The minimal modular form on quaternionic
Abstract.
Suppose that is a simple reductive group over , with an exceptional Dynkin type, and with quaternionic (in the sense of GrossWallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form on quaternionic , and some applications. The valued automorphic function is a weight four, level one modular form on , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic and . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups .
Contents
1. Introduction
This paper is a sequel to the paper [Pol18a]. In [Pol18a], we studied “modular forms” on the quaternionic exceptional groups, following beautiful work of GanGrossSavin [GGS02] and Wallach [Wal03]. We proved that these modular forms possess a refined Fourier expansion, similar to the Siegel modular forms on the symplectic groups . This is, in a sense, a purely Archimedean result: The representation theory at the infinite place on the quaternionic exceptional groups forces the modular forms to have a robust theory of the Fourier expansion.
Suppose that is a quaternionic exceptional group of adjoint type. The maximal compact subgroup is for a certain group . Denote by the representation of that is the symmetric power of the defining representation of and the trivial representation of . Recall from [Pol18a] that if is an integer, a modular form on of weight is a smooth, moderate growth function that satisfies

for all and

.
Here is a certain firstorder differential operator, closelyrelated to the socalled Schmid operator for the quaternionic discrete series representations on . It is that is annihilated by that is the crucial piece of the definition of modular forms.
There is a weight four, level one modular form on quaternionic that is associated to the automorphic minimal representation, studied by Gan [Gan00a, Gan00b], which we denote by . The automorphic minimal representation is spherical at every finite place, but is not spherical at infinity; at the Archimedean place, it has minimal type . If , then pairing with gives the vector in the minimal representation that is at the Archimedean place and spherical at all the finite places. Our main result is the complete and explicit Fourier expansion of this modular form. See Theorem 1.0.1. Using , we construct special modular forms on and ; see Corollary 1.0.2 and Corollary 1.0.3. Moreover, we study a family of absolutely convergent Eisenstein series on the quaternionic exceptional groups, and prove that all their nontrivial Fourier coefficients are Euler products; see Theorem 3.2.5.
To set up the statements of these results, let us recall from [Pol18a] the shape of the Fourier expansion of modular forms on the quaternionic exceptional groups. Thus suppose is a quaternionic exceptional group of adjoint type, associated to a cubic norm structure over with positive definite trace form. Then has a rational Heisenberg parabolic , with Levi subgroup and unipotent radical . The group is a twostep unipotent group, with center and abelianization . Here is Freudenthal’s defining representation of the group (see [Pol18a] and [Pol18b]).
Suppose is a modular form of weight for . Denote by the constant term of along , i.e.,
A simple argument using the leftinvariance of under proves that determines for the groups studied in [Pol18a]. The Fourier expansion of is then given as follows: For and ,
where here the notation is as follows:

denotes the constant term of along ;

is the Fourier coefficient associated to ;

is Freudenthal’s symplectic form on ;

