A note on the growth of nearly holomorphic vector-valued Siegel modular forms
aa r X i v : . [ m a t h . N T ] D ec A NOTE ON THE GROWTH OF NEARLY HOLOMORPHICVECTOR-VALUED SIEGEL MODULAR FORMS
AMEYA PITALE, ABHISHEK SAHA, AND RALF SCHMIDT
Abstract.
Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ withrespect to some congruence subgroup of Sp n ( Q ). In this note, we prove that the function onSp n ( R ) obtained by lifting F has the moderate growth (or “slowly increasing”) property. Thisis a consequence of the following bound that we prove: k ρ ( Y / ) F ( Z ) k ≪ Q ni =1 ( µ i ( Y ) λ / + µ i ( Y ) − λ / ) where λ ≥ . . . ≥ λ n is the highest weight of ρ and µ i ( Y ) are the eigenvalues ofthe matrix Y . Introduction and statement of result
Let G be a connected reductive group over Q and K a maximal compact subgroup of G ( R ).One of the properties that an automorphic form on G ( R ) is required to satisfy is that it shouldbe a slowly increasing function (also referred to as the moderate growth property ). We now recallthe definition of this property following [4].A norm k k on G ( R ) is a function of the form k g k = (Tr( σ ( g ) ∗ σ ( g ))) / where σ : G ( R ) → GL r ( C ) is a finite-dimensional representation with finite kernel and image closed in M r ( C ) andsuch that σ ( K ) ⊆ SO m . For example, if G = Sp n , we may take σ to be the usual embeddinginto GL n ( R ) while for G = GL n we may take σ ( g ) = ( g, det( g ) − ) into GL n +1 ( R ). A complex-valued function φ on G ( R ) is said to have the moderate growth property if there is a norm k k on G ( R ), a constant C , and a positive integer λ such that | φ ( g ) | ≤ C k g k λ for all g ∈ G ( R ). Thisdefinition does not depend on the choice of norm.In practice, automorphic forms on G ( R ) are often constructed from classical objects (suchas various kinds of “modular forms”) and it is not always immediately clear that the resultingconstructions satisfy the moderate growth property. For a classical modular form f of weight k on the upper half plane, one can prove the bound | f ( x + iy ) | ≤ C (1 + y − k ) for some constant C depending on f . Using this bound it is easy to show that the function φ f on SL ( R ) attachedto f has the moderate growth property. More generally, if F is a holomorphic Siegel modularform of weight k on the Siegel upper half space H n , Sturm proved the bound | F ( X + iY ) | ≤ C Q ni =1 (1 + µ i ( Y ) − k ) where µ i ( Y ) are the eigenvalues of Y , which can be shown to imply themoderate growth property for the corresponding function Φ F on Sp n ( R ).Bounds of the above sort are harder to find in the literature for more general modular forms.In particular, when considering Siegel modular forms on H n , it is more natural to considervector-valued modular forms. Such a vector-valued form comes with a representation ρ ofGL n ( C ) corresponding to a highest weight λ ≥ . . . ≥ λ n ≥ H n is a function that transforms like a modular form, but instead of being holomorphic,it is a polynomial in the the entries of Y − with holomorphic functions as coefficients. Thetheory of nearly holomorphic modular forms was developed by Shimura in substantial detail and was exploited by him and other authors to prove algebraicity and Galois-equivariance ofcritical values of various L -functions. We refer the reader to the papers [1, 3, 2, 7, 8, 9] for someexamples.We remark that the moderate growth property for a certain type of modular form is absolutelycrucial if one wants to use general results from the theory of automorphic forms to study theseobjects (as we did in our recent paper [6] in a certain case). It appears that a proof of themoderate growth property, while probably known to experts, has not been formally writtendown in the setting of nearly holomorphic vector-valued forms. In this short note, we fill thisgap in the literature.Consider a nearly holomorphic vector-valued modular form F of highest weight λ ≥ . . . ≥ λ n ≥ F takes values in a finite dimen-sional vector space V . For v ∈ V , denote k v k = h v, v i / where we fix an U ( n )-invariant innerproduct on V . We can lift F to a V -valued function ~ Φ F on Sp n ( R ). For any linear functional L on V consider the complex valued function Φ F = L ◦ ~ Φ F on Sp n ( R ). We prove the followingresult. Theorem 1.1.
