Modular iterated integrals associated with cusp forms
-
Nikolaos Diamantis
Abstract
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.
1 Introduction
This paper deals with two classes of functions that have not been previously studied together, namely modular iterated integrals and higher-order modular forms. We show that they are interrelated in a way that key features of one of them can be elucidated through constructions in the other.
The first class of objects, modular iterated integrals, were introduced recently [2, 4, 3] by Brown in the context of the theory of real-analytic modular forms. Those are real-analytic functions f on the upper half-plane
A transformation law of the form
and for all
An expansion of the form
where
The motivation for their introduction included their possible use towards arithmetic questions involving periods and evidence that the modular graph functions of String Theory are real-analytic modular forms.
A special subclass of the class of real-analytic modular forms consists of the spaces
where
A reason for the special interest in this class is that the modular graph functions are expected to belong to it (cf. [2, Sections 1 and 9], where explicit evidence of this is provided). A second reason is that its structure seems to have arithmetic significance, as indicated by evidence provided by Brown (in [2, 4]) for the motivic nature of the space and by the association to it of classical number theoretic invariants, such as L-functions and period polynomials, by Drewitt and the author [10].
In this paper, we will address two questions that arise from the works mentioned above.
Question 1.
The elements of
where E is the subspace of M generated by Eisenstein series.
What can be said about the remaining modular iterated integrals, i.e. the part originating in cusp forms? This is important not only because an answer describes more completely the structure of
We will provide an answer to this question by constructing (in Section 2.2.2) an explicit family of such functions originating in cusp forms. We denote those functions by
Question 2.
A more explicit characterization of the space
Here we discuss a new approach to this problem: We construct explicit “real-analytic iterated integrals”, denoted by
The tool with which we achieve both of those two aims is based on higher-order forms, the second object we deal with in this work.
The characterizing feature of higher-order modular forms, in the special case of order 2 and weight k, for instance, is the transformation law
where the action of the group on the function is given by
The precise definition will be given in Section 3, where the original notion of higher-order forms will be generalized. Higher-order modular forms have been studied from various perspectives (analytic, adelic, algebraic, spectral etc.) and led to applications to modular symbols, mathematical physics etc. In all these cases, the theory had to be developed on congruence subgroups of level higher than 1, because such forms were parametrized by cusp forms of weight 2, which are trivial in level 1. This was a unnatural constraint because it excluded integrals of higher weight forms from consideration and it also prevented availing oneself of simplifications occurring in
In this paper, we resolve this problem too, by proposing a very general framework within which to consider higher-order forms. The class obtained includes several known and new objects, including higher-order forms for all levels and iterated integrals.
It turns out that the solution to this problem allows us to realize the constructions behind our answers to Questions 1 and 2 above. Firstly, it allows us to produce a family of (extended) modular iterated integrals of length 2 originating in cusp forms (Question 1). These modular iterated integrals, in turn, by their very construction, are obtained from a class of second-order modular forms whose prototypes are exactly the iterated integrals of cusp forms (Question 2). At the same time, these second-order modular forms are not of an ad hoc nature. They form a basis of the class of second-order modular forms they belong to (Theorem 3.8). This suggests a deeper relation between the two objects that are the subject of this paper.
2 Subclasses of the space of real-analytic modular forms
2.1 Review of definitions and notation
We start by introducing some of the notation we will be using and by recalling the definitions of real-analytic modular forms and of modular iterated integrals.
2.1.1 Basic spaces and actions
Let
If
For
Here
We extend the action to
We can now give the definition of a real-analytic modular form.
We call an
For all
For some
(2.1)
We denote the space of real analytic modular (resp. cusp) forms of weights
For
2.1.2 Lie structure
The Lie algebra
(for proofs and further details on this and the rest of this subsection, see [2, Section 2.2]).
These operators induce bigraded derivations on
with a similar definition for
Two pairs of identities we will be using often, taken from [2, Lemma 2.5 and (2.13)], are
To simplify notation, we will omit the index of
2.1.3 Modular iterated integrals
We can now recall the definition of the space
For each integer
where
In [2, Lemma 3.10] it is proved that
Proposition 2.1.
Let
(where
Note that our normalization of
2.2 The space of extended modular iterated integrals
2.2.1 Motivating remarks and definition
With Proposition 2.1 and the definition of
Now, to the best of our knowledge, all explicit examples of elements of
for a suitable choice of indices.
