Formal series of Jacobi forms
Abstract.
We prove for general paramodular level that formal series of scalar Jacobi forms with an involution condition necessarily converge and are therefore the Fourier-Jacobi expansions at the standard 1-cusp of paramodular Fricke eigenforms.
Key words and phrases:
Formal series, Jacobi form, Paramodular form2020 Mathematics Subject Classification:
Primary: 11F46, 11F501. Introduction.
A parmodular form is a Siegel modular form of degree two for the following discrete group
Paramodular forms are a natural generalization of elliptic modular forms and are interesting in many ways. Roberts and Schmidt [35, 36] gave a sophisticated theory of local and global paramodular newforms. Paramodular newforms have applications to modularity; weight two to abelian surfaces [8, 9], and weight three to nonrigid Calabi-Yau threefolds [15] of Hodge type . Conjectures of Ibukiyama [22, 23], and of Ibukiyama and Kitayama [25], connecting paramodular forms to algebraic modular forms on a compact twist of , motivated mainly by independent calculations of dimension formulae, have recently been proven [39, 37]. In [10], this connection was generalized to a correspondence with Fricke eigenspaces. Utilizing results of [10], dimension formulae for Fricke plus and minus spaces of paramodular forms for prime level were computed in [24]. For weights , these recent proofs allow paramodular Hecke eigensystems to also be computed using orthogonal modular forms [10, 3]. Finally, Gritsenko lifts and Borcherds products provide concrete examples of paramodular forms [19].
Paramodular forms have Fourier expansions in three variables but the natural generalization of the Fourier series of an elliptic modular form is perhaps the Fourier-Jacobi expansion of a paramodular form, which, using the components , recollects the Fourier series in powers of ,
Each coefficient is a Jacobi form, , and has its own Fourier expansion . The Fourier-Jacobi expansion thus defines a map to formal series of Jacobi forms . This map is not surjective because we cannot freely select a sequence of Jacobi forms and obtain the convergent Fourier-Jacobi expansion of a paramodular form. One source of consistency conditions among the Fourier-Jacobi coefficients arises from a normalizing involution of , the paramodular Fricke involution , where is the Fricke involution on . This involution splits into plus and minus forms, . The block diagonal form of gives a simple action on the Fourier series and consequently gives the following involution condition on the Fourier-Jacobi coefficients of any , :
| (1) | For all semidefinite with , | |||
Let be the subspace of formal series of Jacobi forms satifying the involution condition:
The Fourier-Jacobi expansion gives . It seems a bit audacious to hope that the involution condition alone forces the convergence of a formal series to a paramodular Fricke eigenform, but theoretical results for low level and computed examples suggest that this is true, and we prove it here.
Theorem 1.1 (Main Theorem).
Let , , and . The map is an isomorphism.
The case is not difficult, and neither is the case , when both the domain and codomain are trivial by a result of Skoruppa [38], but already has important examples. The case of Theorem 1.1 was proven by the first author [1], the cases by Yuen and the second and third authors [26]. These results raised the question, made explicit in [26], of whether the involution condition alone implies the convergence of a formal series of Jacobi forms. Theorem 1.1 resolves this question in the affirmative for the first time.
Bruinier [6], Raum [40], and Bruinier and Raum [7, 5] have approached the theory of formal series of Jacobi forms in a more general setting but with a different set of hypotheses. Bruinier [6] and Raum [40] independently extended Aoki’s result for to vector valued formal series of Jacobi forms, and thereby resolved a conjecture of Kudla [27, 28, 29] concerning generating series of special cycles for degree . In [7], Bruinier and Raum proved that formal series of Jacobi forms for converge under the assumption of all symmetries, thereby proving Kudla’s modularity conjecture for general . Pollack [32] has also given a proof of automatic convergence for cuspidal automorphic forms on that takes Jacobi and Levi symmetries as hypotheses. Bruinier and Raum [5] have recently developed a theory of formal series for subgroups commensurable with ; they prove that compatible formal series of Jacobi forms at every -cusp and Levi symmetries imply convergence. They interpret formal series as sections of a line bundle over a formal complex space. Our Main Theorem 1.1 for paramodular groups in degree two and levels is not a consequence of any of these results. We only use formal series of Jacobi forms at the standard -cusp and only assume symmetry under a single involution.
Our main result has applications to computations. Jacobi restriction [26, 4, 33] is a method that attempts to rigorously compute the space by imposing necessary linear relations on the space of Jacobi forms . Jacobi restriction has provided rigorous upper bounds for , even though the authors could not guarantee in advance that the method would work. The following corollary proves that any schema for computing spaces of paramodular forms that spans spaces of Jacobi forms and imposes the involution condition is in principle sound.
Corollary 1.2.
For , define the -vector space as . The sequence is monotonically decreasing for , and we have .
In particular, we have for , and we have equality for sufficiently large .
The first author was supported by JSPS KAKENHI Grant Numbers JP19K03429 and JP23K03039. The second author was supported by the following grants: JSPS KAKENHI Grant Number JP19K03424, JP20H00115 and JP23K03031. The authors thank the American Institute of Mathematics for its critical support of this research.
2. Notation.
We denote the natural numbers by and the whole numbers by . Throughout this article denotes a level, a weight, and a sign. We write for the transpose of a matrix , and for the transpose inverse. Let for compatibly sized matrices, and for . For , set .
