Singular modular forms on quaternionic E8
Aaron Pollack

January 2021

Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,3

4

Modular forms on exceptional groups

5Singular modular forms on E8,46

Proof of Theorem

7

Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

GoalThis talk is about: The construction of two very niceautomorphic forms on quaternionic E8E8,4 : real reductive group of type E8 with split rank four; thisis quaternionic E8
The symmetric space E8,4 /K does not have Hermitian
structure, but still possesses automorphic forms that behave
similarly to classical holomorphic modular forms
Similarly: They have a ‘robust’ Fourier expansion; called’modular’ forms
There are two modular forms on E8,4 that can write down
explicitly
Theorem: These modular forms have all Fourier coefficients
in QTime permitting: An application to a very interesting
automorphic form on E6,4
Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,3

4

Modular forms on exceptional groups

5Singular modular forms on E8,46

Proof of Theorem

7Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

Siegel modular forms
The symplectic groupSp2n = {g ∈ GL(2n) : t g −1n 1n g = −1n 1n }
 a b
Sp2n ⊇ U(n) ' −ba : a + ib ∈ U(n)The symmetric space
Sn := n × n symmetric matrices
Hn = {Z = X + iY : X , Y ∈ Sn (R), Y > 0} the Siegel upper
half-space
Hn ' Sp2n (R)/U(n) the symmetric space
Sp2n (R) acts on Sp2n (R)/U(n) = Hn via
g · Z = (aZ + b)(cZ + d)−1if g =a bc d

in n × n block form.

Aaron Pollack

Singular modular forms on quaternionic E8

Siegel modular forms: continued
Siegel modular form of weight ` > 0:
Definition and basic propertiesf : Hn → C holomorphic such thatf ((aZ + b)(cZ + d)−1 ) = det(cZ + d)` f (Z ) for all
some congruence subgroup of Sp2n (Z)Fourier expansion:
Xf (Z ) =af (T )e 2πi tr(TZ )a bc dT ∈Sn (Q),T ≥0with af (T ) ∈ C and T ≥ 0 means “T is positivesemi-definite”.
If n = 1, these are classical modular forms for SL2
If f a Siegel modular form, can consider f ∈ H 0 (Γ\Hn , L` )a global section of a holomorphic line bundle L` on Γ\HnAaron Pollack

Singular modular forms on quaternionic E8∈ΓSiegel modular forms automorphicallyϕ : Sp2n (Q)\ Sp2n (A) → C withThe definitiondet
1 ϕ(gk) = z(k)−` ϕ(g ) for all k ∈ U(n), z : U(n) → U(1) ⊆ C×2 DCR,` ϕ ≡ 0: ϕ annihilated by linear differential operator DCR,`
so that fϕ on Hn satisfies the Cauchy-Riemann equations
The Fourier expansion  Y 1/21X
ϕf= ϕf (n(X )m)1Y −1/2X
aϕ (T )e 2πi tr(TX ) e −2π tr(TY )=T ∈Sn (Q),T ≥0where iY = m · i in Hn and aϕ (T ) ∈ C.
Automorphicallyπ = ⊗v πv with π∞ a holomorphic discrete series
representation
Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,34Modular forms on exceptional groups

5

Singular modular forms on E8,4

6

Proof of Theorem

7

Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

A very nice exceptional groupE7,3 : has a symmetric space with Hermitian tube structure
Θ: octonions with positive-definite norm form. This is an
8-dimensional, non-associative R-algebra that comes equippedwith a quadratic form Θ → R and an R-linear conjugation∗ : Θ → Θ.
J = H3 (Θ): Hermitian 3 × 3 matrices with elements in Θ.
 c1 x3 x2∗

J =  x3∗ c2 x1  : ci ∈ R, xj ∈ Θ .

x2 x1∗ c3E7,3 acts on
HJ = {Z = X + iY : X , Y ∈ J, Y > 0}
by “fractional linear” transformations.
Aaron Pollack

Singular modular forms on quaternionic E8

Holomorphic modular forms on E7,3

For an integer ` > 0, f : HJ → C is a holomorphic modular formof weight ` if
f is holomorphic, moderate growthf (γZ ) = j(γ, Z )` f (Z ) for all γ ∈ Γ ⊆ E7,3 a congruencesubgroup
These holomorphic modular forms on E7,3 have a Fourierexpansion:
Xf (Z ) =af (T )e 2πi tr(TZ )T ∈JQ ,T ≥0with the af (T ) ∈ C.

