Correction to: Poles and residues of standard L -functions attached to Siegel modular forms
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Mathematische Annalen
CORRECTION
Correction to: Poles and residues of standard L-functions attached to Siegel modular forms Shin-ichiro Mizumoto1 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 · 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 (s, f , 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 ∈ Z>0 let Skn be the space of holomorphic cusp forms of weight k for (n) := Sp(n, Z) = Sp 2n (Z) . Let f ∈ Skn be a Hecke eigenform and L(s, f , St) be the standard L-function attached to f . Take ε, ν ∈ {0, 1} such that n ≡ ε (mod 2) and k ≡ ν (mod 2). Let s s , C (s) := 2(2π )−s (s) R (s) := π − 2 2
Communicated by Wei Zhang. The original article can be found online at https://doi.org/10.1007/BF01446591.
B 1
Shin-ichiro Mizumoto [email protected] Department of Mathematics, Tokyo Institute of Technology, Oo-okayama, Meguro-ku, Tokyo 152-8551, Japan
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S. Mizumoto
and (s, f , St) := R (s + ε)
n
C (s + k − j)L(s, f , St) .
j=1
By Böcherer [1], (s, f , St) has a meromorphic continuation to the whole s-plane and is invariant under the substitution s → 1 − s. The symbol ords=c stands for the order of zero of a meromorphic function at s = c. The largest integer ≤ 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 (s, f , St) are contained in [−n + k − ν, n − k + ν + 1] ∩ Z and for every m ∈ [1, n − k + ν + 1] ∩ Z we have ords=m (s, f , St) ≥
k−ν m−n + − 1. 2 2
Proof Put n n s + j −1 j −1 n (s) := s+ , Cn (s) := = s− 2 2 n s + j=1 j=1 n
n+1 2
n−1 2
,
and nk 2
μ(n, k, s) := (−1) 2
n 2 +3n 2 −2ns−nk+1
π
n(n+1) 2
n s + k − n+1 2 . n (s + k)
For m ∈ Z>0 let Hm be the Siegel upper half space of degree m. For w ∈ 2Z≥0 , z ∈ Hm and s ∈ C with Re(w + 2s) > m + 1 let (m) Ew (z, s) := det(Im(z))s
det(cz + d)−w | det(cz + d)|−2s
{c,d}
(m) .
∗∗ Here runs cd
be the nonholomorphic Eisenstein series of weight w for ∗ ∗ (m) . The Eisenstein series over a complete set of representatives of 0(m) ∗ (m) E w (z, s) has a meromorphic continuation to the whole s-plane. By Böcherer [1, p. 157],
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Correction to: Poles and residues o
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