We extend some recent work of D. McCarthy, proving relations among some
Fourier coefficients of a degree 2 Siegel modular form $F$ with arbitrary level
and character, provided there are some primes $q$ so that $F$ is an eigenform
for the Hecke operators $T(q)$ and $T_1(q^2)$

Walling, Lynne H.

English

arXiv

We extend some recent work of D. McCarthy, proving relations among some Fourier coefficients of a degree 2 Siegel modular form F with arbitrary level and character, provided there are some primes p so that F is an eigenform for the Hecke operators T(p) and T1(p2)

Walling, Lynne

Explore Bristol Research

                          Walling, L. (2017). Some relations on Fourier coefficients of degree 2 Siegelforms of arbitrary level. Journal of Number Theory.https://doi.org/10.1016/j.jnt.2017.04.003Peer reviewed versionLicense (if available):CC BY-NC-NDLink to published version (if available):10.1016/j.jnt.2017.04.003Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Elsevier at http://www.sciencedirect.com/science/article/pii/S0022314X17301890#. Please refer to anyapplicable terms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-termsSOME RELATIONS ON FOURIER COEFFICIENTS OFDEGREE 2 SIEGEL FORMS OF ARBITRARY LEVELLYNNE H. WALLINGAbstract. We extend some recent work of D. McCarthy, proving rela-tions among some Fourier coefficients of a degree 2 Siegel modular formF with arbitrary level and character, provided there are some primes pso that F is an eigenform for the Hecke operators T (p) and T1(p2).1. IntroductionIn a recent paper [3], McCarthy derives some nice results for Fourier co-efficients and Hecke eigenvalues of degree 2 Siegel modular forms of level 1,extending some classical results regarding elliptic modular forms. In partic-ular, with F a degree 2, level 1 Siegel modular form that is an eigenform forall the Hecke operators T (p), T (p2) (p prime), and a(T ) denoting the T thFourier coefficient of F , McCarthy shows that:(a) provided that a(I) = 1 and p is prime, the T (p)-eigenvalue λ(p) andthe T (p2)-eigenvalue λ(p2) are described explicitly in terms of a(pI)and a(p2I);(b) for r ≥ 1, a(I)a(pr+1I) is described explicitly in terms of a(I), a(pI),a(pr−1I), a(pr−1pr+1), and a(pr((1 + u2)/p uu p))where 1 ≤u < p/2 with u2 6≡ 1 (p);(c) if a(I) = 0 then a(mI) = 0 for all m ∈ Z+; further, if m,n ∈ Z+with (m,n) = 1, then a(I)a(mnI) = a(mI)a(nI).(As defined in Sec. 2, T2(p2) is the Hecke operator associated with thematrix diag(p, p, 1/p, 1/p), T1(p2) is the Hecke operator associated with thematrix diag(p, 1, 1/p, 1), and T (p2) = T2(p2) + pk−3T1(p2) + p2k−6. In [2],for χ = 1,T (p2) is denoted by T˜2(p2).) McCarthy’s approach begins withsome formulas from [1], which are somewhat cumbersome.In this note we use the formulas from [2] that give the action of Heckeoperators on Fourier coefficients of a Siegel modular form F , allowing forarbitrary level and character, and giving a simpler proof of McCarthy’sabove results (with no restriction on the level or character). Here when wesay that a modular form has weight k, levelN and character χ, we mean thatit transforms with weight k and character χ under the congruence subgroupΓ0(N ) ={(A BC D)∈ Sp2(Z) : N|C},Key words and phrases. Hecke eigenvalues, Siegel modular forms.2010 Mathematics Subject Classification: Primary 11F46, 11F1112 LYNNE H. WALLINGwhere Sp2(Z) is the symplectic group of 4 × 4 integral matrices. We workwith “Fourier coefficients” attached to lattices (as explained below), makingit simpler to work with the image of F under a Hecke operator. For p primeand degree 2, the local Hecke algebra is generated by T (p), T1(p2) andT2(p2). When N = 1, Proposition 5.1 of [2] gives a relation between thesegenerators, from which we deduce that with p - N , T (p) and T1(p2) generatethe local Hecke algebra, as do T (p) and T˜2(p2). However, when p|N , we haveT2(p2) = (T (p))2. Hence in this note we use the local generators T (p) andT1(p2); to more easily apply the results of [2], we use the operatorT˜1(p2) = T1(p2) + χ(p)pk−3(p+ 1)in place of T1(p2).Using some rather special aspects of working with degree 2 Siegel modularforms, we prove the following extensions of [3].Theorem 1.1. Suppose that F is a degree 2 Siegel modular form of weightk ∈ Z+, level N and character χ with Fourier expansionF (τ) =∑Ta(T ) exp(2piiTr(Tτ)).Also suppose that p is prime with F |T (p) = λ(p)F and F |T˜1(p2) = λ˜1(p2)F .(a) We haveλ(p)a(mI) = χ(p)pk−2η(p)a(mI) + a(mpI),whereη(p) =1 + χ(−1)(−1)k if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.(Thus when a(mI) 6= 0, λ(p) is given explicitly in terms of p, a(mI)and a(pmI).) As well, we haveχ(p)pk−2λ˜1(p2)a(mI) = χ(p2)p2k−4(α(I; p)− p)a(mI)+ λ(p)a(pmI)− a(p2mI)whereα(I; p) =2 if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.