is a special function on defined in terms of the Bessel functions for and the element .
Denote by the fixed basis of from [Pol18a] so that is a basis of . The basis elements of are essentially characterized by the fact that for a certain compact subgroup of and the factor of automorphy on specified in loc cit; see [Pol18a, section 9]. Then
with
and . Here is the similitude character of . Moreover, the constant term
for some holomorphic modular form of weight on and for a specific element that exchanges the upper and lower halfspaces .
With this result recalled, let us now state the Fourier expansion of . Denote by Coxeter’s integral octonions [EG96, (5.1)] and the associated integral lattice in the exceptional cubic norm structure . The Freudenthal space has a natural integral lattice . For , define to be the largest positive integer so that . For , define analogously.
Recall Kim’s modular form [Kim93] on the exceptional tube domain, which has Fourier expansion
Denote by the automorphic form on so that descends to , is holomorphic on , antiholomorphic on , and on one has if .
Theorem 1.0.1.
Let the notations be as above. Then
with
There is a degenerate Heisenberg Eisenstein series on each of the quaternionic exceptional groups, which we write as . This is a function (depending on ) satisfying for all . When is quaternionic and , it turns out that the Eisenstein series is regular at , and is defined [Gan00a] (up to a nonzero scalar multiple) as the value of this Eisenstein series at this point.
Combining the archimedian results of [Pol18a] with work of Gan [Gan00a, Gan00b, Gan11], Kim [Kim93], GrossWallach [GW96], KazhdanPolishchuck [KP04] and GanSavin [GS05], most of Theorem 1.0.1 was known. See the discussion in subsection 2.2. What is left is to pin down a couple constants. We do this by analyzing the Fourier expansion of the weight Eisenstein series on , and applying the SiegelWeil theorem of [Gan00b] which relates to this Eisenstein series on .
The Eisenstein series is easier to compute with in relation to (which defines ) because is in the range of absolute convergence of but not of . In fact, we study absolutely convergent degenerate Heisenberg Eisenstein series on the quaternionic exceptional groups in general, which is our second main result. More precisely, if is even and the special point is in the range of absolute convergence, the Eisenstein series is a modular form on of weight . We prove that at such an , all of the nontrivial Fourier coefficients of are Euler products. This is the analogue to the exceptional groups of the corresponding classical fact about holomorphic Siegel Eisenstein series on symplectic groups [Sie39]; we defer a precise statement of this result to section 3. The proof is an easy consequence of a weak form of the main result of [Pol18a]: The Fourier expansion of has many terms, some of which are Euler products and some of which are not. However, applying [Pol18a], one can deduce that all of the terms that are not Euler products vanish at for purely Archimedean reasons.
We also give a few applications of Theorem 1.0.1, to modular forms on , and . Namely, one can pull back the minimal modular form on to the simplyconnected quaternionic and . Denote these pullbacks by and , respectively. These pullbacks give interesting singular and distinguished modular forms on and . The modular form is singular in that it has no rank three or rank four Fourier coefficients, but it does have nonzero rank two Fourier coefficients. The modular form is not singular–it has nonzero rank four Fourier coefficients. However, it is distinguished in that it has only one orbit of nonzero rank four Fourier coefficients.
Corollary 1.0.2.
The automorphic functions and define nonzero modular forms on and of weight . Moreover

The modular form has nonzero rank two Fourier coefficients, but all of its rank three and rank four Fourier coefficients are ;

