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CONGRUENCES of SIEGEL EISENSTEIN SERIES of DEGREE TWO 11 Cubic Lifts to Siegel Cusp Forms of Scalar Weight Other Than Three (Cf

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CONGRUENCES of SIEGEL EISENSTEIN SERIES of DEGREE TWO 11 Cubic Lifts to Siegel Cusp Forms of Scalar Weight Other Than Three (Cf CONGRUENCES OF SIEGEL EISENSTEIN SERIES OF DEGREE TWO TAKUYA YAMAUCHI 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 found before. In [24] 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. Herbrand-Ribet’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],[22] 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 H = Z M (C) tZ = Z > 0 be the Siegel upper half space of degree 2 and Γ := 2 { ∈ 2 | } t 02 E2 02 E2 Sp4(Z) = γ GL4(Z) γ γ = where E2 is the identity matrix ( ∈ E 0 ! E 0 ! ) − 2 2 − 2 2 −1 A B arXiv:1912.03847v2 [math.NT] 13 Dec 2019 of size 2. The group Γ acts on H2 by γZ := (AZ + B)(CZ + D) for γ = Γ C D! ∈ and Z H . For such γ and Z we also define the automorphic factor j(γ, Z) := det(CZ + D). ∈ 2 A B Put Γ∞ = Γ . For each positive even integer k 4 let us consider the Siegel ( 02 D! ∈ ) ≥ Eisenstein series of weight k with respect to Γ: 1 E(2)(Z)= ,Z H . k j(γ, Z)k ∈ 2 γ∈XΓ∞\Γ Key words and phrases. Congruences, Siegel Eisenstein series, Galois representations. The author is partially supported by JSPS KAKENHI Grant Number (B) No.19H01778. 1 2 TAKUYA YAMAUCHI This series convergences absolutely and it has the Fourier expansion E(2)(Z)= A(T )qT , qT = exp(2π√ 1tr(TZ)) k − T ∈Sym2(Z)∗ X ≥0 2 ∗ where Sym (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 n-th 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 AT is explicitly given as follows: if rank(T ) = 0, hence T = 0 , then A(0 ) = 1; • 2 2 t n 0 if rank(T ) = 1, there exists S GL2(Z) such that STS = for some positive • ∈ 0 0! σk−1(n) 2k integer n. Then A(T ) = 2 = σk−1(n) where ζ(s) stands for the Riemann ζ(1 k) −Bk zeta function; − if rank(T ) = 2, • L(2 k,χT ) Bk−1,χT k−3 (1.1) A(T ) = 2 − ST = 4k , ST := Fq(T ; q ) ζ(1 k)ζ(3 2k) − BkB2k−2 − − q| det(2Y T ) where Fq(T, X) Z[X] is the Siegel series for T at any prime q dividing det(2T ) and ∈ χT is the quadratic character associated to the quadratic extension Q( det(2T ))/Q. − Notice that ST Z and it is known that ST = 1 for which det(2T ) isp a fundamental ∈ − discriminant. For each rational prime p we fix the embeddings Q ֒ Q , Q ֒ C and an isomorphism C Q → p → ≃ p which is compatible with those embeddings. For any extension K of Q or Qp we denote by = K the ring of integers of K and = K a prime ideal above p. We often drop O O P P the superscript “K” if it is obvious from the context. For a Siegel modular form F of de- T gree 2 with the Fourier expansion F (Z) = AF (T )q we say F is defined over if O T ∈Sym2(Z)∗ X ≥0 A(T ) for any T Sym2(Z)∗ and we often say F is -integral if F is defined over for ∈ O ∈ ≥0 O O some . For two integral Siegel modular forms F, G of degree 2 with the Fourier expansions O T T F (Z)= AF (T )q , G(Z)= AG(T )q , AF , AG for some we say F ∈ O O T ∈Sym2(Z)∗ T ∈Sym2(Z)∗ X ≥0 X ≥0 is congruent to G modulo or simply we write F G mod if AF (T ) AG(T ) mod for P ≡ P ≡ P any T Sym2(Z)∗ . We also say F is a non-trivial -integral Siegel modular form if F 0 mod ∈ ≥0 O 6≡ 2 ∗ × . Hence there exists at least one T Sym (Z) such that AF (T ) . This implies that if P ∈ ≥0 ∈ O CONGRUENCES OFSIEGEL EISENSTEIN SERIES OF DEGREE TWO 3 further F is a Hecke eigen form, then all Hecke eigenvalues are -integral. For two non-trivial O -integral Siegel modular forms F, G, we write F G mod if they share all Hecke eigenvalues O ≡ev P modulo . Obviously if F G mod and either of F, G is non-trivial, then F G mod . P ≡ P ≡ev P We normalize Ek by 1 (1.2) Gk(Z) := ζ(1 k)ζ(3 2k)Ek(Z). 