WEIGHTED SUM OF THE EXTENSIONS OF THE REPRESENTATIONS OF QUADRATIC FORMS

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1 WEIGHTED SUM OF THE EXTENSIONS OF THE REPRESENTATIONS OF QUADRATIC FORMS BYEONG-KWEON OH Abstract Let L, N and M be positive definite integral Z-lattices In this paper, we show some relation between the weighted sum of representations of L and N by gen(m) and the weighted sum of extensions of M σ in the gen(m σ ) via N η when M is even and gcd(dl, dm) = As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (3) in [Bo] for the relation of the Fourier coefficients between Eisenstein series and Jacobi-Eisenstein series Introduction Let M be a positive definite integral Z-lattice It is known to very hard problem to determine the number of representation from L to M for any positive definite integral Z-lattice L whose rank is less than that of M except some particular cases Let r(l, gen(m)) be the weighted sum of representations of L by gen(m) defined by r(l, gen(m)) := R(L, M) w(m) where R(L, M) is the set of all representations from L to M and w(m) := for the isometry group O(M) of M Then, the Minskowski-Siegel formula says that r(l, gen(m)) = ɛ α p (L p, M p ), p including where α p is the local density at p and ɛ = 2 if rankm = rankn and ɛ =, otherwise This formula implies that the weighted sum of representations of L by gen(m) depends only on the local structure of the Z-lattices L and M Let L and M be any Z-lattices and σ be a representation from L to M For any Z-lattice N and any representation η : L N, we say σ is extensible to N via η if there is a representation ρ : N M such that ρ η = ρ Furthermore, ρ is called the extension of σ via η If the representation ρ is bijective, then we say the pair (M, σ) is isometric to the pair (N, η) The existence of such an extension implies that there is an integral solution of the system of diophantine equation including quadratic 99 Mathematics Subject Classification Primary E2, F50 Key words and phrases Extensions of representations; Minkowski-Siegel formula This work was supported by the Korea Research Foundation Grant (KRF C00006)

2 2 BYEONG-KWEON OH equation and linear equation determined by the above Z-lattices and representations (for details, see [CKKO]) For any Z-lattice M and any representation σ : L M, we say the pair ( M, σ) is contained in the genus of the pair (M, σ) if there is an isometry ρ p : Mp M p such that ρ p σ p = σ p for any prime p Throughout this paper, we briefly use the notation M σ instead of the pair (M, σ) if no confusion arises Under this situation, we may naturally define the weighted sum of extensions of M σ in the gen(m σ ) via N η by r(n η, gen(m σ )) := w(m σ ) M σ gen(m σ )/ R(N η, M σ ), O( M σ ) where R(N η, M σ ) is the set of extensions of σ via η and O( M σ ) = R( M σ, M σ ) In this paper, we show some relation between the weighted sum of representations of L and N by gen(m) and the weighted sum of extensions of M σ in the gen(m σ ) via N η when M is even and gcd(dl, dm) = As a consequence of the particular case when M is even unimodular, we give a formula for the Fourier coefficients of the Jacobi-Eisenstein series E k,s,n (τ, z) for any positive definite half integral symmetric matrix S of rank l and positive integers k, n such that k > n + l +, k 0 (mod 4) by using the Fourier coefficients of the Eisenstein series E k,l+n (z) and E k,l (z) This formula can also be obtained from the formula (3) in [Bo] by using Möbius function techniques Throughout this paper, we always assume that every Z-lattice L is integral, ie, the scale s(l) of L is contained in Z and is positive definite unless stated otherwise For any unexplained terminology and basic facts about Z-lattices, we refer the readers to O Meara s book [O M] and Kitaoka s book [K] 2 Weighted Sum of Extensions for gen L (M) Let L be a fixed Z-lattice of rank l For any Z-lattice K, if there exists a representation τ : L M, we will use the notation K τ For two M σ and N η, if there exists a representation ρ : N M such that ρ η = σ, then we say that N η is represented by M σ and we simply write ρ : N η M σ We also define R(N η, M σ ) := {ρ ρ : N η M σ } If N η M σ and M σ N η, then we say N η is isometric to M σ and we write N σ M η The isometry subgroup O(M σ ) of M σ os defined by {ρ O(M) ρ σ = σ} For any Z-lattice M gen(m) and any representation σ : L M, if there exists an isometry ρ p : Mp M p such that ρ p σ p = σ p for any prime p, then we say M σ gen(m σ ) The σ-class number h(m σ ) is defined by the number of isometric classes in gen(m σ ) It is easy to show that the σ-class number h(m σ ) is always finite for any Z-lattice M and a representation σ : L M We also define gen L (M) by the set of all Mη such that M gen(m) and η : L M The L-class number h L (M) is defined by the number of isometric classes in gen L (M) As usual, R(N, M), (R (N, M)) is denoted by the set of all (primitive, respectively) representations from N to M For the local case, we also define similar notations as above In particular, we use a notation (M σ ) p instead of (M p ) σp, if no confusion arises We denote h L p (M) by the number of isometric classes over Z p in {(M p ) ψ ψ R(L p, M p )}

