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Home , Arithmetic group, Automorphic form

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 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 . 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 (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 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 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 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 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 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 of compact groups and a little . 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 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 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 . 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 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 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. Namely, set ilarly defined , and −푘 • Sp (퐑) = {푔 ∈ SL (퐑) ∶ 푔퐽 푔푡 = 퐽 }, the group 휙푓(푔) = 푗(푔, 푖) 푓(푔 ⋅ 푖). 2푛 2푛 푛 푛 of automorphisms of a 2푛-dimensional real vec- Then if 훾 ∈ Γ and 푧 = 푔 ⋅ 푖, tor space preserving a nondegenerate alternating −푘 bilinear form. 휙푓(훾푔) = 푗(훾푔, 푖) 푓(훾 ⋅ 푧) −푘 −푘 푘 It is good to compare and contrast reductive groups = 푗(훾, 푧) 푗(푔, 푖) 푗(훾, 푧) 푓(푧) = 휙푓(푔) with compact Lie groups. Recall that the compact finite- so that 휙푓 is left-Γ-invariant. Similarly, one can easily verify dimensional Lie groups are classified, and many come 푖푘휃 that if 푘휃 ∈ SO(2) as in (2), then 휙푓(푔푘휃) = 푒 휙푓(푔) for in infinite families such as the special orthogonal group all 푔 ∈ SL2(퐑). SO(푛) and the unitary group 푈(푛). If 퐾 is a compact Lie To summarize, set 퐺 = SL2(퐑) and fix Γ ⊆ 퐺 and 푘 group, then an integer. Then one can consider the space 푀푘,∞(Γ) of • any finite-dimensional representation of 퐾 can be smooth functions 휑 ∶ 퐺 → 퐂 defined as follows. written as a finite sum of irreducible representa- tions, and Definition 1. A smooth function 휑 ∶ 퐺 → 퐂 is in 푀푘,∞(Γ) if • every irreducible (unitary) representation of 퐾 is finite dimensional. 1. 휑(훾푔) = 휑(푔) for all 푔 ∈ 퐺 and 훾 ∈ Γ, 푖푘휃 A reductive is a group 퐺 that possesses this first 2. 휑(푔푘휃) = 푒 휑(푔) for all 푔 ∈ 퐺 and 푘휃 ∈ SO(2), 3. 휑 ∈ 퐶∞(Γ\퐺) has “moderate growth” (condition 2 property, but almost never possesses the second: if 퐺 ≠ {1} above) in an appropriate sense. is a noncompact reductive Lie group, there are no faithful finite-dimensional irreducible unitary representations1 of We have just defined an injection 푀푘(Γ) ↪ 푀푘,∞(Γ). 퐺. All compact Lie groups are reductive, but automor- The space 푀푘(Γ) can then be picked out inside 푀푘,∞(Γ) by phic forms are generally only interesting to consider for finding the functions that (essentially) satisfy the Cauchy- noncompact groups 퐺. The reader should keep in mind Riemann equations. 퐺 = SL푛(퐑) and 퐺 = SO(푝, 푞) with 푝푞 ≠ 0 as examples of Put in this context, one can immediately generalize: reductive groups. what are the elements of 푀푘,∞(Γ) (for some 푘) that do not 3.2. Spaces of functions. For reductive groups 퐺, most come from holomorphic functions on 픥? How about anal- of the discrete Γ퐺 that arise in the theory of ogously defined spaces for groups other than SL2(퐑)? Both automorphic forms are what are known as arithmetic sub- of these generalizations are important. The relevant space groups, and they have the property that Γ퐺\퐺 is some- of functions are called “automorphic forms.” We will con- times (but not usually) compact, but always has finite 퐺- centrate our exposition on the latter question of generaliz- invariant volume. Without giving a precise definition, one ing from SL2 to other groups. should think of Γ퐺 as being defined in the same way as 퐺 3. Automorphic Forms except with the real numbers 퐑 replaced by the integers 퐙. For example, if 퐺 = SL (퐑), then Γ could be SL (퐙) So that the reader can get some sense of the broader pic- 푛 퐺 푛 (= SL (퐑) ∩ 푀 (퐙)) or one of its finite index subgroups. ture, let us briefly say a bit about automorphic forms more 푛 푛 With 퐺 and Γ fixed, one can consider various spaces of generally. As mentioned above, one arrives at the no- 퐺 functions on the manifold Γ \퐺, for example 퐿2,∞(Γ \퐺), tion of automorphic forms by appropriately replacing the 퐺 퐺 the space of smooth, complex-valued 퐿2-functions on pair (SL (퐑), Γ) by other pairs (퐺, Γ ) in the definition of 2 퐺 Γ \퐺. One also considers the space 퐿푚푔,∞(Γ \퐺), the 푀 (Γ) above, where 퐺 is a reductive Lie group (reviewed 퐺 퐺 푘,∞ smooth, moderate-growth functions on 퐺. The group 퐺 momentarily) and Γ is an appropriate discrete subgroup acts on 퐿2,∞(Γ \퐺) via right translation: (푔 ⋅ 휑)(푥) = 휑(푥푔), of 퐺. 퐺 and this action affords an infinite-dimensional unitary rep- 3.1. Reductive groups. At first pass, reductive groups are resentation of 퐺. perhaps best understood by example, as opposed to by def- 1푝 1푛 inition. Set 퐼푝,푞 = ( ) and 퐽푛 = ( ). Some exam- 1 −1푞 −1푛 If 퐺 = 퐺1 × 퐺2 with 퐺1 compact, then 퐺 has nonfaithful irreducible unitary ples of reductive Lie groups 퐺 are: representations via projection onto 퐺1.