The function Φ F defined above has the moderate growth property. The above theorem is a direct consequence of the following bound.
Theorem 1.2.
For any nearly holomorphic vector-valued modular form F as above, there is aconstant C (depending only on F ) such that for all Z = X + iY ∈ H n we have k ρ ( Y / ) F ( Z ) k ≤ C n Y i =1 ( µ i ( Y ) λ / + µ i ( Y ) − λ / ) . The proof of Theorem 1.2, as we will see, is extremely elementary. It uses nothing other thanthe existence of a Fourier expansion, and is essentially a straightforward extension of argumentsthat have appeared in the classical case, e.g., in [5] or [10]. This argument is very flexible andcan be modified to provide a bound for Siegel-Maass forms. With some additional work (whichwe do not do here), Theorem 1.2 can be used to derive a bound on the Fourier coefficients of F . We also remark that the bound in Theorem 1.2 can be substantially improved if F is a cuspform. Notations.
For a positive integer n and a commutative ring R , let M sym n ( R ) be the set ofsymmetric n × n matrices with entries in R . For X, Y ∈ M sym n ( R ), we write X > Y if X − Y ispositive definite. Let H n be the Siegel upper half space of degree n , i.e., the set of Z = X + iY ∈ M sym n ( C ) whose imaginary part Y is positive definite. For such Z and g = (cid:20) A BC D (cid:21) ∈ Sp n ( R ),let J ( g, Z ) = CZ + D .For any complex matrix X we denote by X ∗ its transpose conjugate. For positive definite Y = ( y ij ) ∈ M sym n ( R ), let k Y k = max i,j | y ij | . We denote by µ i ( Y ) the i -th eigenvalue of Y , indecreasing order.2. Nearly holomorphic functions and Fourier expansions
For a non-negative integer p , we let N p ( H n ) denote the space of all polynomials in the entriesof Y − with total degree ≤ p and with holomorphic functions on H n as coefficients. The space N ( H n ) = [ p ≥ N p ( H n ) N THE GROWTH OF NEARLY HOLOMORPHIC FORMS 3 is the space of nearly holomorphic functions on H n .It will be useful to have some notation for polynomials in matrix entries. Let R n = { ( i, j ) : 1 ≤ i ≤ j ≤ n } . Let T pn = { b = ( b i,j ) ∈ Z R n : b i,j ≥ , X ( i,j ) ∈ R n b i,j ≤ p } . For any V = ( v i,j ) ∈ M sym n ( R ), and any b ∈ T pn , we define [ V ] b = Q ( i,j ) ∈ R n v b i,j i,j . In particular,a function F on H n lies in N p ( H n ) if and only if there are holomorphic functions G b on H n suchthat F ( Z ) = X b ∈ T pn G b ( Z )[ Y − ] b . Definition 2.1.
For any δ > , we define V δ = { Y ∈ M sym n ( R ) : Y ≥ δI n } . Lemma 2.2.
Given any Y ∈ V δ , we have k Y − k ≤ δ − .Proof. Note that for any positive definite matrix Y ′ = ( y ′ ij ) we have k Y ′ k = max i,j | y ′ ij | =max i y ′ ii . This is an immediate consequence of the fact that each 2 × Y − is less than or equal to δ − . But theassumption Y ≥ δI n implies that Y − ≤ δ − I n , which implied the desired fact above. (cid:3) An immediate consequence of this lemma is that for any δ ≤ Y ∈ V δ and b ∈ T pn , we have | [ Y − ] b | ≤ δ − p . Definition 2.3.
We say that F ∈ N p ( H n ) has a nice Fourier expansion if there exists an integer N and complex numbers a b ( F, S ) for all ≤ S ∈ N M sym n ( Z ) , such that we have an expression F ( Z ) = X b ∈ T pn X S ∈ N M sym n ( Z ) S ≥ a b ( F, S ) e πi Tr( SZ ) [ Y − ] b that converges absolutely and uniformly on compact subsets of H n . Note that a key point in the above definition is that the sum is taken only over positivesemidefinite matrices. The next proposition, which is well-known in the holomorphic case,shows that this implies a certain boundedness property for the function F . Proposition 2.4.