The modular form is distinguished: it has only one orbit of nonzero rank four Fourier coefficients.
The distinguished nature of these Fourier expansions is a moreorless immediate consequence of the results of [Pol18b, section 7 and 8].
Note that theorem 1.0.1 says that (a scalar multiple of) just fails to have integral Fourier coefficients. All the rank one Fourier coefficients are integers, and all the Fourier coefficients of are integers. Thus, it is reasonable to ask for a nonzero modular form on an exceptional group for which all of its Fourier coefficients are integers. Our final application of Theorem 1.0.1 is to produce such a modular form on .
Following [EG96, GG99, GGS02], there is a unique (up to scaling) automorphic function on a certain anisotropic form of , that is right invariant under and orthogonal to the constant functions. Denote by the theta lift of to via . The modular form is discussed in [GGS02] and [GG99]. The following is an essentially immediate corollary of Theorem 1.0.1 and results of loc cit.
Corollary 1.0.3.
The modular form has rational Fourier coefficients with bounded denominators. Its constant term is proportional to Ramanujan’s function .
1.1. Acknowledgments
We thank Wee Teck Gan and Benedict Gross for their encouragement and helpful comments.
1.2. Notation
Throughout the paper, the notation is as in [Pol18a]. In particular, denotes a field of characteristic , denotes a cubic norm structure over , and or the Lie algebra associated to in [Pol18a, Section 4]. The field will frequently be or . We will assume that is either , or with a composition algebra over . Thus is of type or .
The letter or denotes the maximal compact subgroup of defined in loc cit, where is the adjoint group associated to the Lie algebra . We will sometime refer to elements of is the notation of the model, and other times refer to elements of this Lie algebra in the model. Again, see [Pol18a, section 4]. We write for the representation of on . Modular forms on are by definition certain functions satisfying for all and , which are annihilated by a firstorder differential operator .
One defines (or , if is fixed) to be the Heisenberg parabolic of , which by definition is the stabilizer of the line in . We write , which is also the center of . The letter denotes the similitude character of ; one has given by .
We write for the Lie algebra of the Freudenthal group and for the Lie algebra of the group that preserves the cubic norm on up to similitude. Then (see [Pol18a, section 4] for our normalizations)
For we denote by the element of specified in [Pol18a, section 3.4.2].
Finally, if and an integer, is the Pochhammer symbol.
2. Statement of results, and applications
In this section we state our main results more precisely, and give the proofs of Corollary 1.0.2 and Corollary 1.0.3. We begin by defining the degenerate Heisenberg Eisenstein series on the quaternionic groups , as these Eisenstein series are central to everything that follows. We then review what was known about the automorphic form . Finally, we restate Corollary 1.0.2 and 1.0.3 and give the proofs of these results.
2.1. The degenerate Heisenberg Eisenstein series
In this subsection, we define the degenerate Heisenberg Eisenstein series on the quaternionic exceptional groups . By definition, such an Eisenstein series is associated to a section . More precisely, we use the final parameter in to indicate that is valued and satisfies . Throughout, we will assume that is even.
We now construct such an Eisenstein series explicitly; this makes it easier to do computations. Suppose is a SchwartzBruhat function on . We will define a valued Schwartz function on satisfying momentarily. With this definition, we set and then
It is clear that is a section in the induced representation , and we set
We will be interested in this Eisenstein series at the special value . When , the Eisenstein series converges absolutely at and defines a modular form there; see the remarks after Corollary 1.2.4 in [Pol18a].
The special archimedean function is defined as follows. Denote by the part of , the Lie algebra of the maximal compact . Denote by the equivariant projection. For , define . Here , with and defined in [Pol18b, section 4]. It is clear that .
2.2. The minimal automorphic forms on quaternionic
In this subsection, we briefly discuss the automorphic miniminal representation on quaternionc . The reader should see [Gan00a] and [GS05] and the references contained therein for more details.
For this subsection, let with the octonion algebra over whose trace pairing is positive definite. Then is the quaterionic . Suppose that is a flat section, and the associated Eisenstein series. It is proved in [Gan00a] that for appropriate , has a simple pole at . Moreover, this pole can be achieved when is spherical at every finite place. The automorphic minimal representation is defined [Gan00a] to be the space of residues of the at . By e.g. [MS97] and also [GS05], the space of such automorphic forms only have rank and rank Fourier coefficients along ; for instance, this follows by the analogous local fact for one finite place.
Denote by the Eisenstein series associated to the flat section which has the following properties:

is valued in , and satisfies for all and ;

is spherical at every finite place;