(Thus when χ(p)a(mI) 6= 0, λ˜1(p2) is given explicitly in terms of p,a(mI), a(pmI) and a(p2mI).)(b) Set  = 1 + χ(−1)(−1)k. For r ≥ 1, a(mI)a(pr+1I) is given bya(pmI)a(prI)− χ(p2)p2k−3a(mI)a(pr−1I)+ χ(p)pk−2a(mI)a(pr−1mpr+1m)+ χ(p)pk−2a(mI)∑1≤u<p/2u2 6≡−1 (p)a(prm((1 + u2)/p uu p)).SOME RELATIONS ON FOURIER COEFFICIENTS 3(c) Suppose that n a product of powers of primes p so that F is aneigenform for T (p) and T˜1(p2), and that m ∈ Z+ with (m,n) =1. If a(mI) = 0 then a(mnI) = 0. Also, we have a(I)a(mnI) =a(mI)a(nI).We also prove the following modest generalization.Theorem 1.2. Suppose that F is a degree 2 Siegel modular form of weightk ∈ Z+, level N and character χ with Fourier expansionF (τ) =∑Ta(T ) exp(2piiTr(Tτ)).Suppose that p is an odd prime, and set D =(1p). Let S be the set ofodd primes so that for q ∈ S, F is an eigenform for T (q) and T˜1(q2), andeither q = p or(−pq)= −1. Let n be a product of powers of primes in S.Then for any m ∈ Z+ so that (m,n) = 1, we havea(D)a(mnD) = a(mD)a(nD).Also, a(D)a(mnD) = 0 if a(mD) = 0.We note that McCarthy applies his results to compute eigenvalues of thelevel 1 Eisenstein series with regard to the Hecke operators T (pr) (p prime);as he notes, in [5] we computed the Hecke-eigenvalues of Eisenstein seriesof square-free levels for all primes p, allowing nontrivial character (thengeneralized in [6] for arbitrary level N and character χ, but only for primesp so that p2 - N ).We further note that it seems that these results cannot be extended tohigher degrees, as Lemma 3.1 (which is pivotal for our arguments) does notextend to higher degrees.2. PreliminariesWe will use some language and notation commonly used in quadraticforms and modular forms theory. When Λ is a lattice whose quadratic formis given by the matrix T (relative to some Z-basis for Λ), we write Λ ' T .Now suppose that Λ is a lattice with Λ ' T and that m ∈ Q+; we writeΛm to denote the lattice Λ “scaled” by m, meaning that Λm ' mT . Also,the discriminant of Λ is detT . With Λ,Ω lattices on the same unlerlyingquadratic space over Q, we write {Λ : Ω} to denote the invariant factors ofΩ in Λ.We seth(2) = {X + iY : X,Y ∈ R2,2sym : Y > 0 },where R2,2sym denotes the set of 2 × 2 symmetric matrices with real entries,and Y > 0 means that Y represents a positive definite quadatic form. Fora ring R, we write Sp2(R) for the group of 4 × 4 symplectic matrices withentries in R. Fixing a weight k ∈ Z+, for γ =(A BC D)∈ Sp2(Q), we defineF (τ)|γ = (det γ)k/2 det(Cτ +D)−kF ((Aτ +B)(Cτ +D)−1).4 LYNNE H. WALLINGWhen F is a degree 2 Siegel modular form of weight k, level N and characterχ, this means that for γ =(A BC D)∈ Γ0(N ), we haveF (τ)|γ = χ(detDγ)F (τ).We can write F as a Fourier series:F (τ) =∑T≥0a(T ) exp(2piiTr(Tτ))where the sum is over 2 × 2 symmetric, positive semi-definite, half-integralmatrices T (so the entries in T are half-integers with integers on the diago-nal). Given G ∈ GL2(Z), we have γ =(G−1tG)∈ Γ0(N ). Henceχ(detG)F (τ) = F (τ)|γ= (detG)kF (G−1τ tG−1)= (detG)k∑Ta( tGTG) exp(2piiTr(Tτ)).Thus a(tGTG) = χ(detG)(detG)ka(T ). So we can also write F as a “Fourierseries” supported on isometry classes of even integral, positive semi-definitelattices: For Λ an even integral lattice with Z-basis {x, y}, set c(Λ) = a(TΛ)where, relative to the given basis for Λ, we have Λ ' 2TΛ. When χ(−1) 6=(−1)k, we equip Λ with an orientation, meaning that with G ∈ GL2(Z),(x y)G is a basis for the oriented lattice Λ if and only if detG = 1. ThenF (τ) =∑cls Λc(Λ) e∗{Λτ}where cls Λ varies over all isometry classes of (oriented) even integral, posi-tive semi-definite lattices, ande∗{Λτ} =∑Gexp(2piiTr( tGTΛGτ))where G varies over O(Λ)\GL2(Z) when χ(−1) = (−1)k, and G varies overO+(Λ)\SL2(Z) otherwise. (Here O(Λ) denotes the orthogonal group of Λ,and O+(Λ) = O(Λ) ∩ SL2(Z).)Still suppose that F is a Siegel modular form of degree 2, weight k, levelN and character χ. For p prime, we define T (p), T1(p2), and T2(p2) asfollows. Take δ(p) = diag(p, p, 1, 1), δ1(p2) = (p, 1, 1/p, 1), and δ2(p2) =diag(p, p, 1/p, 1/p). With Γ = Γ0(N ), we setF |T (p) = pk−3∑γχ(γ)F |δ(p)−1γwhere γ varies over (δ(p)Γδ(p)−1 ∩ Γ)\Γ, and for j = 1, 2, we setF |Tj(p2) = pj(k−3)∑γχ(γ)F |δj(p2)−1γwhere γ varies over (δj(p2)Γδj(p2)−1 ∩ Γ)\Γ. Note that replacing δ(p) orδj(p2) by a scalar multiple of itself does not change the definition of theassociated Hecke operator. Note also that in [2], we did not normalizeSOME RELATIONS ON FOURIER COEFFICIENTS 5Tj(p2) by pj(k−3), as is usually done in other texts, and has been done in theabove formula for T1(p2). With