The Ramanujan conjecture and its applications

Abstract

In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed.

This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

1. Introduction

Srinivasa Ramanujan was a mathematical genius with broad interests and deep insights. The numerous formulae he left behind have inspired the development of new theories and discoveries of new connections in mathematics, computer science, engineering and physics. This paper concerns a very important topic in automorphic forms and number theory, called the Ramanujan conjecture, and its amazing applications in computer science. The Ramanujan conjecture in general form asserts that a generic cuspidal automorphic irreducible unitary representation of a reductive group over a global field should be locally tempered everywhere. It originated from Ramanujan's prediction [1] on the behaviour of the Fourier coefficients τ(n) of the weight-12 discriminant function Δ. The generalized Ramanujan conjecture is largely unsettled. The appli- cations are based on the representations of GL_n and certain unitary groups for which the conjecture has been established. These include the explicit constructions of infinite families of Ramanujan graphs and Ramanujan complexes. For several decades, computer scientists have intensively studied families of graphs with expansion constants uniformly bounded away from 0, called expanders, which have wide real-world applications (cf. [2]). Ramanujan graphs [3] are spectrally optimal expanders, which make them excellent communication networks, and Ramanujan complexes [4] are higher-dimensional analogues of Ramanujan graphs. These combinatorial objects were never considered by Ramanujan himself; they bear his name because of the Ramanujan conjecture lurking in the background. Similar constructions from a different perspective give rise to points uniformly distributed on unit spheres and Golden-Gate Sets in quantum computing.

The purpose of this survey paper is to discuss the Ramanujan conjecture and its applications sketched above. Section 2 is devoted to the Ramanujan conjecture. Starting from Ramanujan's original predictions on τ(n), we review the progress on classical cusp forms, state the generalized conjecture, and summarize its current status over function fields and number fields. The spectral theory for graphs and complexes is discussed in §3 and 4, respectively, where Ramanujan graphs [3] and Ramanujan complexes [4] are also introduced as spectrally extremal combinatorial objects with respect to the eigenvalues of the adjacency operators on vertices. For graphs/complexes arising as finite quotients of the building of PGL_n over a non-archimedean local field F, these adjacency operators coincide with the Hecke operators of PGL_n(F). In §5, we recall explicit constructions of Ramanujan graphs and Ramanujan complexes in [3–9] as quotients of the building of PGL_n(F) by discrete subgroups of PGL_n(F) arising from suitable subgroups Γ of global points of certain inner forms of PGL_n or certain unitary groups. When the base field is $\mathbb{Q}$, the groups Γ viewed as subgroups at the archimedean place ∞ of $\mathbb{Q}$ give rise to points uniformly distributed on unit spheres [10–12]. The Golden-Gate Sets arise from first representing the Hecke operators of PGL_n(F) by global points and then regarding them locally as points at the place at ∞. (See [5,13,14] for more details.) The role of the Ramanujan conjecture is also explained.

The Ramanujan conjecture has close ties to the Riemann hypothesis. To date, the standard way to prove the Ramanujan conjecture, such as in [10,15–28] for GL_n and certain unitary groups, is to construct the corresponding Galois representation as predicted in the Langlands program [29]. The Ramanujan conjecture then follows from the Riemann hypothesis for the underlying geometric object which affords the Galois representation, like Deligne's proof for the original Ramanujan conjecture for Δ [19,30]. Other cases of the Ramanujan conjecture, such as [31,32] for inner forms of GL_n and [5,33,34] for classical groups and certain unitary groups, were proved by establishing the Langlands functoriality from representations of the target group to those of a group for which the Ramanujan conjecture holds. In [35], Langlands proposed an analytic method, a variation of which was used in [36] to prove the conjecture for certain representations of GL₂ over CM fields. To graphs and complexes, there are associated zeta functions [37–39]. One criterion for a regular graph to be Ramanujan is that its zeta function satisfies the Riemann hypothesis. For complexes, the situation is more complicated because more operators are involved owing to the higher dimension. In the final section, §6, we introduce a stronger concept, called strongly Ramanujan complex [40], in terms of the eigenvalues of an Iwahori-Hecke operator, or equivalently, the adjacency operator on the directed chambers of the complex, and discuss the relationship between Ramanujan complexes, strongly Ramanujan complexes and the Riemann hypothesis (cf. [41]).