For the theory of Jacobi forms see [11]. The upper half plane is . For a Jacobi form of weight and index with we write the Fourier expansion as with and . The order of a nonzero is . For , we set .
For the theory of Siegel modular forms we refer to [12]. For a ring define the positive definite cone with entries in by , and the positive semidefinite cone by . The Siegel upper half space is . An element in the real symplectic group acts on the symmetric space by , and the Siegel factor of automorphy is . For a function the slash -action defines another function . For a discrete group commensurable with , write for the -vector space of Siegel modular forms of weight , and for the subspace of cusp forms.
For the paramodular group in degree , write the Fourier expansion for as , where . For the Fourier-Jacobi expansion of , we write and collect the Fourier expansion in to obtain with Fourier-Jacobi coeffcients . The paramodular group has a normalizing involution, the paramodular Fricke involution, given by , with . Define the Fricke paramodular group . For , let be the Fricke eigenspaces. There are graded rings, , , as well as the graded ring of Fricke plus forms .
for .
3. Formal Series.
A paramodular form has a Fourier series that converges absolutely and uniformly on compact subsets of . The absolute convergence allows rearrangement into a Fourier-Jacobi series where each Fourier-Jacobi coefficient is a Jacobi form . This follows from the fact that the Fourier-Jacobi expansion is term-by-term invariant under , see [18]. We define the formal Fourier-Jacobi series of a paramodular form by
where is a place-holding variable. The Fourier coefficients of a paramodular form also satisfy symmetries determined by the following group:
We call equation (2) below the -symmetries,
| (2) |
The Fourier coefficients of are related to the Fourier coefficients of the Jacobi forms in the Fourier-Jacobi expansion of by
Accordingly, we define a subspace of that satisfies corresponding symmetries. Let be a subgroup.
| (3) | ||||
Thus we have .
We write if we need to indicate the dependence of this Jacobi form on the formal series . It is often helpful to reformulate the definition of by using the notation for . In this way equation (3) shortens to
| (4) |
We may sometimes avoid tracking boundary conditions in summations by setting for .
Lemma 3.1.
Let be a subgroup. The Cauchy product gives the structure of a graded ring, and a graded subring. In particular, , and . The sets , and are graded ideals in , and , respectively. The Fourier-Jacobi map is an injective ring homomorphism of graded rings that sends to , and the inverse image of is .
Proof.
When the formal series , for , are written as then by the definition of the Cauchy product we have where
Thus follows from the grading on Jacobi forms, which shows that is a graded ring. To derive a similar result for we reformulate the Cauchy product on as
| (5) |
To see that (5) follows from the Cauchy product, note the following four equalities.
Conversely, equation (5) implies the second equality, which is equivalent to the Cauchy product. If we assume for , then, for any , . We use to change the indices of summation,
Thus and is a graded ring, noting that a sum of paramodular forms of distinct weights is zero as a holomorphic function if and only if each summand is zero as a holomorphic function. The sets , and are graded ideals simply because . The convergence of the Fourier-Jacobi expansion shows that is injective on each graded piece and hence on . The absolute convergence of the Fourier-Jacobi expansion for proves that is a homomorphism because for ,
and therefore
For a cusp form , the Siegel map gives the containment . By examining the support of ,
we see that , and that , from which follows. The assertion that relies on special properties of the paramodular group. For general subgroups commensurable with it is not true that the Fourier-Jacobi expansion of a Siegel modular form with coefficients that are all cusp forms necessarily comes from a Siegel modular cusp form; however, the inference is valid for paramodular forms because Reefschläger’s double coset decomposition [34] of has representatives of the form with ; see also Corollary 2.5 of Gritsenko [18]. ∎
The involution decomposes paramodular forms into plus and minus forms where . The action of on is if . Therefore the Fourier coefficients of a Fricke eigenform satisfy the involution condition
| (6) |
Accordingly, we define a subspace of formal series satisfying the corresponding involution condition
| (7) |
The map injects. It is helpful to rewrite equation (7) as . If we write we also have . We set , and the more frequently used .
Lemma 3.2.
Let be a subgroup. The Cauchy product gives , , and the structure of graded rings. In particular, we have , and . The subsets , and are graded ideals in and , respectively. The Fourier-Jacobi expansion map is an injective homomorphism of graded rings that sends to , and the inverse image of is .
Proof.
The proof is very similar to that of Lemma 3.1. ∎
Lemma 3.3.
Let . The graded ring of formal series of Jacobi forms, , is an integral domain.
Proof.
An element in is zero precisely when each graded piece is zero. If a product of nonzero elements from is zero then the product of the nonzero graded pieces of highest weight in each factor must be zero. Each nonzero element in each graded piece has a leading term . The product of these leading terms cannot be zero because the ring is an integral domain. ∎
4. Vanishing Order of Jacobi Forms.
In Corollary 3.1 of [2], Aoki significantly improved the known bounds on vanishing orders of Jacobi forms [11, 16], which were in the index for a fixed weight . He proved that a nontrivial Jacobi form of weight and index cannot have a vanishing order greater than . As a consequence the -vector spaces of formal series are finite dimensional. In Theorem 4.3 we state Aoki’s main theorem from [2] in the case of even weights and use it to improve the estimate concerning when enough to get the desired bound on the growth in for dimensions of spaces of formal series of Jacobi forms, namely, for .
