Congruences of Siegel Eisenstein series of degree two
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Takuya Yamauchi
Congruences of Siegel Eisenstein series of degree two Received: 11 February 2020 / Accepted: 30 October 2020 Abstract. In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp4 (Z).
1. Introduction The congruences for automorphic forms have been studied throughly in several settings. The well-known prototype seems to be one related to Ramanujan delta function and the Eisenstein series of weight 12 for SL2 (Z). Apart from its own interests, various important applications, for instance Iwasawa theory, have been made since many individual examples were found before. In [25] Ribet tactfully used Galois representations for elliptic cusp forms of level one to obtain the converse of the Herbrand’s theorem in the class numbers for cyclotomic fields. HerbrandRibet’s theorem is now understood in the context of Hida theory and Iwasawa main conjecture [1]. The notion of congruence modules is developed by many authors (cf. [13,23] among others). In this paper we study the congruence primes between Siegel Eisenstein series of degree 2 and Siegel cusp forms with respect to Sp4 (Z). To explain the main claims we need to fix the notation. Let H2 = t Z = Z > 0} be the Siegel upper half space of degree 2 and {Z ∈ M2 (C) | 02 E 2 02 E 2 := Sp4 (Z) = γ ∈ GL4 (Z) t γ γ = where E 2 is the −E 2 02 −E 2 02 −1 identity matrix of size 2. The group acts on H2 by γ Z := (AZ + B)(C Z + D) A B for γ = ∈ and Z ∈ H2 . For such γ and Z we also define the automorphic C D A B ∈ . For each positive factor j (γ , Z ) := det(C Z + D). Put ∞ = 02 D even integer k ≥ 4 let us consider the Siegel Eisenstein series of weight k with respect to : 1 (2) , Z ∈ H2 . E k (Z ) = j (γ , Z )k γ ∈∞ \
The author is partially supported by JSPS KAKENHI Grant Number (B) No. 19H01778. T. Yamauchi (B): Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, AobaKu, Sendai 980-8578, Japan e-mail: [email protected] Mathematics Subject Classification 11F46 · 11F33 https://doi.org/10.1007/s00229-020-01256-5
T. Yamauchi
This series convergences absolutely and it has the Fourier expansion √ (2) E k (Z ) = A(T )q T , q T = exp(2π −1tr(T Z )) 2
T ∈Sym (Z)∗≥0
where Sym2 (Z)∗≥0 stands for the set of semi-positive 2 by 2 symmetric matrices whose diagonal entries are integers and the others are half integers. Let Bn be the nth Bernoulli number and Bn,χ the generalized n-th Bernoulli number for a Dirichlet character χ . It is well-known (see [15,19], and Corollary 2, p. 80 of [11]) that A T is explicitly given as follows: • If rank(T ) = 0, hence T = 02 , then A(02 ) = 1; • If rank(T ) = 1, there exists S ∈ GL2 (Z) such that t ST S = positive integer n. Then A(T ) = 2 for the Riemann zeta function; • If rank(T ) = 2, A(T ) = 2
n0 for some 00
2k σk−1 (n) = − σk−1 (n) where ζ (s) stands ζ (1 − k) Bk
Bk−1,χT L(2 − k, χT ) ST = −4k , ST := ζ (1 − k)ζ (3 − 2k) Bk B2k−2
Fq (T ; q k−3 )
q| det(2T )
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