TY - JOUR
T1 - A reformulation of the Siegel series and intersection numbers
AU - Cho, Sungmun
AU - Yamauchi, Takuya
N1 - Funding Information:
Sungmun Cho was supported by JSPS KAKENHI Grant No. 16F16316, Samsung Science and Technology Foundation under Project Number SSTF-BA1802-03, and NRF 2018R1A4A 1023590.
Funding Information:
Takuya Yamauchi is partially supported by JSPS KAKENHI Grant Number (B) No. 19H01778.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over Fp and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp 4/ Q of weight 2, which is independent of p(> 2). In addition, we will explain a description of the local intersection multiplicities of the special cycles over Fp on the supersingular locus of the ‘special fiber’ of the Shimura varieties for GSpin (n, 2) , n≤ 3 in terms of the Siegel series directly.
AB - In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over Fp and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp 4/ Q of weight 2, which is independent of p(> 2). In addition, we will explain a description of the local intersection multiplicities of the special cycles over Fp on the supersingular locus of the ‘special fiber’ of the Shimura varieties for GSpin (n, 2) , n≤ 3 in terms of the Siegel series directly.
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U2 - 10.1007/s00208-020-01999-2
DO - 10.1007/s00208-020-01999-2
M3 - Article
AN - SCOPUS:85084999613
VL - 377
SP - 1757
EP - 1826
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 3-4
ER -