HERMITIAN JACOBI FORMS AND U(p) CONGRUENCES OLAV K. RICHTER AND JAYANTHA SENADHEERA. (Communicated by Ken Ono)

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY S XX HERMITIAN JACOBI FORMS AND Up CONGRUENCES OLAV K. RICHTER AND JAYANTHA SENADHEERA Communicated by Ken Ono Abstract. We introduce a new space of Hermitian Jacobi forms, and we determine its structure. Moreover, we characterize Up congruences of Hermitian Jacobi forms, and we discuss an explicit example.. Introduction and statement of main results Jacobi forms appear naturally in various areas of mathematics and physics, and they connect different types of automorphic forms. In particular, they occur as Fourier-Jacobi coefficients of Siegel modular forms of degree, a fact that figured prominently in the solution of the Saito-Kurokawa conjecture see [,8,5 7,9]. Hermitian modular forms are generalizations of Siegel modular forms, and Hermitian Jacobi forms occur as Fourier-Jacobi coefficients of Hermitian modular forms of degree. Haverkamp [0,] systematically studied Hermitian Jacobi forms, and [5 7, 3] contributed further to the theory of such forms. The usual heat operator is an important device in the study of classical Jacobi forms see, for example, [3,4,8], and its action on Jacobi forms can be corrected so that Jacobi forms of weight k are mapped to Jacobi forms of weight k +see []. The heat operator L m := πi 8πim. τ 4 w z is a natural tool in the theory of Hermitian Jacobi forms, and it plays a vital role in Section 3 of this paper. Equation 3.7 of [4] implicitly gives an action of L m on Hermitian Jacobi forms, but unfortunately, 3.7 of [4] is not quite correct. In fact, the action of L m on the Hermitian Jacobi forms in [5, 6, 0, 3] cannot be corrected as in the case of classical Jacobi forms, and one needs a different notion of Hermitian Jacobi form. In this paper, we introduce the more general space Jk,m δ of Hermitian Jacobi forms over Qi ofweightk, index m, and parity δ = ±, and we set J k,m := J + k,m J k,m see Definition.. A direct computation shows that if φ J k,m δ,then. k m L m φ = E φ + 3 φ, where E is the usual quasimodular Eisenstein series, and where φ J δ k+,m.the fact that φ and φ in. are Hermitian Jacobi forms of different parities explains the need for our Definition.. Received by the editors June, Mathematics Subject Classification. Primary F50; Secondary F33. The first author was partially supported by Simons Foundation Grant # c American Mathematical Society

2 OLAV K. RICHTER AND JAYANTHA SENADHEERA We now introduce notation necessary to state our main results. Throughout, k is an even integer, and M k and S k are the weight k spaces of elliptic modular forms and cusp forms, respectively. We denote the Taylor coefficients of a Hermitian Jacobi form of parity δ by χ δ μ,ν see Section.3, and ζμ,ν δ are the combinations of Taylor coefficients in Proposition.7. Our first result gives the structure of Hermitian Jacobi forms over Qi of index. More specifically, it asserts that J k, is a free module of rank 4 over the ring of modular forms, where a set of generators is given by the Hermitian Jacobi forms φ + 4,, φ 6,, φ+ 8,,andφ+,cusp 0, definedin.5 and Lemma.4. Theorem.. If k 0mod4, then both linear maps and ζ : J k, M k S k+ S k+ S k+4, φ χ + 0,0, ζ+,, χ,0, ζ+, χ+ 4,0 η : M k 4 M k 6 M k 8 M k 0 J k,, are isomorphisms. If k mod4, then both linear maps and e, f, g, h eφ + 4, + fφ 6, + gφ+ 8,, hφ+,cusp ζ : J k, M k S k+ S k+ S k+4, φ χ 0,0, ζ,, χ+,0, ζ, χ 4,0 η : M k 4 M k 6 M k 8 M k 0 J k,, are isomorphisms. 0, e, f, g, h hφ +,cusp 0,, eφ + 4, + fφ 6, + gφ+ 8, Theorem. allows us to investigate congruences and filtrations of Hermitian Jacobi forms of index. We introduce more necessary notation. Throughout, p 5 is a prime. Let Z p := Z p Q be the local ring of p-integral rational numbers, Jk, δ Z p the space of forms in Jk, δ that have p-integral rational coefficients, Ω the filtration of a Hermitian Jacobi form in Jk, δ Z p see Definition 3.4, and Up the following analog of Atkin s U-operator: cn, rq n ζ r ζ r Up := cn, rq n ζ r ζ r, n Z,r O # n r 0 n Z,r O # n r 0 p 4n r where here and in the following q := e πiτ, ζ := e πiz, ζ := e πiw,ando # := Z[i] is the inverse different of O := Z[i]. Finally, for convenience, we write L := L. Our next theorem provides a criterion for the existence of Up congruences of Hermitian Jacobi forms of index. Theorem.. Let φ J δ k, Z p such that φ 0modp. Ifp>k,then { Ω L p+ k p +4 k, if φ Up 0modp, φ = p +5 k, if φ Up 0modp.