2 − − Now we are ready to state our main results: Theorem 1.1. Let k be even positive integer and p 7 be a prime number such that p 1 > 2k 2. ≥ − − 2 ∗ Assume that ordp(B k ) > 0 and ordp(Bk ,χ ) = 0 for some T Sym (Z) . Then there exists 2 −2 −1 T ∈ >0 a non-trivial -integral Hecke eigen Siegel cusp form F defined over for some finite extension O O K/Q such that F Gk mod for a prime ideal of dividing p. ≡ev P P O As explained later the condition p 1 > 2k 2 implies that AG (T ) Zp for any T − − k ∈ ∈ 2 ∗ Sym (Z)≥0. It is natural to ask an explicit form of F in the theorem above. In fact, we can realize it as a Saito-Kurokawa lift. Theorem 1.2. Let k be even positive integer and p 7 be a prime number such that p 1 > ≥ − 2k 2. Assume that ordp(B k ) > 0 and ordp(Bk ,χ) = 0 for some quadratic character χ − 2 −2 −1 of GQ. Then there exist K -integral Saito-Kurokawa lifts Fi = SK(fi), (1 i t) for some O ≤t ≤ fi S k (SL (Z)), (1 i t) and for some finite extension K/Q such that F is congruent ∈ 2 −2 2 ≤ ≤ i X=1 to Gk modulo a prime of dividing p. P O Remark 1.3. The number of Saito-Kurokawa lifts in the claim above are exactly given as one (1) of Hecke eigen cusp forms in S k (SL (Z)) which are congruent to G modulo (see the 2 −2 2 2k−2 P proof of Proposition 4.1). Here G(1) is the Eisenstein series of weight 2k 2 for SL (Z) with 2k−2 − 2 the constant term ζ(3 2k). − + We also study a weak version of the converse of Theorem 1.1. Let hp be the plus part of the ideal class group of Q(ζp) where ζp is a primitive p-th root of unity. Theorem 1.4. Let k be even positive integer and p 7 be a prime number such that p 1 > 2k 2. ≥ − − If there exists a non-trivial K-integral Hecke eigen Siegel cusp form F for some number field O + K which is congruent to Gk modulo some prime ideal above p, then ordp(BkB2k−2hp ) > 0. In + particular, if Vandiver conjecture is true (hence p ∤ hp ) and ordp(Bk) = 0, then it follows that ordp(B2k−2) > 0. 4 TAKUYA YAMAUCHI The following claim is an obvious conclusion of the theorem above and it explains Siegel cusp forms tends to be residually far from Siegel Eisenstein series: Corollary 1.5. Let k be an even positive integer and p 7 be a prime number such that ≥ + p 1 > 2k 2. Assume that ordp(BkB k h ) = 0. Then there is no Hecke eigen cusp form − − 2 −2 p which is congruent to Gk modulo a prime over p. In particular if Vandiver’s conjecture is true, then the claim is true under the condition that ordp(BkB2k−2) = 0 which is depending only on k. Some experts may think of the case when k is odd. In fact we can study the congruence of Siegel cusp forms of odd weights by using mod p Galois representations instead of Siegel Eisenstein series. Theorem 1.6. Let k be an odd positive integer and p 7 be a prime number such that p 1 > ≥ − 2k 2. Let F be a non-trivial K -integral Hecke eigen Siegel cusp form whose mod p Galois − O + representation ρF,p is completely reducible. Then ordp(Bk−1B2k−2hp ) > 0. In particular, if Vandiver conjecture is true and ordp(Bk−1) = 0, then it follows that ordp(B2k−2) > 0. In general any genuine Siegel cusp form F of weight k 3 can not be congruent to Siegel ≥ Eisenstein series modulo p for all but finitely many prime p by Theorem B of [34].
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