

For $N\in 2{\mathbf{Z}}_{> 0}$ and $z\in {\mathfrak {H}}_m$ let $E^{*}(z,s;w,{\mathrm {triv}}_N,N)$ be the Eisenstein series of level N defined in Feit [3, p. 11], with ${\mathrm {triv}}_N$ being the trivial Dirichlet character modulo N. By Shimura [8, Proposition 2.1],

$$\begin{aligned} E_w^{(m)}(z,s)={\mathrm {Tr}}_1^N\left( E^{*}(z,s;w,{\mathrm {triv}}_N,N)|_w{\eta }_m \right) , \end{aligned}$$

where ${\eta }_m :=\begin{pmatrix}0&{}1_m\\ -1_m&{}0\end{pmatrix}$ and ${\mathrm {Tr}}_1^N$ is the trace map from ${\Gamma }_0^{(m)}(N)$ to ${\Gamma }^{(m)}$ defined as in [5, p. 592]. Rewriting this in the notation of Feit [3, p. 48], we have

$$\begin{aligned} E_{k-\nu }^{(2n)}&\left( {\mathcal {Z}}^{(2n)},\frac{s-k+\nu +n}{2} \right) \\ =&g_3(s)\zeta (2s)^{-1}{\mathrm {Tr}}_1^N \left( D\left( \mathcal {Z},\frac{s-k+\nu +n}{2};k-\nu , {\mathrm {triv}}_N,N \right) \bigg {|}_{k-\nu } {\eta }_{2n} \right) . \end{aligned}$$

Hence by (a), for every $c\in \mathbf{C}$ with $\mathrm {Re}(c)>0$ we have

$$\begin{aligned} {\mathrm {ord}}_{s=c}\, L(s,f,{\underline{\mathrm {St}}})\ge {\underset{{\mathcal {Z}}\in {\mathfrak H}_{2n}}{\mathrm {inf}}}{\mathrm {ord}}_{s=c}\, \frac{D\left( {\mathcal {Z}},\frac{s-k+\nu +n}{2};k-\nu , {\mathrm {triv}}_N, N\right) }{{\Gamma }_n \left( \frac{s+k+\nu -1}{2}\right) }, \end{aligned}$$

(b)

where $N\in 2{\mathbf{Z}}_{> 0}$ is arbitrary. By [3, p. 49], every possible pole of

$$\begin{aligned} D\left( {\mathcal {Z}}^{(2n)},\frac{s-k+\nu +n}{2};k-\nu , {\mathrm {triv}}_N, N\right) \end{aligned}$$

in $\mathrm {Re}(s)>0$ is simple and contained in $[1, n-k+\nu +1]\cap \mathbf{Z}$. In particular, the possible poles of $\Lambda (s,f,{\underline{\mathrm {St}}})$ are contained in $\mathbf{Z}$. Now let $m\in {\mathbf{Z}}_{> 0}$. Then

$$\begin{aligned}&{\mathrm {ord}}_{s=m}\, {\Gamma }_n \left( \frac{s+k+\nu -1}{2}\right) \\&\quad =-\delta (m\le n-k-\nu ) \left( \left[ \frac{n-m}{2} \right] -\frac{k+\nu }{2}+1 \right) , \end{aligned}$$

where $\delta (C):=1$ or 0 according as the condition C is satisfied or not. Hence from (b) it follows that

$$\begin{aligned} {\mathrm {ord}}_{s=m}\, L(s,f,{\underline{\mathrm {St}}})&\ge \delta (m\le n-k-\nu ) \left( \left[ \frac{n-m}{2} \right] -\frac{k+\nu }{2}+1 \right) \\&\quad -\delta (m\le n-k+\nu +1). \end{aligned}$$

(c)

On the gamma factor of $\Lambda (s,f,{\underline{\mathrm {St}}})$ we have

$$\begin{aligned} {\mathrm {ord}}_{s=m}\, {\Gamma }_{\mathbf{R}}(s+\varepsilon )\prod _{j=1}^n {\Gamma }_{\mathbf{C}}(s+k-j) =\delta (m\le n-k)(m-n+k-1). \end{aligned}$$

(d)

From (c) and (d) the assertion of Proposition 1 follows. $\square$

Remark

• (1) Proposition 1 follows also from Shimura [9, Theorem 6.1], which is much more extensive.

• (2) By the above-mentioned mail from Professor Chenevier I was informed of the following: Let $F\in S_{13}^{24}$ be the Hecke eigenform discovered by Freitag [4], Satz 5.2. Then by the theory of theta lift for the orthogonal group O(24) of Chenevier–Lannes [2] combined with the Eichler commutation relations of Rallis [7], the equality

  $$\begin{aligned} L(s,F,{\underline{\mathrm {St}}})=\prod _{i=0}^{11}L(s+11-i,{\Delta }) \prod _{j=0}^{24}\zeta (s+12-j) \end{aligned}$$

  holds, where $L(s,\Delta )$ is the Hecke L-function attached to Ramanujan’s $\Delta \in S_{12}^1$ which satisfies a functional equation under $s\mapsto 12-s$. This implies in particular that $\Lambda (s,F,{\underline{\mathrm {St}}})$ has a pole of order $7-[m/2]$ at every $s=m\in [1,13]\cap \mathbf{Z}$. Note that Proposition 1 gives the best possible bound in this case.

Professor Chenevier pointed out also that the Langlands standard gamma factor for $L(s,f,{\underline{\mathrm {St}}})$ in case $k<n$ is

$$\begin{aligned} {\gamma }_k^{(n)}(s):={\Gamma }_{\mathbf{R}}(s+\nu ) \prod _{i=1}^k {\Gamma }_{\mathbf{C}}(s+k-i)\prod _{j=1}^{n-k} {\Gamma }_{\mathbf{R}}(s+\nu +j){\Gamma }_{\mathbf{R}}(s+\nu -j) \end{aligned}$$

according to Moeglin–Renard [6], Proposition 9.1. Hence let

$$\begin{aligned} {\Lambda }^{*}(s,f,{\underline{\mathrm {St}}}):={\gamma }_k^{(n)}(s) L(s,f,{\underline{\mathrm {St}}}), \end{aligned}$$

which also satisfies

$$\begin{aligned} {\Lambda }^{*}(s,f,{\underline{\mathrm {St}}}) ={\Lambda }^{*}(1-s,f,{\underline{\mathrm {St}}})\, . \end{aligned}$$

(e)

For every $m\in {\mathbf{Z}}_{> 0}$ we have

$$\begin{aligned} {\mathrm {ord}}_{s=m}\, {\gamma }_k^{(n)}(s) =-\delta (m\le n-k-\nu )\left( \left[ \frac{n-m}{2}\right] -\frac{k+\nu }{2}+1 \right) \, . \end{aligned}$$

(f)

Hence (c), (e), and (f) imply that in terms of ${\Lambda }^{*}(s,f,{\underline{\mathrm {St}}})$ Proposition 1 takes the following form:

Proposition 2

For any Hecke eigenform $f\in S_k^n$ with $k<n$, every possible pole of ${\Lambda }^{*}(s,f,{\underline{\mathrm {St}}})$ is simple and contained in $[-n+k-\nu ,n-k+\nu +1]\cap \mathbf{Z}$.