The motivation for that, apart from the general aim of understanding more fully the space
Before we state our definition, we note that, in contrast to the definition of
We provide a heuristic argument why this should not be possible. Assume, for contradiction, that the “cuspidal part” is indeed restricted to the “first quadrant” and that
where the sum ranges over a finite number of
for some holomorphic function
where, for compactness of notation, we have taken
Definition 2.2.
Let the space
There are two differences between
Secondly, the restriction of the space belonging to the “first quadrant” present in the definition of
Finally, it should be stressed that we do not claim that
2.2.2 An explicit sub-class of ℳ ℐ 2 ′
We will now define an explicit family of elements of
Let
by
We use the same notation for the sub-representations corresponding to
Let now
be a cusp form of weight k for
where the bar means complex conjugation (acting trivially on X).
Let
and
We have the following proposition.
Proposition 2.3.
Suppose that
Proof.
We show it for
By the first equality of (2.7),
To prove the statement about absolute convergence, we first note that
the change of variables
imply that
By applying this decomposition to the defining series for
To analyze the part corresponding to the first term of (2.10),
we note that each of the integrals appearing in the sum
is (up to a power of i) the value at
It is well-known that
Also,
Upon applying the binomial formula to (2.13) and substituting into the polynomial
we get an expansion in
For the last estimate we used (2.12).
The elementary inequality
which converges uniformly for z in a compact set since
The second term of (2.10) gives the polynomial
which, by Proposition 2.1, has real-analytic functions as coefficients since
Therefore, both pieces of
The bound (2.14) and the polynomial growth of
The series
be functions such that
From [2, Proposition 7.1], we know that
we need to show that it has an expansion of the form (2.1). This is part of the content of the next proposition.
Proposition 2.4.
Suppose
and
where
Further, each
Proof.
Replacing
and expanding with the binomial theorem, we see that the coefficients in the right-hand side of (2.15) can be written as
respectively
Upon unraveling the definition of the action
The change of variables
The first integral in the right-hand side is 0 for
Equation (2.16) follows from this and (2.11) combined with (2.18). Equation (2.17) can be deduced from equation (2.16) upon a conjugation.
To show that
Then this becomes
where we used the Poisson formula followed by a change of variables. With [12, Section 3.2, equation (12)], combined with [14, equation 13.14.9], the integral equals
We can apply this with m, n, p, t replaced by
We can now verify an identity for
Proposition 2.5.
Let
Proof.
With (2.2), we have
The definition of
With the transformation law for
In the minus-case, (2.20) equals
This proposition will enable us to define a subset of
We can now define our sub-class of
Theorem 2.6.
The space
Proof.
An elementary computation implies that for all real-analytic
where, for convenience,
Proposition 2.5, combined with (2.21), implies that each
In the same way, we verify the analogous statement for
Finally, by Proposition 2.4,
3 The space of iterated invariants
To define our extended higher-order modular forms, it will be necessary to describe a general framework involving a family of representations; see [8, 9] for two alternative general definitions of higher-order objects, which are built on only one representation and which use the formalism of the augmentation ideal.
Exceptionally, we will give the next definition for general Fuchsian groups Γ of the first kind acting on
Let
In the following definition, if V is a Γ-module, we view
Definition 3.1.
Set
where the implied action is induced by that of
where
We call the elements of
In the next proposition, we show that this definition is well-founded and we give an equivalent formulation of it.
Proposition 3.2.
For each
We have
Hence, composing with a map
The space
and, for each parabolic
Proof.
(i) We show this by induction over n.
Let f be an element of
Suppose that
which, by induction hypothesis, belongs to
Because (3.1) holds for all
(ii) We have
(iii) This is seen by unraveling the definition, which, as shown in (i), is well-founded. ∎
A first result on the structure of the space of iterated invariants is provided by the following lemma.
To state it, we introduce some additional notation for each
With this notation, we have the following lemma.
Lemma 3.3.
Let
In particular, for
Proof.
Fix a basis
for some
For the case
Corollary 3.4.
Let
where
Proof.
This is deduced directly from Lemma 3.3. One can also use the long exact sequence associated with
to deduce the corollary. ∎
Since the maps ψ and
is given,
for each
where the
For
Then
3.1 Extended higher-order modular forms
For
where
and, for all parabolic
where the action of
In particular,
Let
where
Remark.