196 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 2 The automorphic forms 풜(퐺; Γ퐺) are, by definition, a simple poles at 푠 = 1 and 푠 = 0, is analytic elsewhere in 푚푔,∞ certain dense, nice subspace of 퐿 (Γ퐺\퐺). To pick out the complex plane, and satisfies the exact functional equa- this subspace, one imposes two extra conditions on func- tion 휉(푠) = 휉(1 − 푠). 푚푔,∞ tions 휑 ∈ 퐿 (Γ퐺\퐺). One condition is the analogue of More generally, if {푎푛}푛≥1 is a sequence of complex num- −푠 condition 2 in Definition 1 above, and the other says that 휑 bers, and if the sum ∑푛≥1 푎푛푛 converges absolutely for satisfies sufficiently many differential equations that come 푅푒(푠) > 푠0, one can then consider this function of 푠; it is from the of 퐺. called a Dirichlet series. An 퐿-function is a very special For the sake of completeness, we now spell out these type of Dirichlet series, and they are assigned to all sorts conditions, although the reader might wish to skip to sec- of “number-theoretic data,” including modular forms and tion 4. To make explicit the first condition, recall that every automorphic forms. Besides being representable as a sum 2 −푠 reductive Lie group 퐺 has a maximal compact subgroup ∑푛≥1 푎푛푛 , 퐿-functions share two other properties with 퐾, and any two maximal compact subgroups of 퐺 are con- the : jugate. Fixing one such 퐾, we can restrict the 퐺-action to 퐾 1. They can be representable as a product over primes 푝, 푚푔,∞ and find the functions 휑 ∈ 퐿 (Γ퐺\퐺) whose translates called an “Euler product.” by 퐾 make up a finite-dimensional vector space. These 2. They satisfy a functional equation relating 푠 to 푘 − 푠 functions are called 퐾-finite. This is the analogue of con- for some real number 푘. dition 2 in Definition 1 above. The Euler product comes from a multiplicativity prop- The second condition involves the universal enveloping erty of the numbers 푎 . Specifically, suppose that 푎 = 1 algebra 풰(픤 ) of the complexified Lie algebra 픤 of 퐺. This 푛 1 퐂 퐂 and 푎 = 푎 푎 when 푛 and 푚 are relatively prime. Then, infinite-dimensional noncommutative 퐂-algebra acts on 푛푚 푛 푚 ∞ in the range of absolute convergence, one has 퐶 (Γ퐺\퐺) by differentiating the right-translation action of −푠 −푠 −2푠 퐺. The center 풵(픤퐂) of 풰(픤퐂) is a commutative 퐂-algebra ∑ 푎푛푛 = ∏ (1 + 푎푝푝 + 푎푝2 푝 + ⋯) 푛≥1 푝 of finite type. For example, if 픤 = 픤픩푛, then 풵(픤 ) ≃ 퐂[푡 , … , 푡 ]푆푛 , −푘푠 퐂 1 푛 = ∏ (∑ 푎푝푘 푝 ). 푝 푘≥0 the symmetric polynomials in the variables 푡1, 푡2, … , 푡푛. One can further consider functions 휑 that are annihilated 4.1. Automorphic 퐿-functions. Denote by 휒4 the unique × by an ideal of 풵(픤퐂) of finite codimension. Or in other nontrivial character on (퐙/4퐙) , i.e., 휒4(푛) = 1 if 푛 ≡ 1 words, for which 풵(픤퐂)⋅휑 is finite dimensional. Such func- (mod 4) and 휒4(푛) = −1 if 푛 ≡ 3 (mod 4). The simplest tions are said to be 풵(픤퐶)-finite. example of an 퐿-function, beyond the Riemann zeta func- Putting everything together, set 풜(퐺; Γ퐺) as the space of tion, is smooth, moderate growth functions 휑 ∶ Γ퐺\퐺 → 퐂 that −푠 −푠 −푠 −푠 퐿(휒4, 푠) ≔ 1 − 3 + 5 − 7 + 9 + ⋯ (3) are 퐾-finite and 풵(픤퐂)-finite. These are the Γ퐺-invariant au- −푠 −1 −푠 −1 tomorphic forms on 퐺. = ∏ (1 − 푝 ) ∏ (1 + 푝 ) . 푝≡1(4) 푝≡3(4) 4. 퐿-Functions More generally, suppose 푁 ≥ 1 is a positive integer × × One of the most important aspects of automorphic forms and 휒 ∶ (퐙/푁퐙) → 퐂 is a character, i.e., 휒(푚1푚2) = is that they connect apparently disparate areas of mathe- 휒(푚1)휒(푚2) for integers 푚1, 푚2 prime to 푁. Then one can matics. One way of making this connection precise is to extend 휒 to a function 퐙 → 퐂 as 휒(푚) = 0 if 푚 and 푁 share use 퐿-functions. a nontrivial common factor, and define The reader is probably familiar with the Riemann zeta 퐿(휒, 푠) = ∑ 휒(푛)푛−푠 = ∏ (1 − 휒(푝)푝−푠)−1. function 휁(푠). Recall that if 푠 ∈ 퐂 has 푅푒(푠) > 1, then one 푛≥1 (푝,푁)=1 defines 휁(푠) = ∑ 푛−푠 = ∏ (1 − 푝−푠)−1, When 휒 = 휒4 one obtains the 퐿-function (3). 푛≥1 푝 It is an important fact that these 퐿-functions have mero- where the latter product is over the prime numbers 푝. 퐿- morphic (analytic) continuation and functional equation. functions are generalizations of the Riemann zeta function. In fact, Dirichlet [Dir37] used the properties of these 퐿- It is a famous fact that 휁(푠) has a meromorphic continua- functions to prove that if 푎 and 푁 are relatively prime posi- tion to the complex plane with a simple pole at 푠 = 1, and tive integers, then there are infinitely many prime numbers satisfies a functional equation relating 푠 to 1 − 푠. More congruent to 푎 modulo 푁. specifically, define 휉(푠) = 휋−푠/2Γ(푠/2)휁(푠); then 휉(푠) has The above examples have the property that the coeffi- cients 푎푛 are completely multiplicative, i.e., 푎푚푛 = 푎푚푎푛 2I.e., a compact subgroup that is not strictly included in any other compact for all 푚, 푛, not just when 푚 and 푛 are relatively prime. subgroup. A more complicated example comes from the modular

FEBRUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 197 푛 3 forms. Suppose 푓(푧) = ∑푛≥1 푎푛푞 is a modular form of integer 푎푝(퐸) as 푎푝(퐸) = 푝 + 1 − #퐸(퐅푝). One can package weight 푘. For example, 푓(푧) could be Ramanujan’s func- the integers 푎푝(퐸) together as tion Δ(푧). The space of cusp forms 푆푘(SL2(퐙)) has a pre- −푠 1−2푠 −1 ferred basis, called the basis of Hecke eigenforms, and if 퐿(퐸, 푠) = ∏ (1 − 푎푝(퐸)푝 + 푝 ) . (4) 푝 푓(푧) is such an eigenform, then one can associate an 퐿- function to 푓. Specifically, suppose that 푓(푧) is normal- This is the 퐿-function of an 퐸. −푠 Note that the form of the Euler product for 퐿(퐸, 푠), i.e., ized so that 푎1 = 1. Then one defines 퐿(푓, 푠) = ∑푛≥1 푎푛푛 . Setting Λ(푓, 푠) = 휋−푠Γ(푠)퐿(푓, 푠), then this function satisfies the right-hand side of (4), is the same as that of a weight- the functional equation Λ(푓, 푠) = Λ(푓, 푘 − 푠). In the spe- 2 modular form. One says that the modular form 푓 of cial case of modular forms 푓 as above, the Euler product weight 2 is associated to 퐸 if 퐿(푓, 푠) = 퐿(퐸, 푠); equiva- expansion of 퐿(푓, 푠) looks as follows: lently, if 푎푝(푓) = 푎푝(퐸) for all 푝. That every rational el- liptic curve is associated to some modular form of weight −푠 푘−1 −2푠 −1 퐿(푓, 푠) = ∏ (1 − 푎푝푝 + 푝 푝 ) . 2 was called the Taniyama-Shimura-Weil conjecture, and 푝 was proved (for semistable4) elliptic curves in the famous The above 퐿-function is defined as a product over all the papers [Wil95] and [TW95] of Wiles and Taylor-Wiles. We prime numbers 푝. It is technically very convenient in the direct the reader to Ribet’s article [Rib95] for the history of theory of automorphic forms to also consider products as this problem and its connection to Fermat’s Last Theorem. above over all by finitely many primes, whereby obtaining Suppose 푋 is a smooth projective algebraic variety over what are called partial 퐿-functions. Throughout the rest 퐐 of dimension 푛 and 0 ≤ 푘 ≤ 2푛 is a nonnegative integer. of the text we blur the distinction between these different Then one can assign an 퐿-function 퐿(퐻푘(푋), 푠) to the ´etale 푘 scenarios: some of the statements below are only correct cohomology 퐻푒푡(푋퐐, 퐐ℓ). For example, if 푋 = 퐸 is an ellip- when 퐿-functions are replaced by partial 퐿-functions. tic curve and 푘 = 1, then 퐿(퐻1(퐸), 푠) = 퐿(퐸, 푠) as defined 4.2. Motivic 퐿-functions. One can associate 퐿-functions above. not only to modular forms, but more generally, to auto- As mentioned above, it is via 퐿-functions that number morphic forms 휑. However, one of the most basic ways theorists find a bridge between disparate areas of mathe- to construct 퐿-functions is via Galois theory. Suppose matics: algebraic geometry and automorphic forms. For 퐾 is a number field, with 퐾/퐐 a Galois extension, and example, here is an important conjecture. 휌 ∶ 퐺푎푙(퐾/퐐) → GL (퐂) is a representation. One can 푛 Conjecture 3 (Langlands [Lan80]). Suppose 푋 is a smooth construct an 퐿-function 퐿(휌, 푠) associated to this data as projective variety over 퐐 and 0 ≤ 푘 ≤ 2 dim(푋). Suppose follows: 푘 that the ´etalecohomology 퐻푒푡(푋퐐, 퐐ℓ) affords an irreducible −푠 −1 퐿(휌, 푠) = ∏ det(1 − 휌(퐹푟표푏푝)푝 ) . representation of 퐺푎푙(퐐/퐐) of dimension 푁. Then there is an 푝 푘 automorphic form 휑 on GL푁 so that 퐿(퐻 (푋), 푠) = 퐿(휑, 푠). Here 퐹푟표푏 is a certain conjugacy class of 퐺푎푙(퐾/퐐) as- 푝 5. Exceptional Groups sociated to the prime 푝 of 퐙, and note that the factor −푠 Having defined modular forms and automorphic forms, det(1 − 휌(퐹푟표푏푝)푝 ) is well-defined because the determi- nant makes the choice of representative in the conjugacy let us now turn our attention to exceptional groups. The class irrelevant. Lie groups one meets in practice are often defined as au- These 퐿-functions 퐿(휌, 푠) are known to have meromor- tomorphisms of particular algebraic structures. For exam- phic continuation in 푠, as a consequence of results of ple, Sp(2푛) is the fixing a symplectic Dirichlet and Brauer. Their holomorphy in 푠 is wide open. form on a 2푛-dimensional vector space, and SO(푝, 푞) is the automorphism group fixing a quadratic form of signature Conjecture 2 (The Artin conjecture). Suppose 퐾/퐐 is a Ga- (푝, 푞). These algebraic structures and associated Lie groups lois extension and 휌 ∶ 퐺푎푙(퐾/퐐) → GL푛(퐂) is an irreducible, come in infinite families, e.g., Sp(2푛) for 푛 = 1, 2, 3, 4, .... nontrivial representation. Then 퐿(휌, 푠) is an entire holomorphic Because of this fact, they are called classical groups. function of 푠. It turns out that there are very interesting, exceptional al- gebraic structures. The exceptional algebraic groups are 퐿 Algebro-geometric objects also have associated - defined to be automorphism groups of these structures. 퐿 functions. More precisely, one can associate -functions to More precisely, suppose 푉 is a vector space and {휉 } is the ´etalecohomology of smooth projective varieties over ℓ ℓ a number field. The first important example is that ofan 3 It is a theorem of Weil that 푎푝(퐸) satisfies the bound |푎푝(퐸)| ≤ 2√푝. In other elliptic curve 퐸 over 퐐. For all but finitely many primes 1 words, #퐸(퐅푝) is approximately #퐏 (퐅푝), and 푎푝(퐸) measures the discrepancy. 푝, one can reduce 퐸 modulo 푝 to obtain a curve over 4The semistability condition was later removed in work of Breuil-Conrad- the finite field 퐅푝. Counting points modulo 푝, define the Diamond-Taylor.