T˜1(p2) = T1(p2)+χ(p)pk−3(p+1), Theorem6.1 of [2] gives us the following.Theorem 2.1. Let F be a degree 2 Siegel modular form of weight k, levelN , character χ, and lattice coefficients c(Λ). Then for any even integrallattice Λ, the Λth coefficient of F |T (p) isχ(p2)p2k−3c(Λ1/p) + χ(p)pk−2 ·∑{Λ:Ω}=(1,p)c(Ω1/p) + c(Λp),and the Λth coefficient of F |T˜1(p2) isχ(p2)p2k−3 ·∑{Λ:Ω}=(1/p,1)c(Ω) + χ(p)pk−2α(Λ; p)c(Λ) +∑{Λ:Ω}=(1,p)c(Ω).With Q the quadratic form on Λ, we equip Λ/pΛ with the quadratic form12Q, and α(Λ; p) is the number of isotropic lines in the quadratic space Λ/pΛ.There are p+ 1 lines in Λ/pΛ, and each of these lines is generated either byy + pΛ or by (x + uy) + pΛ for some u with 0 ≤ u < p. So with Λ ' 2I,α(Λ; 2) = 1, α(Λ; p) = 2 when p ≡ 1 (4), and α(Λ; p) = 0 when p ≡ 3 (4).When Λ ' 2T with p|T , α(Λ; p) = p+ 1.Note that with p a prime and m ∈ Z+ so that p - m, for any even integralrank 2 lattice Λ we have α(Λ; p) = α(Λm; p) since scaling by m does notchange whether a line is isotropic in Λ/pΛ.3. Proof of Theorem 1.1The next lemma is pivotal in our proof of Theorem 1.1; when this lemmageneralizes, we can generalize this theorem (as seen in Theorem 1.2).Lemma 3.1. Suppose that F is a degree 2 Siegel modular form of weight k,level N , character χ, and lattice coefficients c(Λ). With ∆ ' 2I, p primeand m ∈ Z+ so that p - m, we have∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p) = η(p)c(∆m)where, as in Theorem 1.1,η(p) =1 + χ(−1)(−1)k if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.Proof. Suppose that {∆ : Ω} = (1/p, 1). Then {∆ : pΩ} = (1, p); also, withT a matrix so that Ωm/p ' mp T , we have pΩm/p ' pmT . This proves that∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p).Let {x, y} be a basis for ∆ relative to which ∆ ' 2I, and suppose that{∆ : Ω} = (1, p). Thus Ω = Z(x+uy)⊕Zpy for 0 ≤ u < p or Ω = Zpx⊕Zy.Hence Ωm/p is even integral if and only if Ω = Z(x + uy) ⊕ Zpy with u2 ≡−1 (p). If p ≡ 3 (4), there are no such u. Suppose that p ≡ 1 (4), and fix u6 LYNNE H. WALLINGso that u2 ≡ −1 (p). Set Ωu = Z(x+uy)⊕Zpy and Ω−u = Z(x−uy)⊕Zpy.Then Ω1/pu and Ω1/p−u are integral with determinant 1. Thus by Exercise 5p. 77 of [4], there is some G ∈ GL2(Z) so that tGTG = I. Thereforec(Ωm/pu ) = χ(detG)(detG)kc(∆m). When p = 2, Ωm/2 is even integral onlyfor Ω1 = Z(x + y) ⊕ Z2y ' 2(1 11 2). Since tG(1 11 2)G = I for G =(1 −10 1), we have c(Ω1/21 ) = c(∆). Thus when p = 2, the sum on Ω isc(Ωm/2) = c(∆m). In the next proposition we use Lemma 3.1 to establish some very usefulidentities.Proposition 3.2. Suppose that F is a degree 2 Siegel modular form ofweight k, level N , character χ, and and lattice coefficients c(Λ). Alsosuppose that F |T (p) = λ(p)F and F |T˜1(p2) = λ˜1(p2)F . Set η(1) = 0,κ(1) = 1. With ∆ ' 2I and m ∈ Z+ so that p - m, for r ≥ 1 we in-ductively define η(pr) and κ(pr) as follows: η(p) is as in Proposition 3.1,κ(p) = λ(p)− χ(p)pk−2η(p), and for r ≥ 2,η(pr) = λ˜1(p2)κ(pr−2)− χ(p2)p2k−3η(pr−2)− χ(p)pk−2α(∆pr−2 ; p)κ(pr−2)andκ(pr) = λ(p)κ(pr−1)− χ(p2)p2k−3κ(pr−2)− χ(p)pk−2η(pr).Then we have∑{∆:Ω}=(1/p,1)c(Ωprm) =∑{∆:Ω}=(1,p)c(Ωpr−2m) = η(pr)c(∆m)(1)andc(∆prm) = κ(pr)c(∆m).(2)Proof. Recall that the value of α(∆; p) is computed after Theorem 2.1; notethat for r ≥ 1, α(∆pr ; p) = p + 1 as then ∆pr/p∆pr is totally isotropic andcontains p + 1 lines. Also, note that the first equality in Equation (1) iseasily verified by replacing Ω by pΩ. We now compute η(pr) and κ(pr).(Case r = 0:) With κ(1) = 1, it is clear that c(∆) = κ(1)c(∆). So supposethat we have {∆ : Ω} = (1, p). Then disc Ωm/p2 = 4m2/p2. Hence whenp 6= 2, Ωm/p cannot be integral, so c(Ωm/p) = 0. When p = 2, we see fromthe discussion at the end of the proof of Lemma 3.1 that Ωm/4 is not evenintegral for any Ω with {∆ : Ω} = (1, 2). Thus Equation (1) holds withη(1) = 0.(Case r = 1:) In Lemma 3.1 we showed that Equation (1) holds withη(p) as defined therein. We know that c(∆m/p) = 0 since ∆m/p is not evenintegral, and so by Theorem 2.1 and the above conclusion we haveκ(p)c(∆m)λ(p)c(∆m)− χ(p)pk−2η(p)c(∆m).