3 WEIGHTED SUM OF THE EXTENSIONS OF THE REPRESENTATIONS OF QUADRATIC FORMS3 Lemma 2 With the same notations as above, if p 2dLdM, then h L p (M) = Furthermore, the number of gen( M σ ) in gen L (M) is p hl p (M) Proof Since the first assertion is trivial by Theorem 54 of [K], we only provide the proof of the second assertion Assume that σ : L M and η p : L p M p are given representations for each p 2dLdM Without loss of generality, we may assume that QL QM := V and σ : QL V By Witt s theorem, we may consider the representation σ p ηp : η p (L p ) σ p (L p ) as an element in O(V p ) Now, define a Z-lattice M V such that M p = σ p ηp (M p ) for p 2dLdM and M p = M p for remaining primes p Such a Z-lattice M always exists and is unique by 84 of [O M] Note that σ(l) M Furthermore, one may easily check that ( M σ ) p (M η ) p for all p 2dLdM From this follows the lemma directly Lemma 22 For two M σ and N η, if (N η ) p is represented by (M σ ) p for any prime p, then there exists an M σ gen(m σ ) such that N η M σ Proof We may assume that QL, QN QM := V For any prime p, let π p : (N η ) p (M σ ) p be a representation Define a finite set of primes T := {p ord p (2dLdM) 0 or N p M p } Let M be a Z-lattice in V such that Mp = πp (M p ) for p T and M p = M p otherwise Here, we consider π p as an element in O(V p ) Note that η(l) M Furthermore, one may easily check that N η M η and M η gen(m σ ) Lemma 23 Let N and M be Z-lattices and σ : L M, η : L N be representations Then, we have R(N η, M σ ) R(N, = M) O( M σ ) and M σ gen L (M)/ M σ gen L (M)/ O( M σ ) = R(L, M) Proof Note that the second assertion comes directly from the first assertion by taking N = L Let M be any Z-lattice in gen(m) Since the following map Φ : R(N, M) R(N η, M σ ) defined by Φ(ρ) = ρ R(N η, M ρ η ), is bijective, σ R(L, M) R(N, M) = σ R(L, M) R(N η, M σ ) Now, consider the following group action Ψ : O( M) R(L, M) R(L, M) such that Ψ(ρ, σ) = ρ σ for any ρ O( M) and σ R(L, M) Clearly, the isotropy subgroup of σ R(L, M) is O( M σ ) Furthermore, if M σ is isometric to M σ, then R(N η, M σ ) = R(N η, M σ ) Therefore, R(N, M) = From this follows the lemma σ R(L, M)/ O( M σ ) R(N η, M σ )