198 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 2 ℓ 푘 푐 훼푘 where 푤, 푥, 푦, 푧 ∈ 퐑. Now define Θ = 퐇 ⊕ 퐇 with multi- a finite list of tensors, i.e., 휉ℓ ∈ ⨂푘 푆푦푚 (⋀ 푉). Associ- ated to this data, one can consider the group 퐺 ⊆ GL(푉) plication ∗ ∗ that fixes the tensors 휉ℓ for all ℓ. For example, if 푉 is 2푛- (푥1, 푦1) ⋅ (푥2, 푦2) = (푥1푥2 − 푦2푦1, 푦2푥1 + 푦1푥2). 2 dimensional, then for an appropriate 휔 ∈ ⋀ 푉, Sp(2푛) is The algebra Θ푐 is isomorphic to Θ over 퐂, i.e., Θ푐 ⊗ 퐂 ≃ the subgroup of GL(푉) that fixes 휔. 푐 Θ ⊗ 퐂. The group 퐺2 is If one writes down some random collection of tensors 푐 푐 퐺2 = {푔 ∈ GL(Θ ) ∶ 푔(1) = 1 and 푔(푥 ⋅ 푦) = 푔푥 ⋅ 푔푦}. (6) 휉ℓ as above, then usually the group 퐺 will be trivial, i.e., 퐺 = {1}. However, it sometimes (very rarely) happens that 5.2. The groups 퐹4 and 퐸6. The group 퐺2 is the smallest one can write down tensors 휉ℓ so that 퐺 is nontrivial, and of the exceptional groups. The other exceptional groups in fact has positive dimension. When these are not the are labeled 퐹4, 퐸6, 퐸7, 퐸8. They sit in a chain 퐺2 ⊆ 퐹4 ⊆ classical groups, one gets the exceptional groups. In this 퐸6 ⊆ 퐸7 ⊆ 퐸8, and are of dimensions 14, 52, 78, 133, 248, section, we will define these algebraic structures (the 휉ℓ) respectively. and their automorphism groups. We now define 퐹4 and 퐸6. To do this, let 퐻3(Θ) denote

5.1. The octonions and 퐺2. We begin with the simplest the Hermitian 3 × 3 matrices with coefficients in the octo- case, that of the octonions and the exceptional group 퐺2. nions Θ. That is, The octonions are an 8-dimensional 퐑-vector space Θ, that ∗ comes equipped with a multiplication Θ⊗Θ → Θ, written 푐1 푥3 푥2 ∗ as 푥, 푦 ↦ 푥푦. This multiplication is neither commutative 퐻3(Θ) = {( 푥3 푐2 푥1 ) ∶ 푐푗 ∈ 퐑, 푥푘 ∈ Θ} . ∗ nor associative. However, it does have some redeeming 푥2 푥1 푐3 qualities. Like the octonions, the 27-dimensional vector space 퐻3(Θ) First, there is a multiplicative identity 1 ∈ Θ, and 푥 ⋅ 1 = has an that has surprisingly many auto- 1 ⋅ 푥 = 푥 for all 푥 ∈ Θ. Most importantly, there is a qua- morphisms. More precisely, 퐻3(Θ) has a cubic determi- dratic norm 푛Θ ∶ Θ → 퐑 satisfying 푛Θ(푥푦) = 푛Θ(푥)푛Θ(푦). nant map: if 푋 ∈ 퐻3(Θ), 퐺 The group 2 is defined as the group fixing this multiplica- ∗ Θ 푐1 푥3 푥2 tion on : ∗ 푋 = ( 푥3 푐2 푥1 ) , ∗ 퐺2 = {푔 ∈ GL(Θ) ∶ 푔(1) = 1 and 푔(푥 ⋅ 푦) = 푔푥 ⋅ 푔푦}. (5) 푥2 푥1 푐3 set To make this more concrete, we give the reader an ex- det(푋) = 푐1푐2푐3 − 푐1푛Θ(푥1) − 푐2푛Θ(푥2) − 푐3푛Θ(푥3) plicit description of Θ. Namely, one defines Θ = 푀2(퐑) ⊕ 푀2(퐑), a direct sum of two copies of the 2 × 2-matrices. + trΘ((푥1푥2)푥3). Suppose (푥 , 푦 ) and (푥 , 푦 ) are in Θ. Then 푛 (푥 , 푦 ) = 1 1 2 2 Θ 1 1 The group 퐸6 is defined as the subgroup of GL(퐻3(Θ)) that det(푥1)−det(푦1) is the quadratic norm. The multiplication fixes this determinant map: is defined as 퐸6 = {푔 ∈ GL(퐻3(Θ)) ∶ det(푔푋) = det(푋)∀푋 ∈ 퐻3(Θ)}. (푥 , 푦 ) ⋅ (푥 , 푦 ) = (푥 푥 + 푦∗푦 , 푦 푥 + 푦 푥∗). 1 1 2 2 1 2 2 1 2 1 1 2 The group 퐹4 is the subgroup of 퐸6 that fixes the element 1 = diag(1, 1, 1) ∈ 퐻 (Θ). 푎 푏 ∗ 푑 −푏 3 3 Here if 푚 = ( 푐 푑 ) is a 2 × 2 matrix, then 푚 = ( −푐 푎 ) so ∗ ∗ More precisely, there are different “forms” of these ex- that 푚 + 푚 = tr(푚)12 and 푚푚 = det(푚)12. The involu- ceptional groups. Recall that if 퐻1 and 퐻2 are real Lie tion ∗ on 푀2(퐑) extends to one on Θ, with similar prop- 퐻 퐻 ∗ ∗ ∗ groups, then one says that 2 is a “real form” of 1 if erties: (푥, 푦) = (푥 , −푦). One has (푥, 푦) + (푥, 푦) = tr(푥)1 퐿푖푒(퐻 ) ⊗ 퐂 ≃ 퐿푖푒(퐻 ) ⊗ 퐂 ∗ 1 2 . Every reductive Lie group has and (푥, 푦)(푥, 푦) = 푛Θ((푥, 푦)). a real form that is compact, and also has a real form that The group 퐺2 defined by (5) is a noncompact Lie group is split. The groups 퐺2, 퐹4, 퐸6 that we described above are of dimension 14. There is also a compact form of the group 푐 these split forms. Replacing 퐻3(Θ) with 퐻3(Θ ) in the def- 퐺2, which is defined as the automorphisms of an algebra initions of 퐹4 and 퐸6 yields different real forms of these Θ푐, closely related to Θ. groups. By contrast with the case of 퐺2, the forms of 퐹4 To define Θ푐, first recall Hamilton’s and 퐸6 defined in this way are not compact. 퐇 = 퐑 ⊕ 퐑푖 ⊕ 퐑푗 ⊕ 퐑푘 5.3. The group 퐸7. To define the group 퐸7, one proceeds as follows. For ease of notation, set 퐽 = 퐻3(Θ). This no- with multiplication 푖푗 = 푘, 푗푘 = 푖, 푘푖 = 푗. Denote by tation comes from the fact that 퐽 is what is called a “Jor- ∨ ∨ ∗ ∶ 퐇 → 퐇 the involution on 퐇 defined as dan algebra.” Now set 푊 = 퐑 ⊕ 퐽 ⊕ 퐽 ⊕ 퐑, where 퐽 is the linear dual of 퐽. This space 푊 is 56-dimensional, and (푤 + 푥푖 + 푦푗 + 푧푘)∗ = 푤 − 푥푖 − 푦푗 − 푧푘, comes equipped with a nondegenerate symplectic form

FEBRUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 199 and a particular quartic form 푄 (homogeneous of degree the modular forms—that are close cousins of the classical four) which were defined by Freudenthal. The group 퐸7 modular forms on SL2 defined in section 2. Just as in the can be defined as the subgroup of GL(푊) that fixes these classical case, there is a function 푗 ∶ 퐺(퐑) × 픥퐽 → 픥퐽 sat- two forms on 푊. isfying 푗(푔1푔2, 푍) = 푗(푔1, 푔2푍)푗(푔2, 푍) for all 푔1, 푔2 ∈ 퐺(퐑).

5.4. The exceptional group 퐸8. Every reductive Lie group To define the modular forms, suppose Γ is an arithmetic acts on its Lie algebra by automorphisms. And conversely, subgroup of 퐺(퐑) and 푘 ≥ 0 is an integer. A modular form if 픤 is a (reductive) Lie algebra, then one can define a cor- of weight 푘 for Γ is a holomorphic function 푓 ∶ 픥퐽 → 퐂 푘 responding Lie group 퐺 as on 픥퐽 satisfying 푓(훾푍) = 푗(훾, 푍) 푓(푍) for all 훾 ∈ Γ and for which the Γ-invariant function 퐺 = {푔 ∈ GL(픤) ∶ [푔푋, 푔푌] = 푔[푋, 푌] ∀푋, 푌 ∈ 픤}. 푘/2 −푘 | det(푌) 푓(푍)| = |푗(푔, 푖13) 푓(푔 ⋅ 푖13)| All the reductive Lie algebras with the exception of 퐸8 have a nontrivial finite-dimensional representation on a vector is of moderate growth on 퐺(퐑). These modular forms have a “푞-expansion,” completely space smaller than its Lie algebra. For 퐸8, its smallest non- trivial representation is the adjoint action on its Lie alge- analogous to the Fourier expansion of classical modular forms. Assume for simplicity that Γ = 퐺(퐙) is the maximal bra, so in a sense one must define 퐸8 through this action. Fortunately, it is not so difficult to define the Lie algebra arithmetic subgroup of 퐺(퐑). Fix a certain 퐽퐙 in 퐽, and let 퐽+ denote the intersection of 퐽 with the closure of 픢8. 퐙 퐙 퐽 in 퐽 . If 푓 is a modular form for Γ, then One can do this as follows [Rum97]. Denote by 푉3 the + 퐑 2휋푖 tr(푇푍) three-dimensional representation of 픰푙3. Then 푓(푍) = ∑ 푎푓(푇)푒 (8) ∨ 푇∈퐽+ 픢8 = (픰푙3 ⊕ 픢6) ⊕ 푉3 ⊗ 퐽 ⊕ (푉3 ⊗ 퐽) . (7) 퐙 for complex numbers 푎푓(푇); this is its Fourier expansion. This is a 퐙/3-grading with 픰푙3 ⊕ 픢6 in degree zero, 푉3 ⊗ 퐽 ∨ Some examples of modular forms on 퐺 can be found in degree one, and (푉3 ⊗ 퐽) in degree two. The subalge- ∨ in work of Baily [Bai70], Kim [Kim93], and Kim-Yamauchi bra 픰푙3 ⊕ 픢6 acts on 푉3 ⊗ 퐽 and (푉3 ⊕ 퐽) in the obvious manner. For an analogous construction of the Lie algebra [KY16]. However, let us reiterate that very little is known about them. For example, the 퐿-functions 퐿(푓, 푠) of mod- 픤2, see [FH91, p. 358]. In fact, one can construct all the exceptional Lie algebras using an analogue of (7). ular forms on 퐺 are known to have meromorphic continu- ation in 푠 [Lan71]. However, the conjectured finer analytic 6. Automorphic Forms on Exceptional Groups properties of these 퐿-functions, such as the finiteness of its Having described exceptional groups, and automorphic poles, are not known. forms in general, let us now consider automorphic forms 6.2. Quaternionic modular forms. Of the exceptional on exceptional groups. Dynkin types, 퐸7 is special in that the types 퐺2, 퐹4, 퐸6, and 퐸 do not have real forms 퐺 so that the associated sym- 6.1. Holomorphic modular forms on 퐸7. We first turn 8 metric space 푋 has “modular forms” similar to the clas- our attention to the exceptional group 퐸7, as it possesses 퐺 5 “modular forms” completely analogous to the classical sical holomorphic modular forms. It turns out there is a holomorphic modular forms that we described in the first certain type of nonholomorphic “quaternionic” modular 푐 forms on one of the real forms of each of the exceptional section. Let 퐽 = 퐻3(Θ ), and let 퐺 denote the exceptional Dynkin types group of type 퐸7 defined as in subsection 5.3. The group 퐺 is neither split nor compact. For this group, if 퐾퐺 de- 퐺2 ⊆ 퐹4 ⊆ 퐸6 ⊆ 퐸7 ⊆ 퐸8. (9) notes the maximal compact of 퐺(퐑), the These were defined by Gross and Wallach [GW96] and 퐺(퐑)/퐾 has a 퐺(퐑)-invariant complex structure. 퐺 Gan-Gross-Savin [GGS02], and have subsequently been Denote by 퐽 the subset of 퐽 consisting of elements + studied by Gan, Loke, Weissman, and the author. Among of the form 푌 2 for some 푌 ∈ 퐽 with det(푌) ≠ 0. (If their nice features are that they have a Fourier expansion 푌 ∈ 퐽, then its square 푌 2—in the sense of usual matrix with some of the same good properties of classical mod- multiplication—is also an element of 퐽.) The symmetric ular forms [Pol20], and that they behave well under pull- space 퐺(퐑)/퐾 can be identified with 퐺 back in the sequence (9). 픥퐽 = {푍 = 푋 + 푖푌 ∶ 푋 ∈ 퐽, 푌 ∈ 퐽+}. Instead of being 퐂-valued functions on an exceptional group 퐺, quaternionic modular forms are naturally val- Consequently, there is an action of 퐺(퐑) on 픥퐽 , which one should think of as generalized (exceptional) linear frac- ued in the finite-dimensional representations of the com- tional transformations. pact Lie group SU(2)/{±1}. More precisely, if 푛 ≥ 1 is As the reader will no doubt have noticed, the space 픥 퐽 5There is a real form of 퐸 whose symmetric space has hermitian structure, but 픥 6 is very similar to the complex upper half-space . And in- the automorphic forms on this 퐸6 do not have Fourier expansions similar to that deed, the group 퐺 possesses special automorphic forms— of (8).