3 HERMITIAN JACOBI FORMS AND Up CONGRUENCES 3 Recall that Up congruences of elliptic modular forms have applications in the context of traces of singular moduli and class equations see [, 9, 8]. It would be interesting to see if Up congruences for Hermitian Jacobi forms also find further applications. The paper is organized as follows. In Section., we give a new definition of Hermitian Jacobi forms over Qi. In Section., we discuss the theta decomposition of Hermitian Jacobi forms. In Section.3, we prove Theorem.. In Section 3., we explore congruences of Hermitian Jacobi forms of index, and we provide the ingredients to prove Theorem.. Finally, in Section 3., we present an explicit example to illustrate Theorem.. Many of the results here have also been reported in the second author s University of North Texas doctoral dissertation [4].. Hermitian Jacobi forms over Qi Hermitian Jacobi forms appear as Fourier-Jacobi coefficients of Hermitian modular forms of degree over a complex quadratic field see [7, 0]. In this paper, we restrict ourselves to the case where the complex quadratic field is the Gaussian number field Qi. Throughout, k and m are nonnegative integers, and if s C, then s denotes its complex conjugate... A new definition. Recall from the introduction that O := Z[i] is the ring of Gaussian integers with inverse different O # := Z[i]. Let O := {,,i, i} be the group of units of O and ΓO := { ɛm ɛ O, M SL Z } be the Hermitian modular group. Our following extension of Haverkamp s [0] notion of a Hermitian Jacobi form depends on a parity δ = ±. Definition.. A holomorphic function φ : H C C is a Hermitian Jacobi form of weight k, index m, and parity δ if it satisfies the transformation laws aτ + b φ cτ + d, ɛz cτ + d, ɛ w = σɛɛ k cτ + d k e πimczw cτ+d φτ,z,w, cτ + d for all ɛ ab cd ΓO, { if δ =+ where σɛ := ɛ and if δ =, φτ,z + λτ + μ, w + λτ + μ =e πimλλτ+zλ+λw φτ,z,w, for all [λ, μ] O. Furthermore, one requires that φ has a Fourier series expansion of the form φτ,z,w= cn, rq n ζ r ζ r. n=0 r O # nm r 0 A Hermitian Jacobi form is called a cusp form if cn, r = 0 unless mn r > 0. We denote the space of Hermitian Jacobi forms of weight k, index m, and parity δ by Jk,m δ, and the space of cusp forms in J k,m δ δ,cusp by Jk,m. Finally, the space of Hermitian Jacobi forms of weight k and index m is defined by } J k,m := J + k,m J k,m {φ = +,φ φ + J + k,m,φ J k,m.