Let F ∈ N p ( H n ) have a nice Fourier expansion. Then for any δ > , thefunction F ( Z ) is bounded in the region { Z = X + iY : Y ∈ V δ } .Proof. We may assume that δ ≤
1. By the notion of a nice Fourier expansion, for each b ∈ T pn ,the series R b ( Y ) := X S ∈ N M sym n ( Z ) S ≥ | a b ( F, S ) | e − π Tr( SY ) converges for any 0 < Y ∈ M sym n ( R ). For any Z in the given region, using Lemma 2.2, we get | F ( Z ) | ≤ X b ∈ T pn R b ( Y ) δ − p , (1) AMEYA PITALE, ABHISHEK SAHA, AND RALF SCHMIDT and so to prove the proposition it suffices to show that each R b ( Y ) is bounded in the region Y ≥ δ . By positivity, we have | a b ( F, S ) e − π Tr( SY ) | ≤ R b ( Y )for each Y > S ∈ N M sym n ( Z ). Therefore | a b ( F, S ) | ≤ R b ( δI n / e δπ Tr( S ) . (2)Next, note that if Y ≥ δI n , then Tr( SY ) ≥ δ Tr( S ) for all S ≥
0. To see this, we write Y = δI n + Y where Y ≥ Y − δI n . As S ≥ Y SY ) ≥ SY ) = Tr( SδI n ) + Tr( Y SY ) ≥ Tr(
SδI n ).Using the above and (2), we have for all Y ≥ δI n R b ( Y ) ≤ R b ( δI n / X ≤ S ∈ N M sym n ( Z ) e − δπ Tr( S ) . As the sum P ≤ S ∈ N M sym n ( Z ) e − δπ Tr( S ) converges to a finite value (for a proof of this fact, see [5,p. 185]) this completes the proof that R b ( Y ) is bounded in the region Y ≥ δI n . (cid:3) Bounding nearly holomorphic vector-valued modular forms
Let ( ρ, V ) be a finite-dimensional rational representation of GL n ( C ) and h , i be a (unique upto multiples) U ( n )-invariant inner product on V . In fact, the inner product h , i has the propertythat h ρ ( M ) v , v i = h v , ρ ( M ∗ ) v i for all M ∈ GL n ( C ). (To see this, note that it’s enough to check it on the Lie algebra level. It’strue for the real subalgebra u ( n ) and so by linearity it follows for all of gl(n , C ).) For any v ∈ V ,we define k v k = h v, v i / . As is well known, the representation ρ has associated to it an n -tuple λ ≥ λ ≥ . . . ≥ λ n ≥ ρ . We let d ρ denote the dimension of ρ .We define a right action of Sp n ( R ) on the space of smooth V -valued functions on H n by( F (cid:12)(cid:12) ρ g )( Z ) = ρ ( J ( g, Z )) − F ( gZ ) for g ∈ Sp n ( R ) , Z ∈ H n . (3)A congruence subgroup of Sp n ( Q ) is a subgroup that is commensurable with Sp n ( Z ) andcontains a principal congruence subgroup of Sp n ( Z ). For a congruence subgroup Γ and a non-negative integer p , let N pρ (Γ) be the space of all functions F : H n → V with the followingproperties.(1) For any g ∈ Sp n ( Q ) and any linear map L : V → C , the function L ◦ ( F (cid:12)(cid:12) ρ g ) lies in N p ( H n ) and has a nice Fourier expansion.(2) F satisfies the transformation property F (cid:12)(cid:12) ρ γ = F for all γ ∈ Γ . (4)Let N ρ (Γ) = S p ≥ N pρ (Γ). We refer to N ρ (Γ) as the space of nearly holomorphic Siegel mod-ular forms of weight ρ with respect to Γ. We put N ( n ) ρ = S Γ N ρ (Γ) , the space of all nearlyholomorphic Siegel modular forms of weight ρ .Recall that for any Y > M sym n ( R ), we let µ ( Y ) ≥ µ ( Y ) ≥ . . . ≥ µ n ( Y ) > Y . We can now state our main result. N THE GROWTH OF NEARLY HOLOMORPHIC FORMS 5
Theorem 3.1.