4 4 BYEONG-KWEON OH 3 Weighted Sum of Extensions for gen(m σ ) In this section, we show some relation between the weighted sum of representations of L and N by gen(m) and the weighted sum of extensions of M σ in the gen(m σ ) via N η when M is even and gcd(dl, dm) = Let L be any even Z-lattice We define S(L) := {L L L QL, n(l ) 2Z} Note that S(L) is a finite set and S(L) S(L ) if L L For any even Z-lattice L, we define a function t L : S(L) Z such that t L (L) = and t L (L ) = L L L t L ( L) One can easily show that t L (L ) depends only on the structure of an abelian group L /L, which is called the generalized Möbius function We will usually use the notation t(l /L) instead of t L (L ) As a function defined on the set of abelian groups, t is multiplicative, ie, if L /L L /L L 2 /L and gcd( L /L, L 2 /L ) =, then t(l /L) = t(l /L)t(L 2 /L) Since L /L p L p/l p, t(l /L) = p t(l p/l p ) Furthermore, if p is a prime, t((z/pz) m ) = ( ) m p 2 m(m ) and t(g) = 0 for the other type of p-group G (see [D]) Lemma 3 Let α and β be a function from the set S of even Z-lattices to Q satisfying α(l) := β(l ), for all L S Then, we have β(l) = L S(L) L S(L) t(l /L)α(L ) Proof Let S(L) = {L = L, L 2,, L s } such that L j L i for i j Define A = (a ij ) M s s (Z) such that a ij = if L i L j and a ij = 0 otherwise Note that A = I + U, where U is a upper triangular matrix Since α(l i ) = s j=i a ijβ(l j ) by definition, (t(l /L), t(l 2 /L),, t(l s /L)) is the first row of A From this follows the lemma From now on, we always assume that M and L are even Z-lattices such that gcd(dl, dm) = Let σ : L M be any primitive representation Then, for any M gen(m) and any primitive representation σ : L M, we have M σ gen(m σ ) by Theorems 535 and 54 of [K] Let N be any even Z-lattice and η : L N be a primitive representation Then, there is a Z-sublattice N of N such that N = η(l) N By considering η as an isometry from QL to QN, we define N(L ) := η(l ) N for any Z-lattice L such that L L QL So, it may be possible that s(n(l )) is not contained in Z For any representation κ : L M, we also define R L (N(L ) η, M κ ) := {ρ : N(L ) M ρ η = κ}, where η is considered as a representation from L to N(L )

5 WEIGHTED SUM OF THE EXTENSIONS OF THE REPRESENTATIONS OF QUADRATIC FORMS5 Theorem 32 Under the same notations and assumptions as above, we have R(N η, M σ ) = O( M σ ) M σ gen(m σ)/ L L QL t(l /L) R(N(L ), M) and M σ gen(m σ )/ O( M σ ) = L L QL t(l /L) R(L, M) Proof Note that the second assertion comes directly from the first assertion by taking N = L Let L be any even Z-lattice containing L on QL For any representation κ : L M, we may naturally consider κ : κ (M) QL M as a primitive representation Hence R(L, M) = L L QL R ( L, M) Note that R L (N(L ) η, M κ ) = R κ (M) QL(N(κ (M) QL) η, M κ ) Therefore, by Lemma 23, we have R(N(L ), M) = R L (N(L ) η, M κ ) κ R(L,M) = L L QL = L L QL κ R ( L,M) κ R ( L,M)/ From this and Lemma 3 follows the theorem R L(N( L) η, M κ ) O(M) O(M κ ) R L(N( L) η, M κ ) Let L = Zz +Zz 2 + +Zz l, M = Zx +Zx 2 + +Zx l +Zx l+ + +Zx l+m be even Z-lattices with rank l and l + m, respectively and σ : L M be the primitive representation such that σ(z i ) = x i For τ H n and z M k n (C), the Jacobi-Siegel theta series θ M,L,n (τ, z) is defined by θ M,L,n (τ, z) := r(a, B)e ( tr(aτ + t zb) ) A Sym n (Z) 0,B M k,n (Z) The Fourier coefficient r(a, B) is defined as follows: Let N = Zy + Zy 2 + Zy l + Zy l+ + + Zy l+n be even Z-lattice such that (B(y i, y l+j )) = B and (B(y l+i, y l+j )) = 2A and η : L N be the primitive representation such that η(z i ) = y i Then, r(a, B) = R(N η, M σ ) For a non-vanishing property of r(a, B) under some restrictions, see [CKKO], which is a natural generalization of [HKK] Assume that M is even unimodular Z-lattice Then, it is well known that θ M,L,n (τ, z) is a Jacobi form in J m+l/2,/2(b(zi,z j ))(Γ J n,l ) (for details, see [A] or [Z]) In [A], Arakawa proved that if m > 2n + l + 2, then w(m σ ) O( M σ ) θ M,L,n(τ, z) = E m+l/2,/2(b(zi,z j)),n(τ, z), M σ gen(m σ )/

6 6 BYEONG-KWEON OH where E m+l/2,/2(b(zi,z j)),n is the Jacobi-Eisenstein series We define and T l (Z) := {(t ij ) M l l (Z) 0 t ij < t jj and t ji = 0 for i < j l} l l t(t ) := t Zx + + Zx l /Z t j x j + + Z t jl x j j= for any free Z-module Zx + + Zx l of rank l and T = (t ij ) T l (Z) Note that GL l (Z)M T l (Z) = for any M M l (Z) (see [An]) Corollary 33 Let k > n + l + be any integer satisfying k 0 (mod 4) and S Sym l (Z) >0 Let E k,s,n (τ, z) = r(a, B)e(tr(Aτ + t zb)) A Sym n (Z) 0,B M l n (Z) ( ) S B/2 be the Fourier expansion of the Jacobi-Eisenstein series If t is positive definite, then we have B/2 A (( )) S[T t(t )a ] t (T )B/2 k,l+n t (B/2)T A T T l (Z) r(a, B) =, t(t )a k,l (S[T ]) T T l (Z) where a k, (X) is the Fourier coefficient of the Eisenstein series of weight k and degree Proof This is the direct consequence of Theorem 32 Remark 34 Note that the number of T satisfying S[T ] Sym l (Z) is always finite If S is primitive, ie, S[T ] Sym l (Z) except T = I l (for example, if det(2s) is a square-free integer), then r(a, B) = (( S B/2 a k,l+n t (B/2) A a k,l (S) This formula is also given by Böcherer in [Bo] )) j= References [A] T Arakawa, Siegel s formula for Jacobi forms, Internat J Math 4(993), [An] A Andrianov, Quadratic forms and Hecke operators, Springer-Verlag, 987 [Bo] S Böcherer, Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen, Math Z 83(983), 2 46 [D] S Delsarte, Fonctions de Mobius sur les groupes abeliens finis, Ann of Math (2) 49(948), [HKK] J Hsia, Y Kitaoka and M Kneser, Representations of positive definite quadartic forms, J Reine Angew Math 30(978), 32 4 [K] Y Kitaoka, Arithmetic of quadratic forms, Cambridge Univ Press, 999 [CKKO] W K Chan, B M Kim, M-H Kim and B-K Oh, A local-global principle for extensibility of representations of quadratic forms and ite applications, to appear in Ramanujan J [O M] OT O Meara, Introduction to quadratic forms, Springer-Verlag, 963

7 WEIGHTED SUM OF THE EXTENSIONS OF THE REPRESENTATIONS OF QUADRATIC FORMS7 [S] G L Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math Ann 6(939), [Z] C Ziegler, Jacobi forms of higher degree, Abh Math Sem Univ Hamburg 59(989), Department of Applied Mathematics, Sejong University, Seoul, , Korea address: bkoh@sejongackr