4 4 OLAV K. RICHTER AND JAYANTHA SENADHEERA Remark.. The space of Hermitian Jacobi forms in [5,6,0,3] coincides with the space of Hermitian Jacobi forms of positive parity, i.e., with J + k,m. The next proposition follows exactly as Propositions.3 and.4 of [0] see also Lemma of [3]. We omit the proof, which is contained in [4]. Proposition.3. Let φτ,z,w = cn, rq n ζ r ζ r Jk,m δ. Then we have the following: i The coefficient cn, r depends only on nm r and on r mod mo. ii If ɛ O,thenσɛɛ k cn, r =cn, ɛr. iii If m =, k 0mod4,andδ =+,thencn, r depends only on n r. iv If m =, k mod4,andδ =, thencn, r depends only on n r. v If m =and k is odd, then φ =0... The theta decomposition. Haverkamp [0] establishes the so-called theta decomposition for Hermitian Jacobi forms of positive parity. Set θm,sτ,z,w:= H q r /m ζ r ζ r. r O # r s mod mo Exactly as in [0], one finds that φ = n,r cn, rqn ζ r ζ r Jk,m δ has the theta decomposition. φτ,z,w= h s τθm,sτ,z,w, H s O # /mo where h s are certain vector-valued modular forms of weight k formoreproperties of h s,see[4]. Note that we suppress the dependence on δ and we write h s instead of h δ s. For the remainder, we are only interested in Hermitian Jacobi forms of index. Observe that if m =,then{0,, i, +i } is a set of representatives for the set of cosets O # /mo. We now recall the theta decompositions for important examples of Hermitian Jacobi forms of positive parity. Consider the Jacobi theta function. θ a,b τ,z:= n Z e πia+n τ+πin+az+b and its following specializations theta constants: x := θ 0,0 τ,0=+.3 n= q n, y := θ 0, τ,0=+ n q n n= a, b R, z := θ,0 τ,0 = q 8 It is well known see, for example, [] that.4 x 4 = y 4 + z 4, E 4 = x8 + y 8 + z 8, n=0 q nn+., E 6 = x4 + y 4 x 4 + z 4 y 4 z 4,

5 where E k τ := k HERMITIAN JACOBI FORMS AND Up CONGRUENCES 5 B k n= d n dk q n denotes the usual Eisenstein series. Sasaki [3] provides the theta decomposition of several Hermitian Jacobi forms of index. In particular, he considers up to normalization the following Hermitian Jacobi forms φ + k, J + k, for k =4, 8, and φ+,cusp 0, J +,cusp 0, :.5 φ + 4, := x6 + y 6 θ,0 H + z6 θ H, + θ H, + i x6 y 6 θ H,, +i φ + 8, := x4 + y 4 θ,0 H + z4 θ H, + θ H, + i x4 y 4 θ H,, +i φ +, := x + y θ,0 H + z θ H, + θ H, + i x y θ H,, i+ φ +,cusp 0, := 64 x6 y 6 z 6 θ H, θ H,. i Hermitian Jacobi forms of negative parity have not been studied rigorously in the literature, but they do arise via Fourier-Jacobi coefficients of Hermitian modular forms of degree with certain characters see [7]. Hermitian Eisenstein series are examples of such