(Induction step:) Suppose that r ≥ 2 and that the proposition holds forall ` with 0 ≤ ` < r. First, from Theorem 2.1 and the induction hypothesisSOME RELATIONS ON FOURIER COEFFICIENTS 7we have∑{∆:Ω}=(1,p)c(Ωpr−2m) = (λ˜1(p2)κ(pr−2)− χ(p2)p2k−3η(pr−2))c(∆m)− χ(p)pk−2α(∆pr−2 ; p)κ(pr−2)c(∆m)= η(pr)c(∆m).Hence we also havec(∆prm) = (λ(p)κ(pr−1)− χ(p2)p2k−3κ(pr−2)− χ(p)pk−2η(pr))c(∆m)= κ(pr)c(∆m).Thus induction on r proves the proposition. We also have the following helpful result.Proposition 3.3. Suppose that F is a degree 2 Siegel modular form ofweight k, level N , character χ, and lattice coefficients c(Λ); recall thatc(Λ) = a(TΛ) where Λ ' 2TΛ. Fix a prime p and r ≥ 1; take ∆ ' 2Irelative to a Z-basis {x, y}. Set  = 1 + χ(−1)(−1)k. Then with η(p) asdefined in Lemma 3.1 and η(pr+1) as defined in Proposition 3.2, we haveη(p)a(prI)− η(pr+1)a(I)= −a(pr−1mpr+1m)− ∑1≤u<p/2u2 6≡−1 (p)a(prm((1 + u2)/p uu p)).Proof. By Proposition 3.2, η(pr+1)c(∆) =∑{∆:Ω}=(1,p) c(Ωpr−1). With Ω sothat {∆ : Ω} = (1, p), we either have Ω = Z(x + uy) ⊕ Zpy for 0 ≤ u < p,or Ω = Zpx ⊕ Zy. Then for u 6= 0, we have Ωu = Z(x + uy) ⊕ Zpy '2pr+1(((1 + u2)/p uu p); from our above discussion on Fourier coefficients ofa Siegel modular form F , we have c(Ω1/pu ) = χ(−1)(−1)kc(Ω1/p−u ). Similarly,c((Zpx⊕ Zy)1/p) = χ(−1)(−1)kc((Zx⊕ Zpy)1/p).Further, if p is odd and u2 ≡ −1 (p), then by Exercise 5 p. 77 of [4], thereis some G ∈ GL2(Z) so thattG((1 + u2)/p uu p)G = I;hence with G′ = diag(−1, 1)G, we gettG′((1 + u2)/p −u−u p)G′ = I,and thus c(Ω1/pu ) + c(Ω1/p−u ) = (1 + χ(−1)(−1)k)c(∆pr). Similarly, whenp = 2, Ω1 ' 2r+2m(1 11 2), which can be diagonalized using the matrixG =(1 −10 1), and so c(Ω1/21 ) = c(∆pr). Using the definition of η(p), theproposition now follows. 8 LYNNE H. WALLINGTheorem 1.1 is now easy to prove. Take ∆ ' 2I; recall that c(∆prm) =a(prmI). The first claim of (a) follows immediately from Theorem 2.1 andLemma 3.1. To prove the second claim in (a), we first use Theorem 2.1 togetλ˜1(p2)c(∆m) = χ(p)pk−2α(∆; p)c(∆m) +∑{∆:Ω}=(1,p)c(Ω)(3)andλ(p)c(∆pm) = χ(p2)p2k−3c(∆) + χ(p)pk−2∑{∆:Ω}=(1,p)c(Ω) + c(∆p2).(4)Solving Equation (4) for the sum on Ω and substituting into χ(p)pk−2·Equation(3) yields the second claim in (a).To prove (b), we first use Theorem 2.1 and Proposition 3.2 to obtaina(pr+1I) = λ(p)a(prI)− χ(p2)p2k−3a(pr−1I)− χ(p)pk−2η(pr+1)a(I).Next we multiply this equation by a(mI), use Theorem 1.1(a) to substi-tute for λ(p)a(mI), and use Proposition 3.3 to substitute for η(p)a(prI) −η(pr−1I)a(I); (b) now immediately follows.For (c), suppose that n = pe11 · · · pett where p1, . . . , pt are distinct primesso that F is an eigenform for T (pi) and T˜1(p2i ) (1 ≤ i ≤ t). For any m′ ∈ Z+with (n,m′) = 1, repeated applications of Proposition 3.2 gives usa(m′nI) = κ(pe11 ) · · ·κ(pett )a(m′I).Thus (taking m′ = m) we have a(mnI) = 0 if a(mI) = 0. Further (takingm′ = 1), we havea(nI) = κ(pe11 ) · · ·κ(pett )a(I)and hence a(I)a(mnI) = a(mI)κ(pe11 ) · · ·κ(pett )a(I) = a(mI)a(nI).4. Proof of Theorem 1.2As previously noted, the key to proving Theorem 1.1 is Lemma 3.1. Wecan extend this lemma to some extent, as follows.Lemma 4.1. Suppose that F is a degree 2 Siegel modular form of weightk, level N , and character χ, and let c(Λ) denote the Λth coefficient of F .Suppose that p is an odd prime and ∆ ' 2(1p). For m ∈ Z+ with p - m,we have ∑{∆:Ω}=(1/p,1)c(Ωpm) = χ(−1)(−1)kc(∆m).For q an odd prime with(−pq)= −1 and q - m, we have∑{∆:Ω}=(1/q,1)c(Ωqm) = 0.SOME RELATIONS ON FOURIER COEFFICIENTS 9Proof. Let {x, y} be a Z-basis for ∆ relative to which ∆ '(22p). Thenthe only lattice Ω so that {∆ : Ω} = (1, p) and Ωm/p is even integral ifΩ = Zpx⊕ Zy ' 2p(p1)= 2p(0 11 0)(1p)(0 11 0).With γ = diag((0 11 0),(0 11 0)), we have F |γ = χ(−1)F and conse-quently c(2m(p1))= χ(−1)(−1)kc(2m(1p)). Hence∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p) = χ(−1)(−1)kc(∆m).With q an odd prime with(−pq)= −1 and q - m, there is no lattice Ω sothat {∆ : Ω} = (1, q) and Ωm/q is even integral, and hence∑{∆:Ω}=(1/q,1)c(Ωqm) = 0.To prove Theorem 1.2, we begin by making the following definitions. Setη(1) = 0, κ(1) = 1. For q ∈ S (as defined in the statement of Theorem 1.2),define η(q) as in Lemma 4.1, and set κ(q) = λ(q)−χ(q)qk−2η(q). For r ≥ 2,we define η(qr) and κ(qr) using the inductive formulas from Proposition 3.2(so η(qr), κ(qr) are determined by η(q), λ(q) and λ˜1(q2)). Then mimickingthe proofs of Proposition 3.2 and Theorem 1.1(c) easily yields Theorem 1.2.References[1] A.N. Andrianov, “Euler products that correspond to Siegels modular forms of genus 2”(Russian) Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43110, (English transl.) RussianMath. Surveys 29 (1974).[2] J.L. Hafner, L.H. Walling, “Explicit action of Hecke operators on Siegel modularforms”. J. Number Theory 93 (2002), 34-57.[3] D. McCarthy, “Multiplicative relations for Fourier coefficients of degree 2 Siegel eigen-forms”. J. Number Theory (to appear), 16 p.; arXiv:1505.07049.[4] M. Newman, Integral Matrices, Academic Press, 1972.[5] L. Walling, “Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series”, J.Number Theory 132 (2013), 2700-2723.[6] L. Walling, “Hecke eigenvalues and relations for degree n Siegel Eisenstein series ofsquare-free level”, Int. J. Number Theory (to appear 2017).School of Mathematics, University of Bristol, University Walk, Clifton,Bristol BS8 1TW, United Kingdom; phone +44 (0)117 331-5245, fax +44 (0)117928-7978E-mail address: l.walling@bristol.ac.uk

Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

Walling, L. (2017). Some relations on Fourier coefficients of degree 2Siegel forms of arbitrary level. Journal of Number Theory. Advanceonline publication. https://doi.org/10.1016/j.jnt.2017.04.003Peer reviewed versionLicense (if available):CC BY-NC-NDLink to published version (if available):10.1016/j.jnt.2017.04.003Link to publication record on the Bristol Research PortalPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Elsevier at http://www.sciencedirect.com/science/article/pii/S0022314X17301890#. Please refer to anyapplicable terms of use of the publisher.University of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/SOME RELATIONS ON FOURIER COEFFICIENTS OFDEGREE 2 SIEGEL FORMS OF ARBITRARY LEVELLYNNE H. WALLINGAbstract. We extend some recent work of D. McCarthy, proving rela-tions among some Fourier coefficients of a degree 2 Siegel modular formF with arbitrary level and character, provided there are some primes pso that F is an eigenform for the Hecke operators T (p) and T1(p2).1. IntroductionIn a recent paper [3], McCarthy derives some nice results for Fourier co-efficients and Hecke eigenvalues of degree 2 Siegel modular forms of level 1,extending some classical results regarding elliptic modular forms. In partic-ular, with F a degree 2, level 1 Siegel modular form that is an eigenform forall the Hecke operators T (p), T (p2) (p prime), and a(T ) denoting the T thFourier coefficient of F , McCarthy shows that:(a) provided that a(I) = 1 and p is prime, the T (p)-eigenvalue λ(p) andthe T (p2)-eigenvalue λ(p2) are described explicitly in terms of a(pI)and a(p2I);(b) for r ≥ 1, a(I)a(pr+1I) is described explicitly in terms of a(I), a(pI),a(pr−1I), a(pr−1pr+1), and a(pr((1 + u2)/p uu p))where 1 ≤u < p/2 with u2 6≡ 1 (p);(c) if a(I) = 0 then a(mI) = 0 for all m ∈ Z+; further, if m,n ∈ Z+with (m,n) = 1, then a(I)a(mnI) = a(mI)a(nI).(As defined in Sec. 2, T2(p2) is the Hecke operator associated with thematrix diag(p, p, 1/p, 1/p), T1(p2) is the Hecke operator associated with thematrix diag(p, 1, 1/p, 1), and T (p2) = T2(p2) + pk−3T1(p2) + p2k−6. In [2],for χ = 1,T (p2) is denoted by T̃2(p2).) McCarthy’s approach begins withsome formulas from [1], which are somewhat cumbersome.In this note we use the formulas from [2] that give the action of Heckeoperators on Fourier coefficients of a Siegel modular form F , allowing forarbitrary level and character, and giving a simpler proof of McCarthy’sabove results (with no restriction on the level or character). Here when wesay that a modular form has weight k, levelN and character χ, we mean thatit transforms with weight k and character χ under the congruence subgroupΓ0(N ) ={(A BC D)∈ Sp2(Z) : N|C},Key words and phrases. Hecke eigenvalues, Siegel modular forms.2010 Mathematics Subject Classification: Primary 11F46, 11F1112 LYNNE H. WALLINGwhere Sp2(Z) is the symplectic group of 4 × 4 integral matrices. We workwith “Fourier coefficients” attached to lattices (as explained below), makingit simpler to work with the image of F under a Hecke operator. For p primeand degree 2, the local Hecke algebra is generated by T (p), T1(p2) andT2(p2). When N = 1, Proposition 5.1 of [2] gives a relation between thesegenerators, from which we deduce that with p - N , T (p) and T1(p2) generatethe local Hecke algebra, as do T (p) and T̃2(p2). However, when p|N , we haveT2(p2) = (T (p))2. Hence in this note we use the local generators T (p) andT1(p2); to more easily apply the results of [2], we use the operatorT̃1(p2) = T1(p2) + χ(p)pk−3(p+ 1)in place of T1(p2).Using some rather special aspects of working with degree 2 Siegel modularforms, we prove the following extensions of [3].Theorem 1.1. Suppose that F is a degree 2 Siegel modular form of weightk ∈ Z+, level N and character χ with Fourier expansionF (τ) =∑Ta(T ) exp(2πiTr(Tτ)).Also suppose that p is prime with F |T (p) = λ(p)F and F |T̃1(p2) = λ̃1(p2)F .(a) We haveλ(p)a(mI) = χ(p)pk−2η(p)a(mI) + a(mpI),whereη(p) =1 + χ(−1)(−1)k if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.(Thus when a(mI) 6= 0, λ(p) is given explicitly in terms of p, a(mI)and a(pmI).) As well, we haveχ(p)pk−2λ̃1(p2)a(mI) = χ(p2)p2k−4(α(I; p)− p)a(mI)+ λ(p)a(pmI)− a(p2mI)whereα(I; p) =2 if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.(Thus when χ(p)a(mI) 6= 0, λ̃1(p2) is given explicitly in terms of p,a(mI), a(pmI) and a(p2mI).)(b) Set ε = 1 + χ(−1)(−1)k. For r ≥ 1, a(mI)a(pr+1I) is given bya(pmI)a(prI)− χ(p2)p2k−3a(mI)a(pr−1I)+ εχ(p)pk−2a(mI)a(pr−1mpr+1m)+ εχ(p)pk−2a(mI)∑1≤u<p/2u2 6≡−1 (p)a(prm((1 + u2)/p uu p)).SOME RELATIONS ON FOURIER COEFFICIENTS 3(c) Suppose that n a product of powers of primes p so that F is aneigenform for T (p) and T̃1(p2), and that m ∈ Z+ with (m,n) =1. If a(mI) = 0 then a(mnI) = 0. Also, we have a(I)a(mnI) =a(mI)a(nI).We also prove the following modest generalization.Theorem 1.2. Suppose that F is a degree 2 Siegel modular form of weightk ∈ Z+, level N and character χ with Fourier expansionF (τ) =∑Ta(T ) exp(2πiTr(Tτ)).Suppose that p is an odd prime, and set D =(1p). Let S be the set ofodd primes so that for q ∈ S, F is an eigenform for T (q) and T̃1(q2), andeither q = p or(−pq)= −1. Let n be a product of powers of primes in S.Then for any m ∈ Z+ so that (m,n) = 1, we havea(D)a(mnD) = a(mD)a(nD).Also, a(D)a(mnD) = 0 if a(mD) = 0.We note that McCarthy applies his results to compute eigenvalues of thelevel 1 Eisenstein series with regard to the Hecke operators T (pr) (p prime);as he notes, in [5] we computed the Hecke-eigenvalues of Eisenstein seriesof square-free levels for all primes p, allowing nontrivial character (thengeneralized in [6] for arbitrary level N and character χ, but only for primesp so that p2 - N ).We further note that it seems that these results cannot be extended tohigher degrees, as Lemma 3.1 (which is pivotal for our arguments) does notextend to higher degrees.2. PreliminariesWe will use some language and notation commonly used in quadraticforms and modular forms theory. When Λ is a lattice whose quadratic formis given by the matrix T (relative to some Z-basis for Λ), we write Λ ' T .Now suppose that Λ is a lattice with Λ ' T and that m ∈ Q+; we writeΛm to denote the lattice Λ “scaled” by m, meaning that Λm ' mT . Also,the discriminant of Λ is detT . With Λ,Ω lattices on the same unlerlyingquadratic space over Q, we write {Λ : Ω} to denote the invariant factors ofΩ in Λ.We seth(2) = {X + iY : X,Y ∈ R2,2sym : Y > 0 },where R2,2sym denotes the set of 2 × 2 symmetric matrices with real entries,and Y > 0 means that Y represents a positive definite quadatic form. Fora ring R, we write Sp2(R) for the group of 4 × 4 symplectic matrices withentries in R. Fixing a weight k ∈ Z+, for γ =(A BC D)∈ Sp2(Q), we defineF (τ)|γ = (det γ)k/2 det(Cτ +D)−kF ((Aτ +B)(Cτ +D)−1).4 LYNNE H. WALLINGWhen F is a degree 2 Siegel modular form of weight k, level N and characterχ, this means that for γ =(A BC D)∈ Γ0(N ), we haveF (τ)|γ = χ(detDγ)F (τ).We can write F as a Fourier series:F (τ) =∑T≥0a(T ) exp(2πiTr(Tτ))where the sum is over 2 × 2 symmetric, positive semi-definite, half-integralmatrices T (so the entries in T are half-integers with integers on the diago-nal). Given G ∈ GL2(Z), we have γ =(G−1tG)∈ Γ0(N ). Henceχ(detG)F (τ) = F (τ)|γ= (detG)kF (G−1τ tG−1)= (detG)k∑Ta( tGTG) exp(2πiTr(Tτ)).Thus a(tGTG) = χ(detG)(detG)ka(T ). So we can also write F as a “Fourierseries” supported on isometry classes of even integral, positive semi-definitelattices: For Λ an even integral lattice with Z-basis {x, y}, set c(Λ) = a(TΛ)where, relative to the given basis for Λ, we have Λ ' 2TΛ. When χ(−1) 6=(−1)k, we equip Λ with an orientation, meaning that with G ∈ GL2(Z),(x y)G is a basis for the oriented lattice Λ if and only if detG = 1. ThenF (τ) =∑cls Λc(Λ) e∗{Λτ}where cls Λ varies over all isometry classes of (oriented) even integral, posi-tive semi-definite lattices, ande∗{Λτ} =∑Gexp(2πiTr( tGTΛGτ))where G varies over O(Λ)\GL2(Z) when χ(−1) = (−1)k, and G varies overO+(Λ)\SL2(Z) otherwise. (Here O(Λ) denotes the orthogonal group of Λ,and O+(Λ) = O(Λ) ∩ SL2(Z).)Still suppose that F is a Siegel modular form of degree 2, weight k, levelN and character χ. For p prime, we define T (p), T1(p2), and T2(p2) asfollows. Take δ(p) = diag(p, p, 1, 1), δ1(p2) = (p, 1, 1/p, 1), and δ2(p2) =diag(p, p, 1/p, 1/p). With Γ = Γ0(N ), we setF |T (p) = pk−3∑γχ(γ)F |δ(p)−1γwhere γ varies over (δ(p)Γδ(p)−1 ∩ Γ)\Γ, and for j = 1, 2, we setF |Tj(p2) = pj(k−3)∑γχ(γ)F |δj(p2)−1γwhere γ varies over (δj(p2)Γδj(p2)−1 ∩ Γ)\Γ. Note that replacing δ(p) orδj(p2) by a scalar multiple of itself does not change the definition of theassociated Hecke operator. Note also that in [2], we did not normalizeSOME RELATIONS ON FOURIER COEFFICIENTS 5Tj(p2) by pj(k−3), as is usually done in other texts, and has been done in theabove formula for T1(p2). With T̃1(p2) = T1(p2)+χ(p)pk−3(p+1), Theorem6.1 of [2] gives us the following.Theorem 2.1. Let F be a degree 2 Siegel modular form of weight k, levelN , character χ, and lattice coefficients c(Λ). Then for any even integrallattice Λ, the Λth coefficient of F |T (p) isχ(p2)p2k−3c(Λ1/p) + χ(p)pk−2 ·∑{Λ:Ω}=(1,p)c(Ω1/p) + c(Λp),and