The adjective “extended” in the previous examples aims to distinguish them from the class of (standard) higher-order modular forms (see, e.g., [6]). We can retrieve the standard higher-order modular forms by setting
and the space
The standard higher-order modular forms become trivial in
3.1.1 Iterated Eichler integrals
Important examples and, indeed, some of the prototypes, of the standard higher-order forms mentioned in the closing remark of the last section are the antiderivatives of weight 2 cusp forms and their higher iterated analogues:
In Lemma 3.5 we show that, more generally, Eichler integrals and their iterated counterparts belong to
Lemma 3.5.
Let
Then
Remark.
Though not “visible” in the statement,
Proof.
We first show the assertion for
The action “.” is given explicitly by (3.2). We then see that
where the last integral is obtained by a change of variables and (2.8). Therefore, with (3.3),
The same identity shows that
The proof for general n is an application of the shuffle product formula for iterated integrals, but we give a direct proof by induction. As before, by the definition of the action “.” on
where the sum of integral signs indicates that they are both applied to the integrand following them. Therefore,
This, on the one hand, implies that
belongs to
This completes the proof of the statement. ∎
Upon tensoring with a space of cusp forms, we obtain the following corollary.
Corollary 3.6.
Let
Then
3.2 The space of real-analytic iterated integrals
As mentioned in Section 1, modular iterated integrals are conjectured to be given in terms of
Let
Notice that, although
We also consider the one-dimensional subspace
Explicit examples of elements of
Corollary 3.7.
Suppose that
Then
Proof.
This follows immediately from Lemma 3.5 combined with the identity
The functions of this corollary have been the prototypes for the elements of
Now, with the definition of
We will show that this can be completed to a right exact sequence.
Theorem 3.8.
Suppose that
is exact.
Proof.
The only part remaining to be proved is the surjectivity of
By Proposition 2.3, this is absolutely convergent and its coefficients are of polynomial growth at infinity, and thus they belong to
The image of
For the two equalities, we have used Proposition 2.3 and (2.9). By the explicit formula for the Eichler–Shimura map, we deduce that
This shows that
Remark.
The theorem could be stated in more general form so that the real-analytic analogues of both Eisenstein and Poincaré series are captured. That would have the advantage of accounting for the full space
The family
The family addresses Question 2, inasmuch as it satisfies the following three properties:
Firstly, by Theorem 3.8, the
Secondly, this family is “canonical” in the sense that it induces a generating set for
Thirdly, it is possible to obtain, by a simple process, explicit
This process is given in the following proposition, which also formalizes a link between the two main themes of this note, namely second-order modular forms and iterated integrals.
Proposition 3.9.
Let
Proof.
For each
By Theorem 2.6, the coefficients
defines the sought map. ∎
3.2.1 A classification of holomorphic second-order forms
We close with another implication of Corollary 3.4 to holomorphic extended second-order modular forms. The resulting theorem is the analogue of [6, Theorem 2.3] to the class of extended second-order modular forms.
Let ρ be a complex representation of
For an even
Proposition 3.10.
Suppose that
is exact. In particular,
Proof.
We apply Corollary 3.4 to
The isomorphism
By a direct computation, we see that the invariance of g under S and T is equivalent to the invariance of the associated vector under S and T in terms of the representation ρ.
To prove the surjectivity, let
Assume that
(here
Since
We then have, for all
By the definition of the map
To deduce the dimension formula, we use [5, Theorem 6.3 and Remark 6.4] to compute the dimension of
where
satisfies the recurrence relation
We can also see, by induction, that, for
Since the only eigenvalue of
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/S032460/1
Funding statement: Research on this work was supported in part by EPSRC grant EP/S032460/1.
Acknowledgements
The author is grateful to F. Brown, L. Candelori, C. Franc, G. Mason and F. Strömberg for many helpful comments and suggestions. He also thanks the anonymous referee for a very careful reading of the manuscript and many useful suggestions.
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Articles in the same Issue
- Frontmatter
- Contramodules over pro-perfect topological rings
- Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations
- Resistance scaling on 4N-carpets
- Newton polygons for L-functions of generalized Kloosterman sums
- Embeddings of locally compact abelian p-groups in Hawaiian groups
- A theorem of Roe and Strichartz on homogeneous trees
- Hankel determinants for starlike and convex functions associated with sigmoid functions
- Modular iterated integrals associated with cusp forms
- Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type
- Gompf connected sum for orbifolds and K-contact Smale–Barden manifolds
- Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces
