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Exceptional Groups and Their Modular Forms

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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.
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  • L-Functions and Automorphic Representations*

    L-Functions and Automorphic Representations*

    L-Functions and Automorphic Representations* R. P. Langlands Introduction. There are at least three different problems with which one is confronted in the study of L-functions: the analytic continuation and functional equation; the location of the zeroes; and in some cases, the determination of the values at special points. The first may be the easiest. It is certainly the only one with which I have been closely involved. There are two kinds of L-functions, and they will be described below: motivic L-functions which generalize the Artin L-functions and are defined purely arithmetically, and automorphic L-functions, defined by data which are largely transcendental. Within the automorphic L-functions a special class can be singled out, the class of standard L-functions, which generalize the Hecke L-functions and for which the analytic continuation and functional equation can be proved directly. For the other L-functions the analytic continuation is not so easily effected. However all evidence indicates that there are fewer L-functions than the definitions suggest, and that every L-function, motivic or automorphic, is equal to a standard L-function. Such equalities are often deep, and are called reciprocity laws, for historical reasons. Once a reciprocity law can be proved for an L-function, analytic continuation follows, and so, for those who believe in the validity of the reciprocity laws, they and not analytic continuation are the focus of attention, but very few such laws have been established. The automorphic L-functions are defined representation-theoretically, and it should be no surprise that harmonic analysis can be applied to some effect in the study of reciprocity laws.
  • Uncoveringanew L-Function Andrew R

    Uncoveringanew L-Function Andrew R

    UncoveringaNew L-function Andrew R. Booker n March of this year, my student, Ce Bian, plane, with the exception of a simple pole at s = 1. announced the computation of some “degree Moreover, it satisfies a “functional equation”: If 1 3 transcendental L-functions” at a workshop −s=2 γ(s) = ÐR(s) := π Ð (s=2) and Λ(s) = γ(s)ζ(s) at the American Institute of Mathematics (AIM). This article aims to explain some of the then I (1) (s) = (1 − s). motivation behind the workshop and why Bian’s Λ Λ computations are striking. I begin with a brief By manipulating the ζ-function using the tools of background on L-functions and their applications; complex analysis (some of which were discovered for a more thorough introduction, see the survey in the process), one can deduce the famous Prime article by Iwaniec and Sarnak [2]. Number Theorem, that there are asymptotically x ≤ ! 1 about log x primes p x as x . L-functions and the Selberg Class Other L-functions reveal more subtle properties. There are many objects that go by the name of For instance, inserting a multiplicative character L-function, and it is difficult to pin down exactly χ : (Z=qZ)× ! C in the definition of the ζ-function, what one is. In one of his last published papers we get the so-called Dirichlet L-functions, [5], the late Fields medalist A. Selberg tried an Y 1 L(s; χ) = : axiomatic approach, basically by writing down the 1 − χ(p)p−s common properties of the known examples.