Hermitian modular forms of degree. In particular, there exists such a Hermitian Eisenstein series of weight 6, whose first Fourier-Jacobi coefficient φ 6, is a Hermitian Jacobi form of negative parity, weight 6, and index. It is somewhat demanding to explicitly compute the Fourier series coefficients of the Hermitian Eisenstein series. We determined φ 6, via a different approach. We used SAGE [7] and the SAGE code written by Martin Raum to calculate some Fourier series coefficients of φ 6,, which allowed us to guess and then prove the correct theta decomposition of φ 6,. Lemma.4. Set where φ 6, := h 0θ H,0 + h θh, + h i θh, i + h +i θ H, +i h 0 := x + y x 8 x 6 y x 4 y 4 x y 6 + y 8, h h i h +i := z6 z 4 x 4, := z6 z 4 x 4, := x y x 8 + x 6 y x 4 y 4 + x y 6 + y 8, and where x, y, andz are as in.3. Thenφ 6, J 6,. Proof. Consider ψ, + := E 4φ + 4, + 5 E 4φ + 8, 9 φ+, J, +, and let ψ, + = ĥ0θ,0 H + ĥ θh, + ĥ i θh, + ĥ i +i θ H, +i be its theta decomposition. Then using.4 and.5, one finds that ĥ 0 = E 6 h 0, ĥ ĥ i ĥ +i = E 6h = E 6h i,, = E 6 h +i. Hence ψ, + = E 6φ 6,. Observe that the modular Eisenstein series E 6 can also be viewed as a weight 6 and index 0 Hermitian Jacobi form of negative parity. We conclude that φ 6, J 6,.,

6 6 OLAV K. RICHTER AND JAYANTHA SENADHEERA We end this subsection with the initial Fourier series expansions of the Hermitian Jacobi forms φ + 4,, φ 6,, φ+ 8,,andφ+,cusp 0, for more coefficients of these forms, see [4]. Remark.5. We have the following initial Fourier series expansions: φ + 4, =+q 60+3 ζ ζ + ζ ζ + ζ i ζ i i + ζ ζ i + ζζ + ζ ζ + ζ i ζ i + ζ i ζ i + ζ +i ζ i +ζ +i ζ i +ζ i ζ +i +ζ i +, φ 6, =+q ζ ζ + ζ ζ + ζ i ζ i i + ζ ζ i + ζζ + ζ ζ + ζ i ζ i + ζ i ζ i ζ +i ζ i +ζ +i ζ i +ζ i +, φ + 8, =+q ζζ + ζ ζ + ζ i ζ i + ζ i ζ i φ +,cusp +8 ζ +i ζ i +ζ +i ζ i +ζ i ζ +i ζ +i ζ +i +ζ i +ζ i +, 0, = q ζ ζ + ζ ζ ζ i ζ i i ζ ζ i +q 8 ζ ζ + ζ ζ ζ i ζ i i ζ ζ i ζ +i ζ +i + ζ +i ζ i + ζ +i ζ i + ζ i ζ +i + ζ i ζ +i ζ +i ζ i +i ζ ζ i ζ i ζ +i i ζ ζ +i ProofofTheorem.. We proceed as in [3] see also 3of[8] to prove Theorem.. Consider the Taylor series of a Hermitian Jacobi form φ Jk, δ around z, w =0, 0: φτ,z,w= χ δ μ,ντz μ w ν. μ,ν=0 Then χ δ μ,ν = 0 unless μ ν is even, and if ɛ ab cd ΓO, then one finds that aτ + b χ δ μ,ν cτ + d = σεε k μ+ν cτ + d k+μ+ν χ δ μ,ντ+ πic cτ + d χδ μ,ν τ + πic χ δ! cτ + d μ,ν τ+.... The two following propositions on the Taylor coefficients χ δ μ,ν are easy to verify, and we omit their proofs see also [4]. Proposition.6. Let φτ,z,w= μ,ν=0 χδ μ,ντz μ w ν Jk, δ. If k 0mod4and δ =+,thenχ + 0, τ =χ+,0 τ =0.