200 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 2 an integer, a quaternionic modular form of weight 푛 on a Generic cuspidal automorphic forms are very far from quaternionic exceptional group 퐺 is a function 푓 ∶ Γ\퐺 → being holomorphic: many one-dimensional unipotent av- 푆푦푚2푛(퐂2) satisfying conditions analogous to those of Def- erages of 휑 vanish identically if 휑 is a holomorphic cusp- inition 1, together with being annihilated by a specific first- idal modular form. Similarly, certain three-dimensional linear differential operator 풟푛, i.e., 풟푛푓 = 0. Here unipotent averages of 휑 vanish identically if 휑 is a quater- 푆푦푚2푛(퐂2) is the (2푛+1)-dimensional irreducible represen- nionic cuspidal modular form, in the sense of subsection tation of SU(2). Similar to how the Cauchy-Riemann solu- 6.2. tions force the nice form of the Fourier expansion of holo- The 퐿-functions of generic cuspidal automorphic forms morphic modular forms, being annihilated by 풟푛 forces on the split exceptional groups 퐺2, 퐹4, 퐸6, and 퐸7 have the quaternionic modular form 푓 to have a robust Fourier been studied by Ginzburg, Ginzburg–Rallis, and Piatetski- expansion. The robustness of the Fourier expansion allows Shapiro–Rallis–Schiffmann. For example, in [Gin95] it is one to ask (and answer6) questions such as “Do there exist proved that the so-called standard 퐿-function 퐿(휋, 푆푡푑, 푠) quaternionic modular forms on 퐸8 and 퐺2 whose Fourier of cuspidal generic automorphic representations on split coefficients are all integers?” 퐸7 has meromorphic analytic continuation with at most One beautiful example of such a quaternionic modu- two simple poles. lar form is a function 휃푚푖푛 on 퐸8 of weight 4, defined 6.3.2. Theta functions. As a rule of thumb, constructing and studied by Gan [Gan00]. The function 휃푚푖푛 can be automorphic forms is difficult. The first explicit examples thought of as an 퐸8-analogue of the classical theta func- one tends to write down are often constructed using 휃 func- 푛2 tion 휃(푧) = ∑푛∈퐙 푞 . tions and their generalizations. For example, taking the Another source of examples of these quaternionic mod- 푛2 4 classical 휃 function 휃(푧) = ∑푛∈퐙 푞 and considering 휃(푧) ular forms comes from Eisenstein series associated to holo- gives the modular form 휃4(푧) of section 1. morphic cusp forms. Precisely, if 푛 ≥ 6 and 휑 is a classical The function 휃(푧) has the property that many of its holomorphic modular cusp form of weight 3푛, then one Fourier coefficients are 0: the coefficient of 푞푁 is nonzero can construct out of 휑 an Eisenstein series 퐸(휑) on 퐺2 that only when 푁 is a square. Many reductive groups, and in gives a modular form of weight 푛. The fact that 휑 is holo- particular the split exceptional groups 퐹4, 퐸6, 퐸7, and 퐸8, morphic is crucial to 퐸(휑) being annihilated by 풟푛. possess analogues of this function 휃(푧). Such an analogue 6.3. Automorphic forms on split exceptional groups. goes by the moniker of a minimal automorphic form, and Our treatment of automorphic forms in this article has possesses the property that very many of its generalized been biased towards those that are most similar to the clas- Fourier coefficients are 0. We direct the reader to work of sical holomorphic modular forms that one first meets, in Ginzburg-Rallis-Soudry [GRS97] and Ginzburg for results terms of having a robust Fourier expansion. This bias is a on these minimal automorphic forms. bit out-of-line with the development of the theory of au- Besides 휃 functions themselves, one can use theta func- tomorphic forms. In this final subsection, we go back to tions as integral functions to create other automor- the mainstream and highlight some work on automorphic phic forms. Without going into the details of this proce- forms on split exceptional groups, for which there is not dure, let us simply mention the very interesting examples a similarly robust theory of the Fourier expansion. In no of Rallis-Schiffmann, Li-Schwermer, and Gan-Gurevich- way should our selection of topics here be considered ex- Jiang, which use such a construction to create special au- haustive. tomorphic forms on the split exceptional group 퐺2. 