2. The Ramanujan conjecture

(a). The classical Ramanujan conjecture

Roughly speaking, a modular form is a holomorphic function on the Poincaré upper half-plane which has a lot of symmetry with respect to the modular group ${\mathrm{SL}}_2(\mathbb{Z})$ or a congruence subgroup of ${\mathrm{SL}}_2(\mathbb{Z})$. It is called a cusp form if it vanishes at all cusps of the group of symmetry. The first non-zero cusp form for ${\mathrm{SL}}_2(\mathbb{Z})$ is the weight-12 normalized discriminant function Δ(z). Jacobi showed that it can be expressed as an infinite product in powers of q = e^2πiz related to the partition function

$$\mathrm{\Delta }(z)=q\prod _{n\ge 1}{(1-q^n)}^{24}=\sum _{n\ge 1}\tau (n)q^n.$$

Ramanujan studied the Fourier coefficients τ(n) of Δ. Based on his computation of the first 30 coefficients, in his 1916 paper [1] Ramanujan conjectured the following properties satisfied by τ as an arithmetic function:

• (1)τ is multiplicative, that is, τ(mn) = τ(m)τ(n) for (m, n) = 1. Thus it suffices to know its values at prime powers.
• (2)
  τ satisfies the degree-2 recursive relation for each prime p:

  $$\tau (p^{n+1})-\tau (p)\tau (p^n)+p^{11}\tau (p^{n-1})=0\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}n\ge 1.$$

  Since τ(1) = 1, the τ function is completely determined by its values at the primes p.
• (3)At each prime p, τ(p) satisfies the bound |τ(p)|≤2p^11/2.

As noted by Ramanujan himself, the first two statements can be rephrased in terms of the L-function L(Δ, s) associated with Δ having an Euler product for ℜ(s)≫0 as follows:

$$L(\mathrm{\Delta },s):=\sum _{n\ge 1}\tau (n)n^{-s}=\prod _{p\hspace{0.17em}\mathrm{prime}}\frac{1}{1-\tau (p)p^{-s}+p^{11-2s}}.$$

This assertion, proved by Mordell in 1917, motivated Hecke to introduce the Hecke operator T_p at each prime p so that the product expression is equivalent to Δ being an eigenfunction of T_p with eigenvalue τ(p). In 1937, Hecke [42] extended this result to modular forms for congruence subgroups of ${\mathrm{SL}}_2(\mathbb{Z})$, one of the fundamental contributions of Hecke to the modern theory of automorphic forms. The third assertion can be restated as, in the factorization

$$1-\tau (p)p^{-s}+p^{11-2s}=(1-\alpha (p)p^{-s})(1-\beta (p)p^{-s}),$$

we have

$$|\alpha (p)|=|\beta (p)|=p^{11/2}.$$

This is the classical Ramanujan conjecture, proved by Deligne [19,30] for Δ and cusp forms of weight greater than or equal to 3, Eichler–Shimura [22,27] for weight-2 cusp forms, and Deligne–Serre [20] for cusp forms of weight 1.

(b). Generalized Ramanujan conjecture

Langlands reinterpreted the Hecke eigenforms in terms of automorphic representations for GL₂ over $\mathbb{Q}$ so that both have the same associated L-functions. As such, the cusp forms correspond to cuspidal representations. This viewpoint led Langlands to propose vastly generalized statements for representations of a reductive group G over a global field K [29]. In this context, the classical Ramanujan conjecture has the following generalization. The reader is referred to the survey article [43] by Lomeli and references therein for more details of the topics discussed in this and next subsections.

Let Π be a cuspidal automorphic irreducible unitary representation of G over K. As a global representation of the group $G({\mathbb{A}}_K)$ of adelic points on G, Π = ⊗_{v place of K}′ Π_v is a restricted tensor product of local representations Π_v of the group G(K_v) of points on G over the completion K_v of K at v. The representation Π is said to satisfy the generalized Ramanujan conjecture if every local component Π_v is tempered, in other words, the matrix coefficients of Π_v lie in the space L^2+ϵ(G(K_v)) for all ϵ > 0. There are known examples of representations Π such that none of the unramified local components Π_v are tempered (cf. [44]). The generalized Ramanujan conjecture is predicted to hold for Π generic.

Now we explain the temperedness on more concrete terms. First consider the example where G = GL₂ and $K=\mathbb{Q}$. In this case, a cuspidal automorphic irreducible representation Π is generic. There are two possibilities for the component Π_∞ of Π at the infinite place ∞ of $\mathbb{Q}$. If Π corresponds to a holomorphic cusp form f, then Π_∞ is either a discrete series (if f has weight greater than or equal to 2) or a limit of discrete series representations (if f has weight 1); in both cases, Π_∞ is tempered. The classical Ramanujan conjecture discussed in §2a says that Π_p is tempered for each prime p. On the other hand, if Π corresponds to a real analytic cuspidal Maass form f, then Π_∞ is tempered if and only if f is an eigenfunction of the Laplace operator with eigenvalue greater than or equal to 1/4. This coincides with the celebrated 1/4 conjecture by Selberg [45], which is wide open.

Regarding L(Δ, s) as the L-function attached to the representation of ${\mathrm{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ corresponding to Δ, for G = GL_n, to Π we similarly associate an L-function L(Π, s) which has the form

$$L(\mathrm{\Pi },s)=\prod _{\mathrm{nonarchimedean}\hspace{0.17em}\mathrm{place}\hspace{0.17em}v\hspace{0.17em}\mathrm{of}\hspace{0.17em}K}\hspace{0.17em}\frac{1}{1+a_1(v){(Nv)}^{-s}+\cdots +a_n(v){(Nv)}^{-ns}},$$

such that the factor at a place v is the reciprocal of a polynomial in (Nv)^−s of degree ≤n with complex coefficients a₁(v), …, a_n(v). Here, Nv is the norm of v. Furthermore, a_n(v)≠0 if Π_v is unramified, which is for almost all v. An unramified Π_v is tempered if and only if all roots of

$$1+a_1(v)u+\cdots +a_n(v)u^n$$

have the same absolute value, just like what we saw for the Δ function.

(c). Known cases of the generalized Ramanujan conjecture

In this subsection, we summarize the groups G for which the Generalized Ramanujan Conjecture is established for generic cuspidal irreducible automorphic representations Π of G. To date, the standard way to prove the generalized Ramanujan conjecture for Π is to construct the Galois representation corresponding to Π as predicted by Langlands. Then the Ramanujan conjecture would follow from the Riemann hypothesis for the underlying geometric object which affords the Galois representation, as Deligne's original approach for Δ [19,30]. Another approach is to establish an injective correspondence respecting the Hecke operators from the representations of the target group to those of a group for which the Ramanujan conjecture is known to hold. A third approach is a variation of the analytic method proposed by Langlands in [35].

We distinguish two cases according to the base field K being a function field or a number field.

(I) K is a function field, that is, a finite extension of ${\mathbb{F}}_q(t)$. This is the case for which more results are known.

• —For G = GL_n, the conjecture was proved by Drinfeld [21] for n = 2 and L. Lafforgue [24] for n≥3 by establishing the Langlands conjecture for this group.
• —For G = H^×, the multiplicative group of a central simple division algebra H of dimension n² over K, by Jacquet–Langlands [32] for n = 2 and Badulescu & Roche [31] for n≥3. This is proved by establishing an injective correspondence from automorphic representations of H^× to those of GL_n preserving the associated L-functions.
• —Laumon–Rapoport–Stuhler [25] proved the conjecture for automprphic representations of H^× which are Steinberg at a place where H is unramified by constructing Galois representations.
• —For G = SO_2n+1, Sp_2n or SO_2n, the conjecture was established by Lomeli [34,43], resulting from the globally generic lifting from the classical groups to GL_N.

More generally, without the generic assumption on the representation Π of a classical group G, V. Lafforgue [33] showed that Π is tempered at one unramified place if and only if it is tempered at all unramified places. Recent work of Sawin and Templier [46] proved that Π is tempered at all unramified places under some conditions.

(II) K is a number field, i.e. a finite extension of $\mathbb{Q}$. The knowledge, in this case, is more limited.

• —For holomorphic cuspidal representations of G = GL₂ over $\mathbb{Q}$ by Deligne [19,30], Eichler–Shimura [22,27] and Deligne–Serre [20] as mentioned before, and over totally real fields by Brylinski–Labesse [16] and Blasius [15], proved by constructing Galois representations.
• —For G = H^×, where H is a quaternion algebra over totally real fields ramified at all archimedean places, by Jacquet–Langlands [32] as in function field case.
• —For G = GL_n over K with CM, Π algebraic regular and conjugate self-dual, by Clozel [47], Shin [28] and Chenevier–Harris [17] by constructing Galois representations.
• —For G = GL₂ over K with CM, Π algebraic regular of weight 0, by Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor and Thorne [36] using potential automorphy and invoking the analytic properties of the Rankin–Selberg L-functions.
• —For certain unitary group representations by Rogawski [26], Clozel [10], Clozel–Harris–Labesse [18], Harris–Taylor [23] and Shin [28] by constructing Galois representations, and by Evra–Parzanchevski [5] using functoriality.