Definition 4.1.
For we define by
It is clear that implies .
Definition 4.2.
We define by
It is easy to check that for .
Theorem 4.3 ([2]).
Let . Let have . We have
where runs over all natural numbers satisfying the condition
Lemma 4.4.
Let . If , we have
Proof.
The condition is used in the first equality.
∎
Proposition 4.5.
Let . If then .
Proof.
The case of odd follows from that of even . If then . Since we have and so assuming the result for even weights; hence .
Suppose that there is a nontrivial with vanishing order and even. We will obtain a contradiction to Theorem 4.3. For , we will contradict
Since has a positive minimum, it suffices to show that the set of positive integers satisfying is infinite.
We show that all sufficiently large primes satisfy
We know that implies . Since , we have . Therefore
Lemma 4.4 says that for with we have
We use the linear bound of [16], Proposition 3.2, to check the hypothesis :
Thus by Lemma 4.4 we have
Thus a sufficient condition for a prime to satisfy is
This simplifies to , or , which is true for all sufficiently large because is positive. ∎
Corollary 4.6.
For we have .
Proof.
We have , where is or as is or , by Lemma 3.2 of [26]. By Proposition 4.5 we may cap the summation at and, by Theorem 2.3 in [11], the codimension of in is at most where is the largest integer with . Therefore
It is known that for , . An easy estimate from the dimension formula [11] is . From we have . ∎
Proposition 4.7.
The integral domain is algebraic over its subring . For , each satisfies a polynomial relation of the type
for some , , and some with not identically zero.
Proof.
It suffices to prove the second statement. The group is commensurable with , so by Theorem 6.11 in Freitag [12], the homogeneous quotient field of , , has transcendence degree three. Take three meromorphic functions that are algebraically independent over and select a common denominator . We must have because the are not constant. We obtain four paramodular forms , , , and in that are algebraically independent over . This follows because we may reduce to the case where any putative polynomial relation is homogeneous in the .
Take . We may assume that is nontrivial because otherwise satisfies the conclusion with , , and . Similarly, if then is constant and satisfies the conclusion with , , and . So we assume as well. The four formal series are algebraically independent over because is a monomorphism. Consider the list of five formal series , . For any , there are distinct monomials in five variables with . By substitution of the five formal series into these monomials, we have elements . By Corollary 4.6, however, we have , so for sufficiently large there is a nontrivial -linear dependence relation among these elements. At least one supported monomial in the dependence relation must contain a positive power of because the remaining four formal series are algebraically independent. For the same reason, when a positive power of is supported then its coefficient, after collecting like terms in powers of , must be nontrivial. If is the highest power of that is supported in the dependence relation, then we may write this relation as where the are -linear combinations of monomials in the four . Since is supported, we have nontrivial. Let the weight of be so that . The terms all have the same weight, , which is the weight of . Therefore we may take , as required. ∎
5. Invariance under subgroups of finite index in .
In Proposition 4.7 of the previous section we saw that a formal series of Jacobi forms possessing the involution condition for satisfies a polynomial whose coefficients are formal Fourier-Jacobi expansions of paramodular forms. As a consequence of this polynomial relation we will show in this section that is invariant under a subgroup of finite index in . These arguments best take place inside the ring of formal Fourier series.
The ring structure on the ring of formal Fourier series, , is defined by the Cauchy product, noting that for every the set is finite. We accordingly use a place-holding variable to write an element as
The ring of formal Fourier series, is also an integral domain, which will be proven in Corollary 5.2. The Fourier expansion of a paramodular form defines a map
Given a formal series of Jacobi forms , we may define the associated formal Fourier series, , by
Extending by linearity we have a map from the ring . That is a ring homomorphism is an exercise using the Cauchy product similar to the computations demonstrating equation (5). For we have the compatibility .
Formal Fourier series share some properties with formal power series in three variables due to the following monomorphism that sends to the monomial .
Lemma 5.1.
Set , , . The map
is a ring homomorphism satisfying the following properties.
-
(1)
,
-
(2)
,
-
(3)
is injective.
Proof.
Setting and solving the system of linear equations , , and , the unique solution is , , and . This proves formula if we understand that for . Formula follows from since rearrangements of formal power series are equal. The ring homomorphism property is then formal because the are linear in . The injectivity follows from formula . ∎
Corollary 5.2.
The ring is an integral domain.
Proof.
A ring with a monomorphism to an integral domain is an integral domain. The monomorphism here is . ∎
There is a copy of inside the group of automorphisms of formal Fourier series.
Lemma 5.3.
For define
The map is an automorphism of and the map
is a homomorphism. For , we have if and only if for all . For we have for all .
Proof.
We show is an automorphism. The map has as an inverse and the additivity of is clear so that it suffices to prove . We have
We use the equality to change the index of summation.
We show that is a homomorphism. Take and . We have
For we have where for . We then have and, by definition of formal series, this equals if and only if for all . The final assertion, for all , follows from the -symmetries of in equation (2). ∎
The next proposition shows that a formal series of Jacobi forms satisfying the involution condition necessarily has additional symmetries.
Proposition 5.4.
Let . There is a subgroup of finite index in such that .
Proof.
