Exceptional Groups and Their Modular Forms Aaron Pollack 1. Introduction Then Lagrange’s theorem says that 푟4(푛) ≥ 1 for every pos- It is an old theorem of Lagrange that every nonnegative itive integer 푛. In fact, there is a beautiful formula due to integer can be expressed as the sum of four squares of Jacobi for 푟4(푛). Denote by 휎(푛) = ∑푑|푛 푑 the sum of the integers. That is, if 푛 ∈ 퐙≥0, then there exist integers divisors of the integer 푛. Then 푟4(푛) = 8휎(푛) if 푛 is odd and 2 2 2 2 푥1, 푥2, 푥3, 푥4 so that 푛 = 푥1 + 푥2 + 푥3 + 푥4. For the sake 푟4(푛) = 24휎(푛) if 푛 is even. of comparison, note that 7 is not the sum of three squares, Where does such a nice formula come from? The con- 2 2 2 so not every integer can be expressed as 푥1 + 푥2 + 푥3. temporary explanation is that this formula comes from If 푛 ∈ 퐙≥0 and 푚 > 0 is a positive integer, denote by the theory of modular forms. Consider the power series 푛 푟푚(푛) the number of ways one can express 푛 as the sum of ∑푛≥0 푟4(푛)푞 ∈ 퐙[[푞]] in the variable 푞. If 푧 is in the com- 푚 squares of integers. That is, define 푟푚(푛) to be the size plex upper half-plane 픥 = {푥 + 푖푦 ∶ 푥, 푦 ∈ 퐑, 푦 > 0} and 2휋푖푧 푛 of the set 푞 = 푒 , then 휃4(푧) ≔ ∑푛≥0 푟4(푛)푞 becomes a holomor- {(푥 , 푥 , … , 푥 ) ∈ 퐙푚 ∶ 푥2 + 푥2 + ⋯ + 푥2 = 푛}. phic function on 픥. The function 휃4(푧) is an example of a 1 2 푚 1 2 푚 modular form (defined below), which amounts to the fact 1 2 Aaron Pollack is an assistant professor at the University of California, San Diego. that 휃4(푧 + 1) = 휃4(푧) and 휃4 (− ) = (4푧) 휃4(푧) as func- 4푧 His email address is [email protected]. tions on 픥. The first functional equation is obvious but The author was supported by the Simons Foundation via Collaboration Grant the second is not. These symmetries go a long way to prov- number 585147. ing the relationship between 푟4(푛) and the divisor function Communicated by Notices Associate Editor Steven Sam. 휎(푛). For permission to reprint this article, please contact: 1 Note that 푧 ↦ 푧 + 1 and 푧 ↦ − are the linear frac- [email protected]. 4푧 tional transformations of 픥 associated to the matrices ( 1 1 ) DOI: https://doi.org/10.1090/noti2226 0 1 194 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 2 0 −1 and ( 4 0 ). There is a finite index subgroup Γ of SL2(퐙) 2.1. Examples. The special function 휃4(푧) described in so that 휃4(푧) satisfies a functional equation associated to the introduction is a modular form of weight 2. More gen- 푎 푏 푚 ≥ 0 every element 훾 ∈ Γ. Namely, if 훾 = ( 푐 푑 ) ∈ Γ, then erally, if is a nonnegative integer, then one can de- 푛 푎푧+푏 2 fine 휃푚(푧) = ∑ 푟푚(푛)푞 . When 푚 is even, 휃푚(푧) is a 휃4 ( ) = (푐푧 + 푑) 휃4(푧). In this way, one can think of 푛≥0 푐푧+푑 modular form of weight 푚/2. 휃 (푧) as a special function associated to the group SL . The 4 2 A simpler set of examples of modular forms are the so- reader can see [Zag08] and [Ser73] for an introduction to called Eisenstein series 퐸푘(푧). Suppose Γ ⊆ SL2(퐙) is fixed, modular forms, and in particular [Zag08, Section 3] for a 1 푛 and denote Γ∞ = {( 0 1 ) ∈ Γ}. If 푘 ≥ 4, one defines proof of Jacobi’s formula via 휃4(푧). −푘 The modular form 휃4(푧) is a special example of what 퐸푘(푧; Γ) = ∑ (푐푧 + 푑) . (1) is called an automorphic form. Automorphic forms are 훾∈Γ∞\Γ functions that have large groups of discrete symmetries 푎 푏 In the sum above, 훾 = ( 푐 푑 ) and the condition 푘 ≥ 4 en- and satisfy particular types of differential equations. (The sures that the sum converges absolutely to a holomorphic function 휃4(푧) satisfies the Cauchy-Riemann equations.) ∗ ∗ function on 픥. Note that if 훾∞ ∈ Γ∞, then 훾∞훾 = ( 푐 푑 ), so This article is about automorphic forms, with a special em- that the sum (1) is well-defined. phasis on those automorphic forms whose groups of sym- These functions have simple Fourier expansions. For ex- metries are connected to the exceptional Lie groups. We ample, if Γ = SL2(퐙), then 퐸푘(푧) ≔ 퐸푘(푧; Γ) has Fourier do not suppose the reader knows anything about modu- expansion lar forms or exceptional groups, only the representation theory of compact groups and a little algebraic geometry. 