.
One defines to be a certain nonzero multiple of . It is proved in [GS05] by a somewhat indirect method that is nonzero, i.e. that does have a nontrivial pole at . Below we will give a direct proof of this. More precisely, we shall use the following fact.
Proposition 2.2.1.
The Eisenstein series is regular at , and defines a modular form on of weight . Up to nonzero scalar multiples,
This proposition is essentially contained in [Gan00a], [Gan00b], [GS05], [GW96]. As it is crucial to the main results of this paper, we spell out a direct proof of it in section 4.
Now, because is a modular form on , the results of [Pol18a] imply that its Fourier expansion takes the following shape. Denote by Coxeter’s integral subring [EG96, (5.1)] of , by , and . Then for and one has
with the constant term given by
for a holomorphic weight modular form on . Because is minimal, is nonzero only for rank one. Denote by the largest positive integer so that . By [Gan11] and [KP04], we may scale so that
Moreover, from [Gan00b], is proportional to Kim’s [Kim93] level one, weight modular form on . Thus, applying the results of [GW96], [Gan00a], [Gan00b], [GS05], [KP04], [Kim93] and [Pol18a], what is left is to pin down the constants and . This is precisely what Theorem 1.0.1 does. We restate the result now.
Theorem 2.2.2.
The Eisenstein series is regular at and defines a modular form on of weight at this point. The Fourier coefficient corresponding to the rank one element is nonzero. Denote by the scalar multiple of for which this Fourier coefficient is equal to . Moreover, denote by the spherical automorphic form on so that descends to , is holomorphic on , antiholomorphic on , and on has the Fourier expansion
Then one has
2.3. The singular modular form
In this subsection, we consider the singular modular form on the simplyconnected quaternionic . This modular form is defined as follows. First, fix a quaternion algebra over , which is ramified at the archimedean place. Recall that the quaternionic Lie algebra is , in the notation of [Pol18a]. For ease of notation, we write and .
Now, fix not representing the identity coset in ; in other words, . By the CayleyDickson construction, one can form an octonion algebra out of and . See [Pol18b, section 8]. With such a , is ramified at infinity. The CayleyDickson construction induces an identification , an embedding , and then consequently an embedding .
More precisely, denote by the defining representation of , and define an identification as in [Pol18b, section 8.1.2]. From [Pol18b, Proposition 8.1.5], one gets a group (this is the group in the notation of that proposition) together with maps and where the first map induces an isomorphism of Lie algebras. Consequently, one obtains a map . As , one obtains a specific embedding .
Denote by the connected component of the identity of the subgroup of that preserves . We define a map as follows. First, define via its action on as for , and . Because the quadratic norm on is for , it is easy to see directly that this action preserves the symplectic and quartic form on . Now because , this defines . Finally, it is clear by construction that this preserves , and thus we obtain , as claimed.
Denote by the connected component of the identity of the centralizer of in .
Lemma 2.3.1.
The group is the simply connected quaternionic .
Proof.
Indeed, one verifies without difficulty that , and thus has the correct Lie algebra. Moreover, the centralizes in , and this sits in , see [Pol18b, Proposition 8.1.5]. Because is connected and centralizes , this proves that is in the center of . Thus is connected, has Lie algebra equal to , and contains in its center, so .∎
Denote by the automorphic function that is the pullback of to via the embedding . Compare [GW94] and [Lok00, Lok03]. We have the following result, which is a restatement of Corollary 1.0.2 (1).
Proposition 2.3.2.
The automorphic function is a modular form on of weight . It has nonzero rank two Fourier coefficients, but all of its rank three and rank four Fourier coefficients are .
The definitions and results of [Pol18a] were made for adjoint groups, not simply connected ones. However, it is easy to see that they carry over immediately for the simply connected . Indeed, because the that is the center of acts trivially on , and because the map of real groups is surjective [Ta00], the archimedean theory is identical for modular forms on the adjoint and modular forms on the simply connected .
Proof of Proposition 2.3.2.
To see that is a modular form, one must only check the condition . One way to do this is simply observe that since satisfies the equations of [Pol18a, Theorem 7.3.1 or Theorem 7.5.1], so too does . One can also reason directly with the definition of in terms of a basis of and , or apply results of [GW94, Lok00, Lok03].
For the analysis of the Fourier coefficients, this is a direct consequence of [Pol18b, Theorem 8.1.4]. Namely, if is nonzero, and denotes the Fourier coefficient of the modular form , then
(The sum has only finitely many nonzero terms.) Because all the numbers are nonnegative, it is clear that is nonzero. Finally, because is only nonzero for rank one, all the rest of claims of the proposition follow immediately from [Pol18b, Theorem 8.1.4]. This completes the proof.∎