7 χ, χ +, HERMITIAN JACOBI FORMS AND Up CONGRUENCES 7 If k 0mod4and δ =, thenχ 0,0 τ =χ, τ =χ 4,0 τ =χ 0,4 τ = τ =0. If k mod4and δ =, thenχ 0, τ =χ,0 τ =0. If k mod4and δ =+,thenχ + 0,0 τ =χ+, τ =χ+ 4,0 τ =χ+ 0,4 τ = τ =0. Proposition.7. Let φτ,z,w= μ,ν=0 χδ μ,ντz μ w ν Jk, δ.then χ δ 0,0τ M k, χ δ,0τ,χ δ 0,τ S k+, χ δ 4,0τ,χ δ 0,4τ S k+4, ζ,τ δ :=χ δ,τ πi k χδ 0,0 τ S k+, ζ,τ δ :=χ δ,τ πi k + χδ, πi τ+ k +k + χδ 0,0 τ S k+4. Remember the definition of the Jacobi theta function θ a,b in. to verify the factorization.6 θ H τ,z,w=θ, a+bi a,0 τ,z + wθ b,0τ,iw z. Let φ Jk, δ. Expand the theta decomposition φτ,z,w = h a+bi τθ H τ,z,w, a+bi = a,b {0,} h a+bi.6 a,b {0,} τθ a,0 τ,z + wθ b,0τ,iw z into a Taylor series, and compare its coefficients with the coefficients of the Taylor series φτ,z,w=χ δ 0,0τ+χ δ,τzw + χ δ 0,τ+χ δ,0τ z + w +χ δ,τz w + χ δ 0,4τ+χ δ 4,0τ z 4 + w 4 +. A direct calculation reveals that see [4] for more details χ δ 0,0,χ δ,,χ δ,0, χδ, χ δ 4,0 =.7 h 0,h where.8 with,hi,h+i T 0 T 0 T 0 0 T 0 A := T 0 T T 0 T + T 0 T 4 T T 0 T T 0 T 0 T T 0 T T 0 T + T 0 T 4 T T 0 T T 0 T 0 T T T T 0 T T a := θ a,0 τ, T a := πi d dτ θ a,0τ, T a := πi d dτ θ a,0τ, and where T b, T b,andt b are defined analogously. Moreover, one finds that A, det A = T T 0 T 0 T T T 0 4T 0 T 0 T T 0. Now we are in a position to prove Theorem.. We only consider the case k 0 mod 4, since the proof of the case k mod 4 is completely analogous.

8 8 OLAV K. RICHTER AND JAYANTHA SENADHEERA Note that Proposition.7 shows that the map ζ is well-defined. First, we demonstrate that ζ is injective. Let φ =φ +,φ J k, such that ζφ =0. Then χ + 0,0 = ζ+, = χ,0 = ζ+, χ+ 4,0 =0. Proposition.7 implies that χ+, =0, ζ, + = χ+,, and hence χ+, χ+ 4,0 = 0. Furthermore, Proposition.6 gives that χ +,0 =0andχ 0,0 = χ, = χ, = χ 4,0 =0. Thus,forδ = ± we have 0, 0, 0, 0 = χ δ 0,0,χ δ,,χ δ,0, χδ, χ δ 4,0 = h 0,h.7,hi,h+i where A is as in.8. Since det A 0,weobtainφ =0. Next we show the injectivity of η. Lete, f, g, h M k 4 M k 6 M k 8 M k 0 such that eφ + 4, + fφ 6, + gφ+ 8, + hφ+,cusp 0, = 0. Observe the theta decompositions of φ + 4,, φ+ 8,,andφ+,cusp 0, in.5 and of φ 6, in Lemma.4todiscoverthat where e, f, g, hh =0, 0, 0, 0, x6 + y 6 z6 z6 x6 y 6 H := h 0 h h i h +i x4 + y 4 z4 z4 x4 y x6 y 6 z 6 64 x6 y 6 z 6 0 with h 0, h, h i,andh+i as in Lemma.4. One finds that note that x 4 = y 4 +z 4 which shows that e = f = g = h =0. Finally, det H = 9 8 x6 y 6 z 6 0, dim M k +dims k+ +dims k+ +dims k+4 =dimm k 4 +dimm k 6 +dimm k 8 +dimm k 0, and we conclude that ζ and η are isomorphisms. Remark.8. The proof of Lemma.4 reveals that E 6 φ 6, = E 4 φ+ 4, + 5 E 4φ + 8, 9 φ+,. Thus, the restriction of the maps η and ζ in Theorem. to the case of positive parity yields the structure of J + k, in [3]. 3. Hermitian Jacobi forms modulo p 3.. Congruences and filtrations. In this section, we explore congruences and filtrations of Hermitian Jacobi forms. In particular, we establish the necessary tools to prove Theorem.. For Hermitian Jacobi forms φτ,z,w = cn, rq n ζ r ζ r and ψτ,z,w = c n, rq n ζ r ζ r with p-integral rational coefficients, we write φ ψ mod p whenever cn, r c n, r modp for all n, r. Proposition 3.. If φ Jk, δ Z p such that φ = eφ + 4, + fφ 6, + gφ+ 8, resp. φ = hφ +,cusp 0,, then the elliptic modular forms e, f, andg resp. h have p-integral rational coefficients. Moreover, if φ 0modp, thene f g 0modp resp. h 0modp. A,