푘−1 푛 퐸푘(푧) = 1 + 훼푘 ∑ (∑ 푑 ) 푞 2. Modular Forms 푛≥1 푑|푛 We now dig into the definition of modular forms. Sup- for a nonzero rational number 훼푘. By relating 휃4(푧) with 2 퐸 (푧; Γ) pose that 푘 > 0 is a positive integer and Γ ⊆ SL2(퐙) is a a certain weight- Eisenstein series 2 , one can prove finite index subgroup. We assume that there exists apos- Jacobi’s formula for 푟4(푛). itive integer 푁 so that Γ contains the subgroup Γ(푁) of The examples above all have the feature that the con- 푎 (0) ≠ 0 푦푘/2|푓(푧)| SL2(퐙) consisting of matrices congruent to 1 modulo 푁, stant term 푓 , so that the function grows 푎 푏 as 푦 → ∞. The subspace of modular forms that decay as i.e., Γ(푁) = {( 푐 푑 ) ∈ SL2(퐙) ∶ 푎, 푑 ≡ 1 (mod 푁), 푏, 푐 ≡ 0 (mod 푁)}. A modular form of weight 푘 for Γ is a holomor- 푦 → ∞ occupies a special place in the theory. More pre- phic function 푓 ∶ 픥 → 퐂 that is semi-invariant under Γ and cisely, the noncompact complex curve 푌 Γ can be compact- doesn’t grow too quickly. More precisely, a holomorphic ified to a curve 푋Γ. The set 푋Γ ⧵ 푌 Γ is finite, and is called function 푓 ∶ 픥 → 퐂 is a modular form of weight 푘 for Γ if the cusps of the modular curve 푋Γ. The modular forms 푘/2 푓 of weight 푘 for which the function 푦 |푓(푧)| on 푌 Γ de- 푘 푎 푏 1. 푓(훾푧) = (푐푧 + 푑) 푓(푧) for all 훾 = ( 푐 푑 ) ∈ Γ, cays towards the cusps are called cusp forms, denoted by 푘/2 2. the function 푦 |푓(푧)| grows at most polynomially 푆푘(Γ) ⊆ 푀푘(Γ). Ramanujan defined the following beauti- with 푦 on Γ\픥. (Note that 푦푘/2|푓(푧)| is Γ-invariant, as ful cusp form of weight 푘 = 12: −2 퐼푚(훾푧) = 퐼푚(푧)|푐푧 + 푑| .) Δ(푧) ≔ 푞 ∏ (1 − 푞푛)24 = ∑ 휏(푛)푞푛. 푛≥1 푛≥1 We denote by 푀 (Γ) the space of modular forms of weight 푘 The numbers 휏(푛) are by definition the complex numbers 푘 for Γ. For 푘 and Γ fixed, the space 푀 (Γ) is a finite- 푘 that make this an equality. It is not obvious that Δ(푧) is dimensional (sometimes 0) complex vector space. a modular form. Another fact that is not clear from the Note that if 푓 ∶ 픥 → 퐂 is a holomorphic function, then definition is that 휏(푚푛) = 휏(푚)휏(푛) if the positive integers 푓(푧) 푑푧 is Γ-invariant if and only if 푓(훾푧) = (푐푧 + 푑)2푓(푧), 푚 and 푛 are relatively prime. To give the reader a sense for so weight-2 modular forms are in bijection with a subspace a simple statement which is already not known, we remark of the holomorphic differential-one forms on 푌 ≔ Γ\픥. Γ that it is conjectured that 휏(푛) ≠ 0 for all 푛. One of the first things that must be said about modular forms is that they have a Fourier expansion: suppose for 2.2. The group SL2. As suggested in the introduction, 1 푛 modular forms should be thought of as functions associ- simplicity that Γ = SL2(퐙). Then the elements ( 0 1 ) ∈ Γ for 푛 ∈ 퐙 and thus the condition 푓(훾푧) = (푐푧 + 푑)푘푓(푧) ated to the group SL2. To make this precise, recall that the becomes 푓(푧 + 푛) = 푓(푧). As 푓 is holomorphic, this im- action of SL2(퐑) on 픥 via linear fractional transformations 2휋푖푛푧 is transitive, and that the stabilizer in SL2(퐑) of the point plies that 푓(푧) = ∑푛∈퐙 푎푓(푛)푒 for complex numbers √−1 ∈ 픥 is the special orthogonal group of size two: 푎푓(푛) ∈ 퐂. As the reader can immediately check, condi- tion 2 above implies that 푎푓(푛) = 0 if 푛 is negative. That cos(휃) sin(휃) 2휋푖푛푧 SO(2) = {푘휃 ≔ ( ) ∶ 휃 ∈ 퐑} . (2) is, 푓(푧) = ∑푛≥0 푎푓(푛)푒 . − sin(휃) cos(휃) FEBRUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 195 That is, one has an SL2(퐑)-equivariant identification 픥 ≃ • GL푛(퐑), SL2(퐑)/ SO(2). • SL푛(퐑), 푎 푏 푡 For 푔 = ( 푐 푑 ) ∈ SL2(퐑) and 푧 ∈ 픥, set 푗(푔, 푧) = • SO(푝, 푞) = {푔 ∈ SL푝+푞(퐑) ∶ 푔퐼푝,푞푔 = 퐼푝,푞}, the 푐푧 + 푑. The reader can easily check that for 푔1, 푔2 ∈ SL2(퐑), special orthogonal group of a vector space with a 푗(푔1푔2, 푧) = 푗(푔1, 푔2푧)푗(푔2, 푧). Now if 푓 ∈ 푀푘(Γ) is a non-degenerate quadratic form of signature (푝, 푞), 푘 푡 weight- modular form, then one can define a closely re- • 푈(푝, 푞) = {푔 ∈ GL푝+푞(퐂) ∶ 푔퐼푝,푞푔 = 퐼푝,푞}, the sim- lated function 휙푓 ∶ Γ\ SL2(퐑) → 퐂.