For any F ∈ N ( n ) ρ , there is a constant C F (depending only on F ) such that forall Z = X + iY ∈ H n we have k ρ ( Y / ) F ( Z ) k ≤ C F n Y i =1 ( µ i ( Y ) λ / + µ i ( Y ) − λ / ) . In order to prove this theorem, we will need a couple of lemmas.
Lemma 3.2.
For any v ∈ V , and any Y > in M sym n ( R ) , we have n Y i =1 µ i ( Y ) λ n +1 − i ! k v k ≤ k ρ ( Y ) v k ≤ n Y i =1 µ i ( Y ) λ i ! k v k . Proof.
This follows from considering a basis of weight vectors. Note that it is sufficient to provethe inequalities for Y diagonal as any Y can be diagonalized by a matrix in U ( n ) and our normis invariant by the action of U ( n ). (cid:3) Next, we record a result due to Sturm.
Lemma 3.3 (Prop. 2 of [10]) . Suppose that F is a fundamental domain for Sp n ( Z ) such thatthere is some δ > such that Z = X + iY ∈ F implies that Y ∈ V δ . Let φ : H n → C be anyfunction such that there exist constants c > , λ ≥ with the property that | φ ( γZ ) | ≤ c det( Y ) λ for all Z ∈ F and γ ∈ Sp n ( Z ) . Then for all Z ∈ H n we have the inequality | φ ( Z ) | ≤ c φ n Y i =1 ( µ i ( Y ) λ + µ i ( Y ) − λ ) . Proof of Theorem 3.1.
Let F be as in the statement of the theorem, so that F ∈ N ρ (Γ) forsome Γ ⊂ Sp n ( Z ). We let γ , γ , . . . , γ t be a set of representatives for Γ \ Sp n ( Z ). Fix anorthonormal basis v , v , . . . , v d of V and for any G ∈ N ( n ) ρ define G i ( Z ) := h G ( Z ) , v i i . Notethat k G ( Z ) k = (cid:16)P di =1 | G i ( Z ) | (cid:17) / .Let F be as in Lemma 3.3. By Proposition 2.4, it follows that there is a constant C dependingon F such that | ( F | ρ γ r ) i ( Z ) | ≤ C for all 1 ≤ r ≤ t , 1 ≤ i ≤ n , and Z ∈ F . Moreover,for any Z = X + iY ∈ F we have each µ j ( Y / ) ≥ δ / and therefore (cid:16)Q nj =1 µ j ( Y / ) λ j (cid:17) ≤ det( Y ) λ / δ P nj =2 ( λ j − λ ) . Now consider the function φ ( Z ) = k ρ ( Y / ) F ( Z ) k . For any γ ∈ Sp n ( Z ), there exists γ ∈ Γ and some 1 ≤ r ≤ t such that γ = γ γ r . An easy calculation showsthat k φ ( γZ ) k = k ρ ( Y / )( F | ρ γ r )( Z ) k . So for all Z ∈ F , γ ∈ Sp n ( Z ) we have, using Lemma 3.2 and the above arguments, k φ ( γZ ) k ≤ det( Y ) λ / δ P ni =2 ( λ i − λ ) d / C. So the conditions of Lemma 3.3 hold with λ = λ /
2. This concludes the proof of Theorem3.1. (cid:3)
Corollary 3.4.
For any F ∈ N ( n ) ρ , there is a constant C F (depending only on F ) such that forall Z = X + iY ∈ H n we have k ρ ( Y / ) F ( Z ) k ≤ C F (1 + Tr( Y )) nλ (det Y ) − λ / . AMEYA PITALE, ABHISHEK SAHA, AND RALF SCHMIDT
Proof.