9 HERMITIAN JACOBI FORMS AND Up CONGRUENCES 9 Proof. The initial Fourier series expansions in Remark.5 imply that the generators φ + 4,, φ 6,, φ+ 8,,andφ+,cusp 0, are linearly independent over the field Z /pz see [4] for more details. Suppose that φ = eφ + 4, +fφ 6, +gφ+ 8, the case φ = hφ+,cusp 0, is analogous. Note that the elliptic modular forms e, f, andg have bounded denominators. If e, f, or g do not have p-integral rational coefficients, then there exists some integer t such that 0 p t φ p t eφ + 4, + pt fφ 6, + pt gφ + 8, mod p. This yields a nontrivial linear dependence relation for φ + 4,, φ 6,,andφ+ 8,, which contradicts the above. Similarly, if φ 0modp such that e, f, org do not vanish modulo p, then one also obtains a nontrivial linear dependence relation for φ + 4,, φ 6,,andφ+ 8,,which is again a contradiction. An argument as in the proof of Lemma. of Sofer [6] shows that if two Hermitian Jacobi forms of indices m and m are congruent modulo p, thenm = m. We now give an analog of Sofer s Lemma. in the case m =. Corollary 3.. Let φ Jk, δ Z p and ψ Jk δ, Z p such that 0 φ ψ mod p. Thenk k mod p. Proof. Recall that if two elliptic modular forms f i M ki i =, have p-integral rational coefficients such that 0 f f mod p, then k k mod p see [5, 8]. This fact in combination with Proposition 3. implies the claim. We also record the following consequence of Theorem. and Proposition 3.. Corollary 3.3. Let φ Jk, δ Z p and ψ Jk δ, Z p such that φ ψ mod p. If δ δ and k k mod 4, thenφ ψ 0modp. Corollary 3. shows that there are congruences among Hermitian Jacobi forms of different weights, and one wishes to find the smallest weight in which the coefficientwise reduction of a Hermitian Jacobi form modulo p exists. Definition 3.4. Set J } k, {φ δ := mod p :φ Jk, δ Z p. For Hermitian Jacobi forms with p-integral rational coefficients, we define the filtration modulo p by { Ωφ :=inf k : φ mod p J } k, δ. Next we generalize Proposition of [] see also Proposition.5 of [0] to the case of Hermitian Jacobi forms of index. Proposition 3.5. If φ Jk, δ Z p, thenlφ modp is the reduction of a Hermitian Jacobi form modulo p. Moreover, we have with equality if and only if p Ωφ. Ω Lφ Ωφ+p +, Proof. It is easy to adapt the proofs in [0,] to the case of Hermitian Jacobi forms. Specifically, one employs., Proposition 3., Corollary 3., and Theorem and Lemma 5 of [8]. We omit further details, which are contained in [4]. Tate s theory of theta cycles see 7 of[3] was extended to Jacobi forms see [] and Jacobi forms of higher degree see [0]. The arguments in [0, ] apply also to Hermitian Jacobi forms, and Corollary 3. and Proposition 3.5 are the key ingredients in proving Theorem.. We omit the detailed proof of Theorem., which is contained in [4].