2.4. The distinguished modular form
Pulling back to the semisimple simplyconnected quaternionic , we obtain a modular form . In this subsection, we discuss the automorphic form , and explain why it is distinguished.
Fix a quadratic imaginary extension of . Recall that the Lie algebra is the quaternionic Lie algebra of type . Using and some additional data, the socalled second construction of Tits produces an exceptional cubic norm structure . We will use this construction to define the map and analyze the Fourier coefficients of .
Thus, suppose , and that . In this subsection, we let denote and denote . Set . Then one can make into a cubic norm structure using and ; see, e.g. [Pol18b, section 7.1]. If is positive definite, then so is the trace pairing on , and thus the Lie algebra is of type and quaternionic at infinity. We will choose and , so that over , but other choices of should yield interesting results^{1}^{1}1At this point, we have only understood the automorphic form when , because in the computations above and below we used that was spherical at every finite place..
Now, via this construction of Tits, we obtain and then and then finally . More precisely, from [Pol18b, section 7.2] there is an identification . From [Pol18b, Proposition 7.2.2], there is a group (denoted in that proposition) that comes with maps and , and it is easy to see that the first map induces an isomorphism of Lie algebras. Consequently, as above, these constructions define an embedding .
Denote by the connected component of the identity of the subgroup of that preserves . We will construct explicitly the simplyconnected quaternionic inside , just as we did for in the previous subsection. More precisely, consider the subgroup of defined as the with and . Let this act on as for , , and . It is clear from the formulas defining the second construction of Tits [Pol18b, section 7.1] that this action preserves the norm and pairing on . It acts on as , for and . Consequently, one obtains maps , and this lands in because its action on fixes .
Denote by the connected component of the identity of the centralizer of this in .
Lemma 2.4.1.
The group is the simplyconnected quaternionic .
Proof.
Indeed, one verifies quickly that , and thus the Lie algebra of is . Moreover, the that is the center of is identified with the diagonal in . Because is connected and in , this is in . Hence is connected, has Lie algebra and contains in its center, which proves that .∎
Denote by the automorphic function that is the pullback to via the embedding . The following result proves Corollary 1.0.2 (2).
Proposition 2.4.2.
The automorphic function is a modular form on of weight . It has nonzero rank four Fourier coefficients. However, is distinguished in the following sense: if denotes the Fourier coefficient associated to and is rank four, then implies that for some with .
Again, because the quaternionic adjoint group is connected [Ta00], there is literally no difference between the archimedean theory for the simply connected quaternionic and the adjoint form. Thus, the definitions and results of [Pol18a]–which were proved in the adjoint case–apply immediately to the group .
Proof of Proposition 2.4.2.
To see that is a modular form of weight , again it suffices to check that , which may be done by, e.g., applying the results of [Pol18a, section 7]. Alternatively, one can apply results of [Lok03]. The Fourier coefficients of are controlled by [Pol18b, Theorem 7.3.1], which gives the result.
More precisely, suppose , and denote by the Fourier coefficient of associated to . As explained in [Pol18b, section 7], if and , then can be regarded as an element of , by applying the second Tits construction. Then for ,
Again, the sum is only has finitely many nonzero terms. It follows immediately from [Pol18b, Theorem 7.3.1 part (1)] that for to be nonzero and rank four, we need for some with .
To see that has nonzero rank four Fourier coefficients for our particular choice , , one may proceed as follows. Suppose with . Then set and . Then
is rank one in . As is rank , this completes the proof. ∎
2.5. The integral modular form on
Recall from above that denotes the automorphic function on that takes on just two values and is orthogonal to the constant function [GG99, GGS02]. More precisely, as mentioned above and following [EG96], [GG99], [GGS02], one has
where is the stabilzer of the element . The two double cosets we denote by and . Here
with . See e.g. [EG96]. The set has measure and the set has measure [GG99]. The function takes the value on and value on .
Denote by Ramanujan’s elliptic modular cusp form of weight , and recall that we set
the lift of to . The rank four Fourier coefficients of are discussed in [GGS02], and it is explained there that is a level one, weight modular form on . The constant term is essentially in [GG99, EG96]. Thus much of the following result is contained in [GGS02] and [GG99].
To state the Fourier expansion of , we make some notations. Suppose . As in [Gan00b], define