By Proposition 4.7, satisfies a polynomial relation of the type
for some , , and some with not identically zero. Apply the monomorphism to obtain
The polynomial has a root . Noting by Lemma 5.3 that for every , each element of the orbit is a root. However, the ring is an integral domain and a polynomial of positive degree over an integral domain has at most roots. If we want to be definite we can find with such that . The natural homomorphism from to permutations of the orbit is specified in this labeling by for . Let be the kernel of . Then is a normal subgroup of of index at most . For every we have so that, by Lemma 5.3, for all . Thus as claimed. ∎
6. Specialization.
Let be a subgroup of finite index. A formal series of Jacobi forms has the defining symmetries for all and all . We will construct a dense subset of where the formal series specializes to a holomorphic function of one variable. More precisely, for each , the series will converge to a holomorphic function on the neighborhood of infinity . The formal series thus converges on a dense subset of .
Definition 6.1.
For , let be the minimal positive denominator of .
Definition 6.2.
Let be a subgroup of finite index. Let . Define
We follow Lehner [30] for the theory of Fuchsian groups. We view the Riemann sphere as the extended complex plane. The Riemann sphere is the disjoint union of the extended real numbers, , and the upper and lower half planes. The groups and act on by Möbius transformations and preserve this disjoint union. We will only consider subgroups and the corresponding transformation groups .
Definition 6.3.
Let . The limit set of is the set of such that there exists a and a sequence of distinct with .
For subgroups , if has finite index in then ; this is the theorem in section 2C of [30], page 11. The corollary in section 2F, page 14, is that either or . We mention some terminology to assist readers who use a different reference. A group is Fuchsian when . The theorem in section 2F of [30], page 13, characterizes Fuchsian groups as the discrete subgroups of . For us, the salient result is Theorem 3 in section 3E, page 21.
Theorem 6.4 ([30]).
Let be a subgroup. If is a closed set containing at least two points, such that , then .
As Lehner comments on page 21, this theorem may be rephrased: when has more than one point, is the smallest closed -invariant set containing at least two points.
An example of a Fuchsian group is . The orbit of is and . This implies that any subgroup of finite index also has . For a ring , set . The stabilizer of in is and we have an orbit-stabilizer bijection given by sending . This bijection shows that a subgroup of finite index in cannot stabilize . We require the following ergodic corollary of Theorem 6.4.
Lemma 6.5.
Let be a subgroup of finite index. The -orbit of is dense in .
Proof.
We have because has finite index in . Furthermore, cannot stabilize for the same reason. Therefore, the -orbit of has at least two points, as does its closure . We check that the closure remains -invariant. Take and . We have for some , so that for . Hence and is -invariant. Thus, by Theorem 6.4 we have . The -orbit of is thus dense in . ∎
Lemma 6.6.
Let be a subgroup of finite index. The sets , , and are dense in , , and , respectively.
Proof.
It suffices to prove is dense in . Take and any neighborhoods of and of . We just need to find an element of in . Pick an irrational number .
The group has finite index in and so . By Lemma 6.5, the -orbit is dense in , and hence -orbit-orbit is dense in . Accordingly there is a sequence -orbit with , and we have for all sufficiently large . Set . Since is irrational we have . Set so that and . For all sufficiently large we have , which shows that is nonempty, and hence that is dense in . ∎
Let be a subgroup of with finite index. For , define the Hecke bound . The Hecke bound is a norm on the vector space and the Fourier coefficients , given by , become bounded linear functionals. Hence the Fourier coefficients are continuous in in the topology on induced by this norm. For Jacobi cusp forms the Hecke bound [11], compare page , is given by
Lemma 6.7.
Let with . For all we have the bound .
Proof.
Let be the Fourier series with and . By the familiar formula for the Fourier coefficients, we have
Choosing we have . Thus we have . For this is minimized by proving . This proves the result when , and when we have so the Lemma’s conclusion holds as well. ∎
The following proposition gives a specialization of a cuspidal formal series for each . Each specialization depends only upon a finite number of the Fourier-Jacobi coefficients of , although this finite number may increase with the denominator of .
Proposition 6.8.
Let be a subgroup of finite index. Let . For , set . We have The sequence of Hecke bounds satisfies
where the implied constant depends only on and .
Proof.
Take . For each there is a positive constant such that for all . Here we use the assumption that each is a cusp form and Lemma 6.7. By Proposition 2.2 of [20] or Theorem 4.2 of [2], for example, we have because . This elliptic modular cusp form has the Fourier expansion
for . Next we check . For any we first note that is determined by , and that satisfies
| . |
Thus and the number of integers in this interval is at most the length plus one, showing . Pick an such that the Fourier coefficients for determine . By virtue of the valence inequality, the choice will work. The following supremum exists because the denominator is nonzero, the quotient is continuous in , and the supremum may be taken over a compact sphere.
Therefore we have
for or, equivalently, . Each has at most elements and there are at most of them so we have . Note that implies that . This is because
| . |
We now come to the main point where we use the -conditions for . Our assumption gives us -orbit. Accordingly there exists a matrix with . Letting , the conditions and imply and . Moving this matrix to we define
We note that implies . For , compute , where
For we have so that , as well as and . By the -conditions of equation (3) for we see
if we remember that implies and note . Therefore, for , we have
since . The constant grows with the denominator of but in each individual case controls the growth of all in terms of the finite number of with . Therefore we have
Thus as was to be shown. ∎
Consider a formal series . For a point , if the series converges absolutely, we say that the formal series converges absolutely at .
Corollary 6.9.
Let be a subgroup of finite index. Let . The formal series converges absolutely on . For set . The function defined by
is holomorphic on .
Proof.
Pick with . By Proposition 6.8, for some constant and the elliptic forms given by . From we infer
The radius of convergence of the series thus satisfies , and so this series converges absolutely for . In particular the series converges at if . The condition , however, is just .
Consider so that for some . We have seen that the power series in ,
| (8) |
converges when . A power series that converges on an open set is holomorphic there. Thus is holomorphic on the open set . ∎
7. Locally Bounded.
Locally bounded families of holomorphic functions possess remarkable convergence properties. In Theorem 7.4 we show that, for subgroups of finite index, the partial sums of the formal series are locally bounded on if is integral over . The following homomorphism respects the ring structure but forgets the grading on .
Lemma 7.1.
For each there is a ring homomorphism defined by sending to and extending additively to .
Proof.
The ring structures are defined by the Cauchy product rule, so substitution is a homomorphism. ∎
Let be the open disk about the origin in with positive radius . Let be the ring of power series that converge on . The map that sends a holomorphic function on to its Taylor series about the origin is a ring isomomorphism. For and , let be the sum of at . The evaluation map sending to is a ring homomorphism with the property that .
Lemma 7.2.
For each , define , a radius , and a disk . The ring homomorphism sends into . For , set . For and , we have .
Let be a subgroup of finite index. For , sends into . For and we have for the holomorphic function defined in Corollary 6.9. The map that sends to is linear and multiplicative.
Proof.
We know that is a ring homomorphism from Lemma 7.1. We need to check that the relevant power series converge in and correctly label their values. For the Fourier-Jacobi expansion converges for . The formal series is and the value of on it is . For any , there is an with and . Accordingly, is convergent and, since was arbitrary, . For any we have and .
Assume that . Since , we can apply Corollary 6.9 to assure the convergence of the series for . Any may be written for , so the power series converges on and so . The value of the convergent series at is given by .
Both and the evaluation map are ring homomorphisms so their composition, restricted to the ideal is linear and multiplicative. This shows, for all , that we have
which is what means. ∎
Lemma 7.3.
Let and be given. Let a monic polynomial of degree be given by for . If then .
Proof.
Apply the Triangle Inequality to the case . ∎
Theorem 7.4.
Let . Let . Let be a subgroup of finite index. Let satisfy the monic polynomial relation in for
with for .
Then the sequence of partial sums for is locally bounded on .
Proof.
Pick . We need to make a neighborhood of and a positive constant such that for all and all we have . Let be an open Euclidean ball centered at with closure . We can push the closed ball down by so that this translation is still contained in . We do this in detail. For any , the translated ball lies in the space . Define the function for any by
The function is continuous on and positive on and so has a positive minimum on . Set . We remark that only depends on the choice of the ball . Translate the ball by to obtain the ball with closure in given by . We have
Therefore for and . By this process, is completely determined by . Next define a compact set that contains . Use the continuity of the to define . We will show that works as the local bound at .
The main step will be to show for every and every . The conclusion on the neighborhood then follows because is continuous on and is dense in . The assumption implies . We may rewrite the hypothesis
as where . Apply the ring homomorphism to the given relation in to obtain a relation among formal power series
| (9) |
By Lemma 7.2, we know for , the radius , and the disk . Moreover, the power series converges in . Therefore the formal power series in equation (9) is in the subring of convergent power series on .
For , we have and so we may use the evaluation homomorphism to obtain a relation among complex numbers
| (10) |
For , define . By Lemma 7.2, for any , we have both and . We rewrite equation (10) as
By Lemma 7.3 we have
| (11) |
By the definition of , we have
| (12) |
We have because . Furthermore, for , we have because , and the function is holomorphic on . Thus we may compute the Fourier coefficients of
by the familiar formula
We calculate, for any ,
As mentioned above, we have for , so that by equation (12). Thus
This completes the main step and, as explained, consequently shows that on the neighborhood of for all . Since was arbitrary, the sequence of partial sums is locally bounded on . ∎
8. Holomorphicity.
In Theorem 8.2 we show that the divisibility of paramodular Fricke eigenforms is implied by the cuspidality of the quotient of their formal series. Later, after proving our main result, Theorem 10.1 improves Theorem 8.2 by dropping the cuspidality assumption.
We review divisor theory following Gunning [21]. Let be a domain. For open , let be the ring of holomorphic functions on . For , let be the ring of germs of holomorphic functions, and the field of germs of meromorphic functions at . A holomorphic subvariety is a subset such that each point has a neighborhood where is the zero set of finitely many functions holomorphic on . A holomorphic subvariety is irreducible if for holomorphic subvarieties , , implies or . The regular points of , , are the points where is a complex manifold for some neighborhood of . The regular points are an open dense subset of , locally finitely connected, and is connected precisely when is irreducible. The dimension, , of an irreducible is the dimension of the complex manifold , and the codimension of is .
We let be the set of irreducible, codimension one holomorphic subvarieties of . For a function , the support of is . We say is locally finite if, for all open with , the set is finite. The group of divisors in , , is the abelian group of locally finite functions . Since is countable, we often write a global divisor as , for and distinct . The semigroup of effective divisors, , is defined by .
For the germ of a holomorphic subvariety at , , the dimension, , is the maximal dimension of the finitely many connected components of whose closure contains . We say is pure dimensionsal when all these connected components have the same dimension. The definition of irreducibility for is as before. Let be the set of germs of holomorphic subvarieties at that are irreducible and have codimension one. A function has support . The group of local divisors at , , is the free abelian group on . We write local divisors as with and distinct but, fundamentally, a local divisor at is a function with finite support. The semigroup of effective local divisors, , is defined by . If then is irreducible. For and general , decomposes into a finite union of distinct .
The ring is a unique factorization domain and noetherian. The fundamental correspondence of algebraic geometry holds between germs of holomorphic subvarieties and germs of holomorphic functions at . The ideal consists of the germs that vanish on and, for an ideal , is the germ of the holomorphic subvariety defined by the simultaneous vanishing of the elements of . The germ is irreducible precisely when is prime, and has pure codimension one precisely when is principal. Therefore, for and , there is a neigborhood of and a , with prime in , such that . We call this a local equation at and refer to as a uniformizer. As a consequence of Cartan’s Theorem, see Theorem F6 of [21], there is a neighborhood of such that generates at all points . Cartan’s Theorem asserts the existence of a neighborhood of and of functions such that at all points . Since divides each there is a neighborhood of where at all points . Hence and is prime for all .
For a ring , let be the group of units and . The sequence of semigroups is exact, where is the semigroup homomorphism determined by the factorization in into powers of nonassociate irreducibles , unique up to order and units. The rule defines a group homomorphism . The local divisor is effective precisely when . To each global divisor and point , we associate a local divisor, , where is the finite decomposition of into irreducibles. For each , the global divisor is the unique such that for all . The rule extends to a group homomorphism , where the field of meromorphic functions is the quotient field of . The global assertion that is effective precisely when follows from the corresponding local assertion.
For and , a direct way to compute is to take a regular point and a local equation at , , so that is irreducible and is prime with . The factor in the unique factorization of in defines . This is independent of the choice of the regular point and of the uniformizer .
We use and in the following lemma. The usefulness of the following lemma is that a point on the -orbit of a codimension one irreducible holomorphic subvariety of can be found where the uniformizer is regular in at that point. This allows us to use the Weierstrauss preparation theorem on in the proof of Theorem 8.2. In Lemma 8.1, the term “regular” regettably has two distinct meanings. A holomorphic is regular in at means in all neighborhoods of , whereas a point of a holomorphic subvariety is regular if is a complex manifold for some neighborhood of .
Lemma 8.1.
Let , and . Let . Let be an irreducible component of . For any , is also an irreducible component of , and we have .
There is a , a regular point , a neighborhood of , and a function with an irreducible germ , and with regular in at , such that .
Proof.
Each is biholomorphic on so is an irreducible holomorphic subvariety of codimension one if and only if is. The automorphy of shows that is supported in if and only if is, although we remark that and might be equal. Since is biholomorphic, we have . The automorphy of gives us so that and differ by a multiplicative unit and .
Every regular point has a local defining equation for given by for some neighborhood of , and some uniformizer with an irreducible germ represented by . We may select so that every point of is also a regular point of because is a complex manifold at . By Cartan’s Theorem, we may also assume that a single function element satisfies for all . For the proof of the second part, we break the discussion into two mutually exclusive cases. Case . Some regular point has regular in or at . If in for some regular then we take to satisfy our conclusion. Whereas if in we take and consider . The point is a regular point of because is a regular point of and is biholomorphic. The local uniformizer at may be represented by . In this case
is regular in at . From we deduce so that and play the role of and in the statement of the lemma. Case II. Every regular point has not regular in and not regular in at . Consider any regular point . Take a polydisk about and use the function element . We will ultimately show that Case II is very special and that . We claim that for all . If not there is some with . Since is not regular in at we have for in some neighborhood of , hence necessarily for all . Therefore . The setting of Case II applies equally well to , which is still a regular point of , so that is not regular in at , and for in a neighborhood of , and hence for all . Thus , contrary to our supposition. Thus for all . Therefore vanishes on and so the irreducible divides in ; however, the germ of at is irreducible and so and differ by a multiplicative unit in , after possibly shrinking further. Therefore in Case II we have the possible but special circumstance . Without loss of generality we may adjust the selection of by a unit and assume that in . Now set and consider . The local uniformizer at may be represented by . For this choice we have
which is regular in of order at . Thus the conclusion of the lemma holds for and in the role of and . ∎
Theorem 8.2.
Let and . Let be a subgroup of finite index. Let . Let be nontrivial and . Let be an irreducible component of .
If in then . The meromorphic is holomorphic and .
Proof.
By Lemma 8.1 we can find a point , for some , where the local defining equation has the following nice properties. The point is a regular point of , the uniformizer is regular in at , and the germ is prime in . By Lemma 8.1, in order to prove it suffices to prove , so we may rename as without loss of generality.
In this paragraph we outline the remainder of the proof. We will find a neighborhood of such that every point has the same nice properties that does. We use this to twice move to nearby points. First we select so that is the only irreducible factor of that vanishes at . Then we use the Weierstrauss preparation theorem on to find a , and a holomorphic curve inside and passing through , which guarantees that the formal series converges on . The conclusion about the orders of vanishing readily follows from the convergence of on the holomorphic curve.