For any number field and G = GL_n, Luo–Rudnick–Sarnak [48] gave the best known subconvexity bound for general n, and Blomer–Brumley [49] further improved this bound for n = 2, 3, 4.

3. Ramanujan graphs

In this section, we recall the spectral theory of regular graphs and provide different viewpoints of Ramanujan graphs. The reader is referred to the monograph [39, Ch. 4-5] for more details.

(a). Spectral theory of regular graphs

Let X be a finite connected undirected graph on n vertices. Assume that X is (q + 1)-regular, that is, each vertex has q + 1 neighbours. Identify the adjacency matrix A = A(X) of X with the adjacency operator on L²(X) which maps a function f on vertices of X to the function Af whose value at a vertex x is the sum of f(y) over the vertices y adjacent to x. The eigenvalues of A, called the spectrum of X, are real and satisfy

$$q+1={\lambda }_1>{\lambda }_2\ge \cdots \ge {\lambda }_n\ge -(q+1).$$

Furthermore, −(q + 1) is an eigenvalue if and only if X is a bipartite graph, in which case the eigenvalues are symmetric with respect to 0. Call ± (q + 1) trivial eigenvalues of X and the remaining eigenvalues non-trivial. When X is used as a communication network, its efficiency is measured by the expanding constant h(X) defined as

$$h(X)=\underset{Y\subset X,\hspace{0.17em}|Y|\le |X|/2}{min}\frac{|E(Y,X\setminus Y)|}{|Y|},$$

where $E(Y,X\setminus Y)$ is the collection of edges from Y to its complement in X. In general, it is very time-consuming to compute h(X). Tanner [50] and Alon-Milman [51] showed that h(X) is bounded from above and below in terms of the spectral gap λ₁ − λ₂ = q + 1 − λ₂. More precisely,

$$\frac{q+1-{\lambda }_2}{2}\le h(X)\le \sqrt{2(q+1)(q+1-{\lambda }_2)}.$$

Hence the larger the spectral gap, the more efficient the network. To determine how large the spectral gap can be, we study the distribution of the non-trivial eigenvalues λ.

Let {X_j} be a family of (q + 1)-regular graphs with |X_j| → ∞. The celebrated Alon-Boppana Theorem [3] asserts that

$$lim\underset{j\to \mathrm{\infty }}{inf}\hspace{0.17em}max\hspace{0.17em}\lambda \hspace{0.17em}\mathrm{of}\hspace{0.17em}{X}_j\ge 2\sqrt{q}.$$

The natural counter-statement

$$lim\underset{j\to \mathrm{\infty }}{sup}\hspace{0.17em}min\hspace{0.17em}\lambda \hspace{0.17em}\mathrm{of}\hspace{0.17em}{X}_j\le -2\sqrt{q},$$

does not hold unconditionally, but, as shown by Li [52] and Serre [53], it is valid under a mild assumption, such as X_j contains few short cycles of odd length, or the length of shortest odd cycles in X_j tends to infinity as j does. The interval $[-2\sqrt{q},\hspace{0.17em}2\sqrt{q}]$ is the spectrum of the (q + 1)-regular tree, the universal cover of connected (q + 1)-regular graphs.

(b). Ramanujan graphs

A (q + 1)-regular X is called a Ramanujan graph according to Lubotzky–Phillips–Sarnak [3] if its non-trivial eigenvalues λ satisfy

$$|\lambda |\le 2\sqrt{q},$$

or equivalently, for each non-trivial eigenvalue λ, the two roots of 1 − λu + qu² have the same absolute value q^−1/2. In view of the discussion in §3a, Ramanujan graphs are spectrally extremal, and they are spectrally optimal expanders.

(c). The Ihara zeta function of a graph

A cycle in a graph X is a closed loop with a starting point and an orientation. It is a geodesic if there are no backtracks regardless of the starting point. A closed geodesic cycle is primitive if it is not a repetition of a shorter cycle more than once. Two closed cycles on the same loop are equivalent if they differ by starting points and have the same orientation. Given a primitive closed geodesic cycle C, the collection of cycles equivalent to C is a prime [C] of X.

Motivated by the Selberg zeta function, Ihara [37] introduced the zeta function of X which counts the number N_n(X) of closed geodesic cycles of length n in X (see also [54]); it can also be expressed as a product over the primes of X:

$$Z(X;u)=\exp \left(\sum _{n\ge 1}\frac{N_n(X)}{n}{u}^n\right)=\prod _{\mathrm{prime}\hspace{0.17em}[C]}\frac{1}{1-u^{l(C)}}.$$

Here, l(C) is the length of a cycle C in [C].