Aaron Pollack

Singular modular forms on quaternionic E8

Kim’s modular forms on E7,3Rank
Note that J ⊇ S3 the symmetric 3 × 3 matrices. There is afunction rank : J → {0, 1, 2, 3} extending the rank of symmetricmatrices on S3 .
Theorem 1 (H. Kim)
There exists holomorphic modular forms ΘKim,4 and ΘKim,8 forE7,3 with the following properties:
1 ΘKim,4 is a weight 4, level 1 modular form with Fourier
coefficients in Z. Moreover, the Fourier coefficients aΘKim,4 (T )are 0 unless rank(T ) ∈ {0, 1}.
2 ΘKim,8 is a weight 8, level 1 modular form with Fourier
coefficients in Z. Moreover, the Fourier coefficients aΘKim,8 (T )are 0 unless rank(T ) ∈ {0, 1, 2}.
The modular forms ΘKim,4 , ΘKim,8 are said to be singular.
Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2

Siegel modular forms

3The exceptional group E7,3

4Modular forms on exceptional groups

5Singular modular forms on E8,46

Proof of Theorem

7

Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

Exceptional groups have ‘modular forms’
The groups
G : G2 ⊆ D4 ⊆ F4 ⊆ E6,4 ⊆ E7,4 ⊆ E8,4
K ⊆ G the maximal compact. K  SU(2)/µ2 .
G /K : no Hermitian structure
Definition of modular forms on GLet ` ≥ 1 be an integer. A modular form on G of weight ` isan automorphic form ϕ : Γ\G → Sym2` (C2 )satisfying ϕ(gk) = k −1 · ϕ(g ) for all g ∈ G , k ∈ Kand D` ϕ = 0 for a certain special linear differential operator
D`
Definition due to Gross-Wallach, Gan-Gross-Savin

Aaron Pollack

Singular modular forms on quaternionic E8

These modular forms have nice properties
Theorem 2The modular forms of weight ` ≥ 1 on G have a robust Fourier
expansion, normalized over the integers, that is compatible withpullbacks between groups G above.
The theorem means:
Given a modular ϕ form of weight `, one can ask the question“Are all of ϕ’s Fourier coefficients in some ring R ⊆ C?”
If ι : G1 ⊆ G2 in the above sequence of groups, and if ϕ ismodular form on G2 of weight `, then the pullback ι∗ (ϕ) on
G1 is a modular form of weight `.
Moreover, the Fourier coefficients of ι∗ ϕ are finite sums ofthe Fourier coefficients of ϕAaron Pollack

Singular modular forms on quaternionic E8

More precise Fourier expansion
HaveG ⊇ P = MN ⊇ [N, N] := Z
the Heisenberg parabolic. SetZ
ϕZ (g ) =ϕ(zg ) dz.

Z (Q)\Z (A)Can Fourier expand ϕZ along N/[N, N], and via Theorem 2 thisFourier expansion isFourier expansion
ϕZ (nmf m∞ ) =X

χ(n)cχ,ϕ (mf )Wχ (m∞ )χ∈(N/Z )∨for certain completely explicit functions Wχ : M(R) → Sym2` (C2 ).
The point is that the functions Wχ are independent of ϕThe functions Wχ are 0 unless χ satisfies a certain positivity
condition
Aaron Pollack

Singular modular forms on quaternionic E8

Fourier coefficients

Fourier coefficients
If R ⊆ C is a subring, one says that ϕ has Fourier coefficients in R
if all the functions cχ,ϕ : M(Af ) → C are in fact valued in R.
If χ is non-degenerate in a certain sense, these Fourier
coefficients were defined by Gan-Gross-Savin, using a
multiplicity one result of Wallach.
Motivating question
Fix G and ` ≥ 1. Does there exist a basis of the modular forms onG of weight `, all of whose Fourier coefficients are in Q?

Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,3

4

Modular forms on exceptional groups

5Singular modular forms on E8,4

6

Proof of Theorem

7

Application of ΘminAaron Pollack

Singular modular forms on quaternionic E8

Let P = MN ⊆ E8,4 be the Heisenberg parabolic subgroup,
M = GE7,3 .
Theorem 3 (Gan,P,Savin)
There exists square integrable automorphic forms Θmin and Θntm
on E8,4 with the following properties.
1 Θ
min is a weight 4 modular form with all Fourier coefficientsin Z. Its constant term along N, Θmin,N is essentially ΘKim,4 .
2 Θntm is a weight 8 modular form with all Fourier coefficientsin Q. Its constant term along N, Θntm,N is essentially ΘKim,8 .
These modular forms are singular in the sense that many of their
Fourier coefficients are 0.
The Fourier coefficients are parametrized by elements in a lattice inW = (N/[N, N])∨ . There is a function rank : W → {0, 1, 2, 3, 4}.
The Fourier coefficients aΘmin (w ) of Θmin are 0 unlessrank(w ) ∈ {0, 1}
The Fourier coefficients aΘntm (w ) of Θntm are 0 unlessrank(w ) ∈ {0, 1, 2}
Aaron Pollack

Singular modular forms on quaternionic E8

Remarks
12

Gross-Wallach constructed unitary representations π4 and π8of the real group E8,4 that are small in the sense of GKdimension. The automorphic forms Θmin , Θntm should be1
thought of as globalizations of these representations.
On split E8 there are analogues of Θmin and Θntm . These arecompletely spherical automorphic forms
constructed by Ginzburg-Rallis-Soudry, in the case of theminimal;
constructed by Green-Miller-Vanhove, Ciubotaru-Trapa in thecase of next-to-minimal;
next-to-minimal recently studied byGourevitch-Gustafsson-Kleinschmidt-Persson-Sahi.