This follows immediately from Theorem 3.1 and the following elementary inequality,which holds for all positive integers λ, n and all positive reals y , . . . , y n : n Y i =1 (1 + y λi ) ≤ (1 + y + . . . + y n ) nλ . To prove the above inequality, note that 1 + y λi ≤ (1 + y + . . . + y n ) λ for each i . Now take theproduct over 1 ≤ i ≤ n . (cid:3) The moderate growth property
Given any F ∈ N ( n ) ρ , we define a smooth function ~ Φ F on Sp n ( R ) by the formula ~ Φ F ( g ) = ρ ( J ( g, I )) − F ( gI ) , where I := iI n . Proposition 4.1.
Let F ∈ N ( n ) ρ and ~ Φ F be defined as above. Then there is a constant C suchthat for all Z = X + iY ∈ H n we have (cid:13)(cid:13)(cid:13)(cid:13) ~ Φ F (cid:18)(cid:20) Y / XY / Y − / (cid:21)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C n Y i =1 ( µ i ( Y ) λ / + µ i ( Y ) − λ / ) . Proof.
This follows immediately from Theorem 3.1. (cid:3)
A complex-valued function Φ on Sp n ( R ) is said to be slowly increasing if there is a constant C and a positive integer r such that | Φ( g ) | ≤ C (Tr( g ∗ g )) r for all g ∈ Sp n ( R ). Theorem 4.2.
Let F ∈ N ( n ) ρ and ~ Φ F be as defined above. For some linear functional L on V ,let Φ F = L ◦ ~ Φ F . Then the function Φ F has the moderate growth property.Proof. Note that | Φ F ( g ) | ≤ k L k k ~ Φ F ( g ) k . So it suffices to show that there is a constant C anda positive integer r such that k ~ Φ F ( g ) k ≤ C (Tr( g ∗ g )) r (5)for all g ∈ Sp n ( R ). Since both sides of this inequality do not change when g is replaced by gk , where k is in the standard maximal compact subgroup of Sp n ( R ), we may assume that g is of the form (cid:20) Y / XY / Y − / (cid:21) . Then the existence of appropriate C and r follows easily fromProposition 4.1. Indeed, we can take any r ≥ nλ / C the same constant as in Proposition4.1. (cid:3) References [1] Antonia Bluher. Near holomorphy, arithmeticity, and the theta correspondence. In
Automorphic forms, au-tomorphic representations, and arithmetic (Fort Worth, TX, 1996) , volume 66 of
Proc. Sympos. Pure Math. ,pages 9–26. Amer. Math. Soc., Providence, RI, 1999.[2] Siegfried B¨ocherer and Bernhard Heim. Critical values of L -functions on GSp × GL . Math. Z. , 254(3):485–503, 2006.[3] Siegfried B¨ocherer and Rainer Schulze-Pillot. On the central critical value of the triple product L -function.In Number theory (Paris, 1993–1994) , volume 235 of
London Math. Soc. Lecture Note Ser. , pages 1–46.Cambridge Univ. Press, Cambridge, 1996.
N THE GROWTH OF NEARLY HOLOMORPHIC FORMS 7 [4] Armand Borel and Herv´e Jacquet. Automorphic forms and automorphic representations. In
Automorphicforms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore.,1977), Part 1 , Proc. Sympos. Pure Math., XXXIII, pages 189–207. Amer. Math. Soc., Providence, R.I.,1979. With a supplement “On the notion of an automorphic representation” by R. P. Langlands.[5] Hans Maass. Siegel’s modular forms and Dirichlet series . Lecture Notes in Mathematics, Vol. 216. Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl LudwigSiegel on the occasion of his seventy-fifth birthday.[6] Ameya Pitale, Abhishek Saha, and Ralf Schmidt. Lowest weight modules of Sp ( R ) and nearly holomorphicSiegel modular forms (expanded version) . arXiv:1501.00524.[7] Abhishek Saha. Pullbacks of Eisenstein series from GU(3 ,
3) and critical L -values for GSp × GL . Pacific J.Math. , 246(2):435–486, 2010.[8] Goro Shimura. On Hilbert modular forms of half-integral weight.
Duke Math. J. , 55(4):765–838, 1987.[9] Goro Shimura.
Arithmeticity in the theory of automorphic forms , volume 82 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, 2000.[10] Jacob Sturm. The critical values of zeta functions associated to the symplectic group.
Duke Math. J. ,48(2):327–350, 1981.
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
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