6.3.1. 퐿-functions. Relatively speaking, much more is 6.3.3. Further directions. Finally, we mention that there known about the 퐿-functions of automorphic forms on has been some activity to relate the Fourier coefficients split exceptional groups than on their nonsplit counter- of automorphic functions on split exceptional groups to parts. To explain the state of affairs in just a little more quantities that arise in string theory. We direct the inter- detail, recall that split reductive algebraic groups 퐺 possess ested reader to the book [FGKP18] and the numerous ref- generic cusp forms. Roughly speaking, a generic cusp form erences contained therein. is a cuspidal automorphic form which possesses nontrivial averages over large unipotent groups 푈′. That is, if 휑 is cus- A. Technical Details on Exceptional Groups In this appendix we include a few technical details on ex- pidal and generic, then the integral ∫(푈′∩Γ)\푈′ 휑(푢푔) 푑푢 is not identically 0 as a function of 푔 ∈ 퐺. Here, for the more ceptional groups that might be useful to the interested knowledgeable reader, we point out that 푈′ = [푈, 푈] is the reader. commutator subgroup of a maximal unipotent group 푈. A.1. Real forms of exceptional groups. In section 5 we defined various forms of the exceptional groups. Were- mark that the forms of 퐺2, 퐹4, 퐸6, 퐸7, 퐸8 that are defined 6The answer is “Yes.” there are the simply connected algebraic groups. (The

FEBRUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 201 algebraic groups 퐺2, 퐹4, and 퐸8 are both adjoint and sim- by ply connected, while the simply connected 퐸6 and 퐸7 have 2 centers the cyclic groups of orders 3 and 2, respectively.) 푄((푎, 푏, 푐, 푑)) = (푎푑 − (푏, 푐)) + 4푎 det(푐) We connect what was written above to other standard no- + 4푑 det(푏) − 4(푐#, 푏#). tation for these groups. A.3. The definition of 퐸8. It is also possible to concisely • The real form of 퐹 defined in section 5 using 4 define the Lie bracket on 픢8. Because 푉3 is a representation 퐻3(Θ) is the split form 퐹4(4) if Θ is the split octo- 2 ∨ 2 ∨ of 픰푙3, there are identifications ⋀ 푉3 ≃ 푉3 and ⋀ 푉3 ≃ nions and is the compact form 퐹4(−52) if defined ′ ′ ′ ′ 푐 푉3. Now if 푢, 푢 ∈ 푉3 and 푋, 푋 ∈ 퐽, then [푢⊗푋, 푢 ⊗푋 ] = using 퐻3(Θ ). ′ ′ ∨ (푢∧푢 )⊗(푋×푋 ), considered as an element of (푉3⊗퐽) . The • The real forms of 퐸6 defined in section 5 are the ∨ ∨ Lie bracket [⋅, ⋅] ∶ (푉3 ⊗ 퐽) ⊗ (푉3 ⊗ 퐽) → 푉3 ⊗ 퐽 is defined split form if defined using 퐻3(Θ) and the form ∨ 푐 analogously. Finally, the map (푉3 ⊗ 퐽) ⊗ (푉3 ⊗ 퐽) induced 퐸6(−26) if defined using 퐻3(Θ ). ∨ by the Lie bracket is defined using the maps 푉3 ⊗ 푉3 ≃ • The real forms of 퐸7 defined in section 5 are the 0 ∨ 퐸푛푑(푉3) → 픰푙3 and a bilinear map Φ ∶ 퐽 ⊗퐽 → 픢6 defined split form if defined using 퐽 = 퐻3(Θ) and the form 푐 as 퐸7(−25) if defined using 퐽 = 퐻3(Θ ). 1 Φ0 (푋) = −푐 × (푏 × 푋) + (푐, 푋)푏 + (푏, 푐)푋. • The real forms of 퐸8 defined in section 5 are the 푐,푏 3 split form if defined using 퐽 = 퐻3(Θ) and the form ∨ 푐 Here 푐 ∈ 퐽 and 푏, 푋 ∈ 퐽. 퐸8(−24) if defined using 퐽 = 퐻3(Θ ). In section 5 it was mentioned that every exceptional ACKNOWLEDGMENT. It is a pleasure to thank the ref- Lie algebra can be constructed in a form analogous to (7). erees for their careful reading of this manuscript and To do this, one lets the Jordan algebra (or precisely, cubic thoughtful suggestions. norm structure) 퐽 vary, and the Lie algebra 픢6 is replaced with the Lie algebra 픪(퐽) of the group of determinant- preserving linear maps on 퐽. One obtains a Lie algebra References [Bai70] Walter L. Baily Jr., An exceptional and ∨ 픤(퐽) = (픰푙3 ⊕ 픪(퐽)) ⊕ 푉3 ⊗ 퐽 ⊕ (푉3 ⊗ 퐽) , (10) its Eisenstein series, Ann. of Math. (2) 91 (1970), 512–549, DOI 10.2307/1970636. MR269779 where now 퐽 is an arbitrary cubic norm structure, instead [Dir37] P. G. L. Dirichlet, Beweis des satzes, dass jede unbegren- 푐 of just the exceptional cubic norm structure 퐻3(Θ ). If zte arithmetische progression, deren erstes glied und differenz 퐽 varies over cubic norm structures with positive-definite ganze zahlen ohne gemeinschaftlichen factor sind, unendlich trace forms, one constructs (uniformly) the so-called viele primzahlen enthält, Abhandlungen der Königlichen quaternionic forms of the exceptional groups. Details can Preußischen Akademie der Wissenschaften zu Berlin 48 (1837), 45–71. be found in, for example, [Pol20]. With 