3

Gan constructed Θmin as a special value of an Eisenstein seriesEassociated to IndP 8,4 (δPsmin ), proved it’s square integrable.

1Proved by Gan-Savin for Θmin and π4 . Should be true but not proved forΘntm and π8 .
Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,3

4

Modular forms on exceptional groups

5

Singular modular forms on E8,4

6

Proof of Theorem

7

Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

Heisenberg Eisenstein series
Suppose G = E8,4 , P Heisenberg parabolic.
ν : P → GL1
generating the character group of P. On G = E8,4 ,|ν(p)|29 = δP (p)for p ∈ P. Suppose` ≥ 1 even
G (A)f (g , `; s) ∈ IndP(A) (|ν|s ), certain Sym2` (V2 )-valued section.
PE (g , `; s) = γ∈P(Q)\G (Q) f (γg , `; s) absolutely convergent
for Re(s) > 29.
If s = ` + 1 in range of absolute convergence, E (g , s = ` + 1)
a modular form of weight ` for G
Question
Does E (g , s = ` + 1) have rational Fourier coefficients?
Aaron Pollack

Singular modular forms on quaternionic E8

Next to minimal
Motivated by work of Gross-Wallach on continuation ofquaternionic discrete series, take ` = 8 and G = E8,4 .
Proposition
The Eisenstein series E (g , ` = 8; s) is regular at s = 9 (eventhough outside the range of absolute convergence), and defines
square integrable weight 8 modular form at this point.
Setθntm (g ) = E (g , ` = 8; s = 9)
Theorem 4 (Savin)
The spherical constituent of the degenerate principal series
G (Q )
IndP(Qpp) (|ν|9 ) is “small”, i.e., many twisted Jacquet modules are
0. Consequently, the rank three and rank four Fourier coefficientsof θntm are 0.
Aaron Pollack

Singular modular forms on quaternionic E8

More on next-to-minimal modular form

Theorem 5
The weight 8 modular form θntm has rational Fourier coefficients.
Proof.
1 Savin’s result gives vanishing of rank three and four Fourier
coefficients
2 Explicit computation (outside range of abs. convergence)
gives rationality of rank 1 and rank 2 Fourier coefficients
3 Constant term analyzed using work of H. Kim on weight 8
singular modular form on GE7,3

Aaron Pollack

Singular modular forms on quaternionic E8

Explicit computation of θntm
1Define special Sym2` (V2 )-valued Eisenstein series E` (g ) on
SO(3, 4k + 3)

2

Prove that the constant term θntm from E8,4 down toSO(3, 11) is E8 (g )3Theorem: the E` (g ) have rational Fourier coefficients (in aprecise sense)

4

The Fourier coefficients of E8 (g ) can be identified with rank 1
and rank 2 Fourier coefficients of θntm .

To prove the E` (g ) have rational Fourier coefficients:
Jacquet integral
Explicit computation of certain Archimedean Jacquet integralZe 2πi(v ,x) f` (wn(x)) dx.
V2,4k+2 (R)Aaron Pollack

Singular modular forms on quaternionic E8

Contents
1Introduction

2Siegel modular forms

3The exceptional group E7,3

4

Modular forms on exceptional groups

5

Singular modular forms on E8,46

Proof of Theorem

7

Application of Θmin

Aaron Pollack

Singular modular forms on quaternionic E8

A distinguished modular form
Let G be one of the quaternionic exceptional groups, P = MN its
Heisenberg parabolic, W (R) = (N/[N, N])∨ . Globally, there is anarithmetic invariant on the orbits of M(Q) on W (Q):q : W (Q)rk=4 → Q× /(Q× )2 = { quadratic etale extensions of Q}.
Fact: If F a modular form on G , ω ∈ W (Q) and q(ω) > 0 thenaF (ω) = 0. In other words, only ω corresponding to imaginary
quadratic fields can have associated nonzero Fourier coefficientsFix an imaginary quadratic extension E /Q. Associated to E , there
is a group GE over Q of type E6,4 .
Theorem 6
There is a weight 4 modular form θE on GE with Fourier
coefficients in Z such that θE has nonzero Fourier coefficients of
all ranks and1 If ω ∈ W (Q)rk=4 and q(ω) ∈ Q× /(Q× )2 does not represent
E , then the Fourier coefficient aθE (ω) = 0Aaron Pollack

Singular modular forms on quaternionic E8

Proof of Theorem 6

Proof of Theorem 6:
1Define GE , which is simply-connected of type E6,4

2

Carefully embed GE in E8,4 via ιE : GE → E8,4

3Define θE = ι∗E (θmin ), the pull-back of the modular form
generating the minimal representation on E8,4

4The Fourier coefficients of θE can then be computed fromthose of θmin5

θmin only has nonzero Fourier coefficients for the mostdegenerate ω, those ω of rank 1

6This vanishing of aθmin (ω) imposes a strong arithmetic
condition on the Fourier coefficients of θE .

Aaron Pollack

Singular modular forms on quaternionic E8

Thank you

Thank you for your attention!

Aaron Pollack

Singular modular forms on quaternionic E8