The set of regular points is open in . Regularity in at is a local condition in because regularity of order less than or equal to at is implied by the nonvanishing of a partial derivative . By Cartan’s Theorem, there is a neighborhood of such that generates at all points . Accordingly there is a smaller neighborhood of where every has the nice properties , regular in at , and prime in .
In the local ring , factor and , where are finite products of irreducibles that are not associate to , and where and by the direct way of computing the vanishing order on . We select a neighborhood where the germs of the above factors all have representative function elements. If for all then vanishes on and divides in because is prime, contradicting the fact that is a finite product of irreducibles that are not associate to in . Hence there is a point such that . By shrinking , we may assume that is nonzero on .
We apply the Weierstrauss preparation theorem to , which is regular in at . There exist neighborhoods of and of such that, and
for some , for , and . The set is dense in by Lemma 6.6, so by taking close enough to so that we may choose a root of close enough to , we have
From we inherit the representative function elements , , , , and , so that and in , and is nonzero on .
The formal series of is given by . Recalling that , and using Lemma 7.2 to apply , we have the equality of convergent power series on for and . Also set for . For we have and may apply the evaluation homomorphism to obtain
Making use of Lemma 7.2, this may be written on , which contains a neighborhood of , as
By specializing the factorizations of and in to the holomorphic curve we have
Note that has at most zeros on and is hence nontrivial. Of the two cases , and , the first is our conclusion, so we will conclude the proof by showing that the second does not occur. In the second case, by cancelling powers of the nontrivial , we have
We evaluate these at to obtain
which contradicts . Thus we have .
A meromorphic function with an effective divisor is holomorphic. Therefore is holomorphic and continuous on . There is an open dense subset of where transforms like an element of . By the continuity of and of the factor of automorphy, transforms like an element of on and hence . ∎
9. Main Theorem.
Theorem 9.1.
Let be open. Let be a locally bounded sequence of holomorphic functions on that converges on a dense subset of . Then the sequence converges on and uniformly on compact subsets of .
Proof.
This is Exercise a of section in Chapter of [14].
The proof given here imitates the proof of Lemma IV.4.8 in [13] for the one-dimensional case. It suffices to show that is uniformly Cauchy on compact subsets of . Pick a compact and an . There is no loss of generality in assuming that is a closed ball. We will construct an from the given data , , and .
The family is equicontinuous on because it is locally bounded, see pages - of [31]. Select to enforce this equicontinuity on for .
| (13) |
Cover with open balls inside at each point of with radii that are less than . By the compactness of we have for some , and . Since is dense in , we can pick an ; this uses that is a closed ball. Note that for every point we have and therefore by (13). By hypothesis each sequence converges and so is Cauchy. Select so that
| (14) |
Let be the promised natural number. We have
for any , , and . If , then and so by (14). Choose so that ; then for all by (13). Therefore, for all with , and all , we have , and the sequence is uniformly Cauchy on . ∎
The essence of the proof of the main result, Theorem 9.6, lies in the argument for the following special case of Fricke plus cusp forms. The general case will be reduced to Theorem 9.2.
Theorem 9.2.
The map is an isomorphism for .
Proof.
Since the map is injective, the only issue is surjectivity. Take a nontrivial . By Proposition 5.4, there is a subgroup of finite index in such that , which implies . Completely separately, by Proposition 4.7, satisfies a polynomial relation in ,
for some , , and some with not identically zero. Since , , this polynomial relation is actually in . Since is a graded ideal in , set
Then is integral over and satisfies a monic polynomial relation in given by
with for .
By Theorem 7.4, the sequence of partial sums is locally bounded for . By Corollary 6.9, we also know that the formal series converges on the dense subset of , noting . Therefore, by Theorem 9.1, the sequence of partial sums converges uniformly on compact sets of . The limiting function is given by and satisfies for all because each term is so invariant. The holomorphic function is hence periodic with respect to the translation lattice and has a Fourier series that converges absolutley and uniformly on compact sets of . This absolute convergence of the Fourier series of is the crux of the matter because it will allow us to rearrange the order of summation and thereby deduce the crucial invariance property from the involution condition on . Since the convergence of the sequence of partial sums is uniform on compact subsets, the Fourier coefficients of are supported on and, for , satisfy . For in this proof we have . Recall that is an involution of satisfying , and that absolutely convergent summations over may instead be rearranged to be taken over .
Combining these automorphic properties of we obtain for all , see Gritsenko [17, 18]. Therefore and . Together these imply that by Reefschläger’s decomposition [34], as in Lemma 3.1.
We have for the formal series , and the paramodular forms , . By Theorem 8.2, . From we see that . Combining this with , we have , which implies since is an integral domain and is nontrivial. By Reefschläger’s result, and imply , which proves that is surjective. ∎
Lemma 9.3.
There is a nontrivial cusp form for .
Proof.
There are a number of possible approaches to proving the existence of nontrivial minus cusp forms. We might use an asymptotic trace formula or the oldform theory of Roberts and Schmidt [35, 36]. Perhaps the briefest is to use the injectivity of the Gritsenko lift [18], for . An estimate from the dimension formula [11] is for . Considering , we see for . Thus there is a nontrivial Gritsenko lift in for . ∎
Theorem 9.4.