10 0 OLAV K. RICHTER AND JAYANTHA SENADHEERA 3.. An example. As in the case of elliptic modular forms, one does not know if a given Hermitian Jacobi form has Up congruences for only finitely many primes. In the following example, we consider primes 5 p < 00, and we show that E4 φ+,cusp 0, J 8, + has Up congruences for p =5, 7, 3, 3, 79, and for no other primes 5 p<00 see [4] for more examples. If p 5, 7, 3, 3, 79, then the table of Fourier series coefficients of E4φ +,cusp 0, in the Appendix of [4] guarantees that E4 φ+,cusp 0, Up 0modp. Recall the Ramanujan theta operator Θ := q d dq = d πi dτ, and Ramanujan s [9] identities up to a factor of 4 3. LE = 4ΘE = 3 E E 4, LE 4 = 4ΘE 4 = 4 3 E E 4 E 6, LE 6 = 4ΘE 6 =E E 6 E 4. Moreover,. together with Theorem. and Remark.5 yields the following identities Lφ + 4, =E φ + 4, φ 6,, 3. Lφ 6, = 5 3 E φ 6, 8 3 E 4φ + 4, + φ+ 8,, Lφ + 8, = 7 3 E φ + 8, 4 9 E 6φ + 4, 7 9 E 4φ 6,, Lφ +,cusp 0, =3E φ +,cusp 0,. We employ 3. and 3. in combination with the well-known congruences E p modp ande p+ E mod p to establish with the help of Mathematica the Up congruences for E4φ +,cusp 0, for p =5, 7, 3, 3, 79. Let p = 5, 7, or 3. We cannot apply Theorem., since p < 8. However, straightforward calculations show that L p E4φ +,cusp 0, E4φ +,cusp 0, mod p, which implies the desired Up congruences. If p = 3 or 79, then we apply Theorem.. Let p = 3. One finds that E 0E4E E 4 E6 3 mod 3. Moreover, p + 8 = 7, and a direct calculation reveals that L 7 E4φ +,cusp 0, 0E 4 4 E 6 +5E 4 E6 3 φ +,cusp 0, E φ +,cusp 0, φ +,cusp 0, mod 3. Hence Ω L 7 E4φ +,cusp 0, =0=p +5 8, and E4φ +,cusp 0, U3 0 mod 3. Let p = 79. One finds that E 78 6E4 8 E 6 +0E4 5 E E4 E E4E E4E E4E E6 3 mod 79. Moreover, p + 8 = 63, and a direct calculation shows that L 63 E4φ +,cusp 0, 73E4 3 E 6 +46E4 9 E E4 6 E6 5 +E4 3 E E4 0 E E E4 4 E6 3 +9E4 E6 5 +6E4E E4E E4E 6 0, E 78 8E E4 E6 +7E4E E4E E4E 6 8 φ +,cusp 0, 8E4 4 +7E4 E6 +7E4E E4E E4E 6 8 φ +,cusp 0, mod E φ +,cusp Hence Ω L 63 E4φ +,cusp 0, =66=79+5 8, and E4φ +,cusp 0, U79 0 mod 79.

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12 OLAV K. RICHTER AND JAYANTHA SENADHEERA [3] Ryuji Sasaki, Hermitian Jacobi forms of index one, Tsukuba J. Math , no., MR j:055 [4] J. Senadheera, Hermitian Jacobi forms and congruences, Ph.D. thesis, University of North Texas, 04. [5] Jean-Pierre Serre, Formes modulaires et fonctions zêta p-adiques French, Modular functions of one variable, III Proc. Internat. Summer School, Univ. Antwerp, 97, Springer, Berlin, 973, pp Lecture Notes in Math., Vol MR #7949a [6] Adriana Sofer, p-adic aspects of Jacobi forms, J.NumberTheory63 997, no., 9 0, DOI 0.006/jnth MR b:058 [7] W.A. Stein et al., Sage Mathematics Software Version 5.7, The Sage Development Team, 03, [8] H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III Proc. Internat. Summer School, Univ. Antwerp, 97, Springer, Berlin, 973, pp. 55. Lecture Notes in Math., Vol MR #077a [9] D. Zagier, Sur la conjecture de Saito-Kurokawa d après H. Maass French, Seminar on Number Theory, Paris , Progr. Math., vol., Birkhäuser, Boston, Mass., 98, pp MR b:003 Department of Mathematics, University of North Texas, Denton, Texas address: richter@unt.edu Department of Mathematics, University of North Texas, Denton, Texas address: JayanthaSenadheera@my.unt.edu Current address: Department of Mathematics and Computer Science, Faculty of Natural Sciences, The Open University of Sri Lanka, Nawala 050, Sri Lanka