the Λth coefficient of F |T̃1(p2) isχ(p2)p2k−3 ·∑{Λ:Ω}=(1/p,1)c(Ω) + χ(p)pk−2α(Λ; p)c(Λ) +∑{Λ:Ω}=(1,p)c(Ω).With Q the quadratic form on Λ, we equip Λ/pΛ with the quadratic form12Q, and α(Λ; p) is the number of isotropic lines in the quadratic space Λ/pΛ.There are p+ 1 lines in Λ/pΛ, and each of these lines is generated either byy + pΛ or by (x + uy) + pΛ for some u with 0 ≤ u < p. So with Λ ' 2I,α(Λ; 2) = 1, α(Λ; p) = 2 when p ≡ 1 (4), and α(Λ; p) = 0 when p ≡ 3 (4).When Λ ' 2T with p|T , α(Λ; p) = p+ 1.Note that with p a prime and m ∈ Z+ so that p - m, for any even integralrank 2 lattice Λ we have α(Λ; p) = α(Λm; p) since scaling by m does notchange whether a line is isotropic in Λ/pΛ.3. Proof of Theorem 1.1The next lemma is pivotal in our proof of Theorem 1.1; when this lemmageneralizes, we can generalize this theorem (as seen in Theorem 1.2).Lemma 3.1. Suppose that F is a degree 2 Siegel modular form of weight k,level N , character χ, and lattice coefficients c(Λ). With ∆ ' 2I, p primeand m ∈ Z+ so that p - m, we have∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p) = η(p)c(∆m)where, as in Theorem 1.1,η(p) =1 + χ(−1)(−1)k if p ≡ 1 (4),0 if p ≡ 3 (4),1 if p = 2.Proof. Suppose that {∆ : Ω} = (1/p, 1). Then {∆ : pΩ} = (1, p); also, withT a matrix so that Ωm/p ' mp T , we have pΩm/p ' pmT . This proves that∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p).Let {x, y} be a basis for ∆ relative to which ∆ ' 2I, and suppose that{∆ : Ω} = (1, p). Thus Ω = Z(x+uy)⊕Zpy for 0 ≤ u < p or Ω = Zpx⊕Zy.Hence Ωm/p is even integral if and only if Ω = Z(x + uy) ⊕ Zpy with u2 ≡−1 (p). If p ≡ 3 (4), there are no such u. Suppose that p ≡ 1 (4), and fix u6 LYNNE H. WALLINGso that u2 ≡ −1 (p). Set Ωu = Z(x+uy)⊕Zpy and Ω−u = Z(x−uy)⊕Zpy.Then Ω1/pu and Ω1/p−u are integral with determinant 1. Thus by Exercise 5p. 77 of [4], there is some G ∈ GL2(Z) so that tGTG = I. Thereforec(Ωm/pu ) = χ(detG)(detG)kc(∆m). When p = 2, Ωm/2 is even integral onlyfor Ω1 = Z(x + y) ⊕ Z2y ' 2(1 11 2). Since tG(1 11 2)G = I for G =(1 −10 1), we have c(Ω1/21 ) = c(∆). Thus when p = 2, the sum on Ω isc(Ωm/2) = c(∆m). In the next proposition we use Lemma 3.1 to establish some very usefulidentities.Proposition 3.2. Suppose that F is a degree 2 Siegel modular form ofweight k, level N , character χ, and and lattice coefficients c(Λ). Alsosuppose that F |T (p) = λ(p)F and F |T̃1(p2) = λ̃1(p2)F . Set η(1) = 0,κ(1) = 1. With ∆ ' 2I and m ∈ Z+ so that p - m, for r ≥ 1 we in-ductively define η(pr) and κ(pr) as follows: η(p) is as in Proposition 3.1,κ(p) = λ(p)− χ(p)pk−2η(p), and for r ≥ 2,η(pr) = λ̃1(p2)κ(pr−2)− χ(p2)p2k−3η(pr−2)− χ(p)pk−2α(∆pr−2 ; p)κ(pr−2)andκ(pr) = λ(p)κ(pr−1)− χ(p2)p2k−3κ(pr−2)− χ(p)pk−2η(pr).Then we have∑{∆:Ω}=(1/p,1)c(Ωprm) =∑{∆:Ω}=(1,p)c(Ωpr−2m) = η(pr)c(∆m)(1)andc(∆prm) = κ(pr)c(∆m).(2)Proof. Recall that the value of α(∆; p) is computed after Theorem 2.1; notethat for r ≥ 1, α(∆pr ; p) = p + 1 as then ∆pr/p∆pr is totally isotropic andcontains p + 1 lines. Also, note that the first equality in Equation (1) iseasily verified by replacing Ω by pΩ. We now compute η(pr) and κ(pr).(Case r = 0:) With κ(1) = 1, it is clear that c(∆) = κ(1)c(∆). So supposethat we have {∆ : Ω} = (1, p). Then disc Ωm/p2 = 4m2/p2. Hence whenp 6= 2, Ωm/p cannot be integral, so c(Ωm/p) = 0. When p = 2, we see fromthe discussion at the end of the proof of Lemma 3.1 that Ωm/4 is not evenintegral for any Ω with {∆ : Ω} = (1, 2). Thus Equation (1) holds withη(1) = 0.(Case r = 1:) In Lemma 3.1 we showed that Equation (1) holds withη(p) as defined therein. We know that c(∆m/p) = 0 since ∆m/p is not evenintegral, and so by Theorem 2.1 and the above conclusion we haveκ(p)c(∆m)λ(p)c(∆m)− χ(p)pk−2η(p)c(∆m).(Induction step:) Suppose that r ≥ 2 and that the proposition holds forall ` with 0 ≤ ` < r. First, from Theorem 2.1 and the induction hypothesisSOME RELATIONS ON FOURIER COEFFICIENTS 7we have∑{∆:Ω}=(1,p)c(Ωpr−2m) = (λ̃1(p2)κ(pr−2)− χ(p2)p2k−3η(pr−2))c(∆m)− χ(p)pk−2α(∆pr−2 ; p)κ(pr−2)c(∆m)= η(pr)c(∆m).Hence we also havec(∆prm) = (λ(p)κ(pr−1)− χ(p2)p2k−3κ(pr−2)− χ(p)pk−2η(pr))c(∆m)= κ(pr)c(∆m).Thus induction on r proves the proposition. We also have the following helpful result.Proposition 3.3. Suppose that F is a degree 2 Siegel modular form ofweight k, level N , character χ, and lattice coefficients c(Λ); recall thatc(Λ) = a(TΛ) where Λ ' 2TΛ. Fix a prime p and r ≥ 1; take ∆ ' 2Irelative to a Z-basis {x, y}. Set ε = 1 + χ(−1)(−1)k. Then with η(p) asdefined in Lemma 3.1 and η(pr+1) as defined in Proposition 3.2, we haveη(p)a(prI)− η(pr+1)a(I)= −εa(pr−1mpr+1m)− ε∑1≤u<p/2u2 6≡−1 (p)a(prm((1 + u2)/p uu p)).Proof. By Proposition 3.2, η(pr+1)c(∆) =∑{∆:Ω}=(1,p) c(Ωpr−1). With Ω sothat {∆ : Ω} = (1, p), we either have Ω = Z(x + uy) ⊕ Zpy for 0 ≤ u < p,or Ω = Zpx ⊕ Zy. Then for u 6= 0, we have Ωu = Z(x + uy) ⊕ Zpy '2pr+1(((1 + u2)/p uu p); from our above discussion on Fourier coefficients ofa Siegel modular form F , we have c(Ω1/pu ) = χ(−1)(−1)kc(Ω1/p−u ). Similarly,c((Zpx⊕ Zy)1/p) = χ(−1)(−1)kc((Zx⊕ Zpy)1/p).Further, if p is odd and u2 ≡ −1 (p), then by Exercise 5 p. 77 of [4], thereis some G ∈ GL2(Z) so thattG((1 + u2)/p uu p)G = I;hence with G′ = diag(−1, 1)G, we gettG′((1 + u2)/p −u−u p)G′ = I,and thus c(Ω1/pu ) + c(Ω1/p−u ) = (1 + χ(−1)(−1)k)c(∆pr). Similarly, whenp = 2, Ω1 ' 2r+2m(1 11 2), which can be diagonalized using the matrixG =(1 −10 1), and so c(Ω1/21 ) = c(∆pr). Using the definition of η(p), theproposition now follows. 8 LYNNE H. WALLINGTheorem 1.1 is now easy to prove. Take ∆ ' 2I; recall that c(∆prm) =a(prmI). The first claim of (a) follows immediately from Theorem 2.1 andLemma 3.1. To prove the second claim in (a), we first use Theorem 2.1 togetλ̃1(p2)c(∆m) = χ(p)pk−2α(∆; p)c(∆m) +∑{∆:Ω}=(1,p)c(Ω)(3)andλ(p)c(∆pm) = χ(p2)p2k−3c(∆) + χ(p)pk−2∑{∆:Ω}=(1,p)c(Ω) + c(∆p2).(4)Solving Equation (4) for the sum on Ω and substituting into χ(p)pk−2·Equation(3) yields the second claim in (a).To prove (b), we first use Theorem 2.1 and Proposition 3.2 to obtaina(pr+1I) = λ(p)a(prI)− χ(p2)p2k−3a(pr−1I)− χ(p)pk−2η(pr+1)a(I).Next we multiply this equation by a(mI), use Theorem 1.1(a) to substi-tute for λ(p)a(mI), and use Proposition 3.3 to substitute for η(p)a(prI) −η(pr−1I)a(I); (b) now immediately follows.For (c), suppose that n = pe11 · · · pett where p1, . . . , pt are distinct primesso that F is an eigenform for T (pi) and T̃1(p2i ) (1 ≤ i ≤ t). For any m′ ∈ Z+with (n,m′) = 1, repeated applications of Proposition 3.2 gives usa(m′nI) = κ(pe11 ) · · ·κ(pett )a(m′I).Thus (taking m′ = m) we have a(mnI) = 0 if a(mI) = 0. Further (takingm′ = 1), we havea(nI) = κ(pe11 ) · · ·κ(pett )a(I)and hence a(I)a(mnI) = a(mI)κ(pe11 ) · · ·κ(pett )a(I) = a(mI)a(nI).4. Proof of Theorem 1.2As previously noted, the key to proving Theorem 1.1 is Lemma 3.1. Wecan extend this lemma to some extent, as follows.Lemma 4.1. Suppose that F is a degree 2 Siegel modular form of weightk, level N , and character χ, and let c(Λ) denote the Λth coefficient of F .Suppose that p is an odd prime and ∆ ' 2(1p). For m ∈ Z+ with p - m,we have ∑{∆:Ω}=(1/p,1)c(Ωpm) = χ(−1)(−1)kc(∆m).For q an odd prime with(−pq)= −1 and q - m, we have∑{∆:Ω}=(1/q,1)c(Ωqm) = 0.SOME RELATIONS ON FOURIER COEFFICIENTS 9Proof. Let {x, y} be a Z-basis for ∆ relative to which ∆ '(22p). Thenthe only lattice Ω so that {∆ : Ω} = (1, p) and Ωm/p is even integral ifΩ = Zpx⊕ Zy ' 2p(p1)= 2p(0 11 0)(1p)(0 11 0).With γ = diag((0 11 0),(0 11 0)), we have F |γ = χ(−1)F and conse-quently c(2m(p1))= χ(−1)(−1)kc(2m(1p)). Hence∑{∆:Ω}=(1/p,1)c(Ωpm) =∑{∆:Ω}=(1,p)c(Ωm/p) = χ(−1)(−1)kc(∆m).With q an odd prime with(−pq)= −1 and q - m, there is no lattice Ω sothat {∆ : Ω} = (1, q) and Ωm/q is even integral, and hence∑{∆:Ω}=(1/q,1)c(Ωqm) = 0.To prove Theorem 1.2, we begin by making the following definitions. Setη(1) = 0, κ(1) = 1. For q ∈ S (as defined in the statement of Theorem 1.2),define η(q) as in Lemma 4.1, and set κ(q) = λ(q)−χ(q)qk−2η(q). For r ≥ 2,we define η(qr) and κ(qr) using the inductive formulas from Proposition 3.2(so η(qr), κ(qr) are determined by η(q), λ(q) and λ̃1(q2)). Then mimickingthe proofs of Proposition 3.2 and Theorem 1.1(c) easily yields Theorem 1.2.References[1] A.N. Andrianov, “Euler products that correspond to Siegels modular forms of genus 2”(Russian) Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43110, (English transl.) RussianMath. Surveys 29 (1974).[2] J.L. Hafner, L.H. Walling, “Explicit action of Hecke operators on Siegel modularforms”. J. Number Theory 93 (2002), 34-57.[3] D. McCarthy, “Multiplicative relations for Fourier coefficients of degree 2 Siegel eigen-forms”. J. Number Theory (to appear), 16 p.; arXiv:1505.07049.[4] M. Newman, Integral Matrices, Academic Press, 1972.[5] L. Walling, “Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series”, J.Number Theory 132 (2013), 2700-2723.[6] L. Walling, “Hecke eigenvalues and relations for degree n Siegel Eisenstein series ofsquare-free level”, Int. J. Number Theory (to appear 2017).School of Mathematics, University of Bristol, University Walk, Clifton,Bristol BS8 1TW, United Kingdom; phone +44 (0)117 331-5245, fax +44 (0)117928-7978E-mail address: l.walling@bristol.ac.uk

Lynne H. Walling

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Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

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