The map is an isomorphism for , and .
Proof.
The case is Theorem 9.2, so assume and . Take . There is a nontrivial cusp form by Lemma 9.3. Multiply by to obtain a new formal series . By Theorem 9.2, there exists a such that . Both and have -symmetries and we will show that does too. Apply the ring homomorphism of section 5 to to obtain . Now, for , apply the automorphism of Lemma 5.3, which fixes and to obtain . Since is an integral domain and is nontrivial, we have for all . By Lemma 5.3, we conclude that . By Theorem 8.2 with , the equation implies that is holomorphic, and, consequently, that . From and we obtain . Since is an integral domain and is nontrivial, we have . By Reefschläger’s result and we have . Thus is surjective. ∎
Lemma 9.5.
Let and . Let . There is a cusp form such that .
Proof.
The divisor of is the reducible locus ,
For , we have for the nonvanishing . Thus and have no common zeros on . Pick coset representatives for . For any , consider the nontrivial norm
We have . Pick any that is not on any of the finitely many codimension one complex lines through the origin . The cusp form is as claimed. ∎
Theorem 9.6 (Main Theorem).
Let , , and . The map is an isomorphism. The map is also an isomorphism.
Proof.
The case is easy so assume . Take . For any point there is a cusp form such that , as in Lemma 9.5. Define a corrsponding product of formal series . By Theorem 9.4, there exists a such that . Note that where is the character defined by and on .
In the same way, for any point distinct from we also have . Since is an integral domain we necessarily have , or . Therefore we have whenever both quotients are defined. We may define by setting in neighborhoods where the denominator does not vanish, and by noting this is independent of the choice of . Therefore we have , , and , for any point and any . Thus . Since is an integral domain and is nontrivial, we have and . It remains to show . This follows from , , and the fact that is an integral domain and is nontrivial. The result for cusp forms follows from the general result because, by Lemma 3.2, the inverse image of under is . ∎
To help prove Corollary 1.2 we use a lemma from linear algebra.
Lemma 9.7.
Let , , and be finite dimensional vector spaces. Let and be linear maps. Define a subspace by . Define the projection by . If is injective then is injective, and conversely.
Proof.
We have . Therefore if and only if . ∎
We now prove Corollary 1.2 from the Introduction.
Proof of Corollary 1.2. To prove that the sequence is monotonically decreasing for , it suffices to prove that the projection , sending to , injects. To employ the notation of Lemma 9.7, we set and . To define , first define the finite set and set . Define a linear map by , and a linear map by . From the definition of we see that the kernel of consists of the Jacobi forms vanishing to order , so that . From equation (1) we see that for
Since , Proposition 4.5 implies , so that injects by Lemma 9.7. Since is the restriction of to , we have shown that the projection injects for .
For the inverse system we have . The reason for this is that any equation imposed by equation (1) on is also imposed on for all .
Since the sequence is eventually monotonically decreasing, it is eventually constant, and the injective are eventually isomorphisms. Thus . From Theorem 9.6 we have and this completes the proof.
10. Concluding Remarks.
The Main Theorem 9.6 gives an quick proof, without needing to check convergence, of the existence of the Gritsenko lift [18], , for . For , let be the index raising operator from [11]. For , the formal series is directly checked to be in . Here, the first term with the elliptic Eisenstein series is zero unless is even. By the Main Theorem there is a with .
The Main Theorem 9.6 can be used to construct the global level raising operators in the paramodular newform theory of Roberts and Schmidt [35, 36]. For a prime , the three level raising operators , and , are used to create the oldforms. We can directly define the global level raising operators by giving their action on Fourier-Jacobi expansions. For , let be the index raising operator from [11]. The simplest to define is . For with , define . One directly checks so that by the main result there is an with . This defines since the entire space is the direct sum of the plus and minus forms.
In order to define and , set , and . One directly checks so that by the main result there are with . Defining and correspondingly , we have with , , and . Since these constructions work for , this gives a generalization of the global level raising operators.
The theory of formal series for arithmetic subgroups developed in [5] frames different hypotheses than we do in the case of the paramodular groups. Still, in their Theorem 4.8, Bruinier and Raum considered the implications of their general theory for paramodular groups when a single formal series at the standard -cusp is given. In our notation, they proved the following. For squarefree , let be the extension of by all the Atkin-Lehner involutions. Let be a character trivial on . If a formal series satisfies for all , then converges on . Using simple generators for , this reproves the cases from [26]. In comparison, our main result allows general and assumes a single symmetry under the paramodular Fricke involution.
Theorem 8.2 only needed to be proven for cuspidal quotients because the Main Theorem 9.6 was reduced to the case of formal series of Jacobi cusp forms. With the Main Theorem in hand, however, we may improve Theorem 8.2 to obtain a general criterion for the divisibility of paramodular Fricke eigenforms that is interesting in its own right because it avoids all discussion of divisors.
Theorem 10.1.
Let and . Let , be nontrivial, and . If we have in then the meromorphic is holomorphic and .
Proof.
We have if and only if . Applying to we obtain . Applying we have . Since is an integral domain and the element is nontrivial, we have , and thus . By the Main Theorem 9.6 there is an such that . Thus the quotient is holomorphic. ∎
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