퐽 = 퐻3(퐇), 픤(퐽) is the quaternionic 픢 (also known as 픢 ); if 퐽 = 퐻 (퐂), [FGKP18] Philipp Fleig, Henrik P. A. Gustafsson, Axel Klein- 7 7(−5) 3 schmidt, and Daniel Persson, Eisenstein series and automor- then 픤(퐽) is the quasisplit and quaternionic 픢 (also known 6 phic representations: With applications in string theory, Cam- as 픢6(2)); and if 퐽 = 퐻3(퐑), then 픤(퐽) is the split 픣4. bridge Studies in Advanced , vol. 176, Cam- A.2. The definition of 퐸7. It is not difficult to be even bridge University Press, Cambridge, 2018. MR3793195 more precise about the definition of the exceptional [FH91] William Fulton and Joe Harris, Representation theory: groups 퐸7 and 퐸8 given above. To do so, we require one ad- A first course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. Readings in Mathemat- ditional piece of notation for the Jordan algebra 퐽 = 퐻3(Θ); namely, a symmetric bilinear map × ∶ 퐽 ⊗ 퐽 → 퐽∨ de- ics. MR1153249 fined as follows. Polarizing the determinant map det ∶ [Gan00] Wee Teck Gan, A Siegel-Weil formula for exceptional groups, J. Reine Angew. Math. 528 (2000), 149–181, DOI 퐽 → 퐑, there is a symmetric trilinear map (⋅, ⋅, ⋅) on 퐽 sat- 10.1515/crll.2000.088. MR1801660 isfying (푋, 푋, 푋) = 6 det(푋). Now, for 푋, 푌 in 퐽, define [GGS02] Wee Teck Gan, Benedict Gross, and Gordan Savin, 푋 × 푌 ∈ 퐽∨ 퐽 (푋 × 푌)(푍) = (푋, 푌, 푍) (the linear dual of ) via . Fourier coefficients of modular forms on 퐺2, Duke Math. J. 1 Set 푋# = 푋 × 푋. There is an identically defined map 115 (2002), no. 1, 105–169, DOI 10.1215/S0012-7094-02- 2 × ∶ 퐽∨ ⊗ 퐽∨ → 퐽. 11514-2. MR1932327 퐿 퐸 With this bit of notation, we can now precisely define [Gin95] David Ginzburg, On standard -functions for 6 and 퐸7, J. Reine Angew. Math. 465 (1995), 101–131, DOI the symplectic and quartic forms on 푊 퐽 fixed by 퐸7. For 퐸7, ∨ 10.1515/crll.1995.465.101. MR1344132 suppose (푎, 푏, 푐, 푑) ∈ 푊, so that 푎, 푑 ∈ 퐑, 푏 ∈ 퐽, and 푐 ∈ 퐽 , [GRS97] David Ginzburg, Stephen Rallis, and David ′ ′ ′ ′ and similarly for (푎 , 푏 , 푐 , 푑 ). Then the symplectic form Soudry, On the automorphic theta representation for simply ⟨⋅, ⋅⟩ on 푊 is given by ⟨(푎, 푏, 푐, 푑), (푎′, 푏′, 푐′, 푑′)⟩ = 푎푑′ − laced groups, Israel J. Math. 100 (1997), 61–116, DOI (푏, 푐′) + (푏′, 푐) − 푑푎′ and the quartic form 푄 on 푊 is given 10.1007/BF02773635. MR1469105

202 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 2 [GW96] Benedict H. Gross and Nolan R. Wallach, On quater- nionic discrete series representations, and their continua- tions, J. Reine Angew. Math. 481 (1996), 73–123, DOI 10.1515/crll.1996.481.73. MR1421947 [Kim93] Henry H. Kim, Exceptional modular form of weight 4 on an exceptional domain contained in 퐂27, Rev. Mat. Iberoamer- icana 9 (1993), no. 1, 139–200, DOI 10.4171/RMI/134. MR1216126 [KY16] Henry H. Kim and Takuya Yamauchi, Cusp forms on the exceptional group of type 퐸7, Compos. Math. 152 (2016), no. 2, 223–254, DOI 10.1112/S0010437X15007538. Aaron Pollack MR3462552 [Lan80] R. P. Langlands, 퐿-functions and automorphic represen- Credits tations, Proceedings of the International Congress of Math- The opening is courtesy of brightstars via Getty. ematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, Photo of the author is courtesy of the author. 1980, pp. 165–175. MR562605 [Lan71] Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whit- temore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1. MR0419366 [Pol20] Aaron Pollack, The Fourier expansion of modular forms on quaternionic exceptional groups, Duke Math. J. 169 (2020), no. 7, 1209–1280, DOI 10.1215/00127094-2019-0063. MR4094735 [Rib95] Kenneth A. Ribet, Galois representations and mod- ular forms, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 4, 375–402, DOI 10.1090/S0273-0979-1995-00616-6. MR1322785 [Rum97] Karl E. Rumelhart, Minimal representations of excep- tional 푝-adic groups, Represent. Theory 1 (1997), 133–181, DOI 10.1090/S1088-4165-97-00009-5. MR1455128 [Ser73] J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. When was the last time you visited the MR0344216 [TW95] Richard Taylor and Andrew Wiles, Ring-theoretic prop- erties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), AMS Bookstore? no. 3, 553–572, DOI 10.2307/2118560. MR1333036 [Wil95] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551, DOI 10.2307/2118559. MR1333035 Spend smart. [Zag08] Don Zagier, Elliptic modular forms and their applica- tions, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103, DOI 10.1007/978-3-540-74119- 0_1. MR2409678 Search better. Stay informed.

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