(d). Ramanujan graphs and the Riemann hypothesis

In 1966, Ihara gave a closed form for the zeta function of a regular graph.

Theorem 3.1 ([37]) —

Let X be a finite connected undirected (q + 1)-regular graph. Then

$$Z(X;u)=\frac{{(1-u^2)}^{\chi (X)}}{det(I-Au+q{u}^{2}I)},$$

where χ(X) is the Euler characteristic of X and A = A(X) is the adjacency operator.

Since χ(X) < 0, Z(X;u) is in fact the reciprocal of a polynomial in u. Similar to the Riemann zeta function, Z(X;u) is said to satisfy the Riemann hypothesis if its non-trivial poles, namely, those arising from the non-trivial eigenvalues of X in $det(I-Au+q{u}^{2}I)$, all have the same absolute value q^−1/2. Hence one obtains another criterion for Ramanujan graphs in terms of zeta functions, namely X is a Ramanujan graph if and only if Z(X, u) satisfies the Riemann hypothesis.

4. Ramanujan complexes

Graphs are one-dimensional simplicial complexes. In this section, we consider their generalization to higher-dimensional simplicial complexes, arising as finite quotients of the buildings attached to PGL_n. Parallel to §3, we discuss the spectral theory and Ramanujan complexes. More details can be found in [4] and [39, Ch. 8].

(a). The Bruhat-Tits building of PGL_n

Let F be a non-archimedean local field with q elements in its residue field. It is a finite extension of ${\mathbb{Q}}_p$ or ${\mathbb{F}}_q((t))$. Denote by ${\mathcal{O}}_F$ its ring of integers. The Bruhat–Tits building attached to PGL_n(F) is a contractible (n − 1)-dimensional simplicial complex ${\mathcal{B}}_{n,F}$. Thus it is the universal cover of its finite quotients. Sometimes we denote ${\mathcal{B}}_{n,F}$ by its vertex set ${\mathrm{PGL}}_n(F)/{\mathrm{PGL}}_n({\mathcal{O}}_F)$ when there is no risk of confusion. The group PGL_n(F) acts transitively on the vertices of ${\mathcal{B}}_{n,F}$ by left transformation, and preserves simplices of all dimensions. The vertices of ${\mathcal{B}}_{n,F}$ are partitioned into n different types, marked by $\mathbb{Z}/n\mathbb{Z}$. Those of the same type are not adjacent so that the 1-skeleton of ${\mathcal{B}}_{n,F}$ is an n-partite graph. The Hecke algebra of PGL_n(F) is generated by n − 1 operators A_n,1, …, A_n,n−1 on $L^2({\mathrm{PGL}}_n(F)/{\mathrm{PGL}}_n({\mathcal{O}}_F))$, where A_n,i sends a function f on vertices to the function A_n,if whose value at a vertex x is the sum of f(y) over the neighbours y of x with type(y) = type(x) + i. As shown in [4], the operators A_n,1, …, A_n,n−1 can be simultaneously diagonalized. For each i, the spectrum Ω_n,i of A_n,i is known [55,56]. When n = 2, the building ${\mathcal{B}}_{2,F}$ is a (q + 1)-regular tree, and the Hecke operator A_2,1 is the adjacency operator on the tree.

(b). Spectral theory for finite quotients of ${\mathcal{B}}_{n,F}$ and Ramanujan complexes

The following higher-dimensional analogue of the Alon–Boppana theorem was proved by Li.

Theorem 4.1 ([4]) —

Let {X_j} be a family of finite quotients of ${\mathcal{B}}_{n,F}$ such that each X_j contains a ball isomorphic to a ball in ${\mathcal{B}}_{n,F}$ with radius approaching ∞ as j → ∞. Then for 1≤i≤n − 1, the closure of the set {eigenvalues of A_n,i on X_j:j≥1}contains Ω_n,i.

This spectral behaviour motivates the following definition of higher-dimensional analogue of Ramanujan graphs, called Ramanujan complexes. Let Γ be a discrete torsion-free cocompact subgroup of PGL_n(F) so that the quotient $X_{\mathit{\Gamma }}=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{n,F}$ is a finite (n − 1)-dimensional simplicial complex. We say X_Γ is a Ramanujan complex if, for each 1≤i≤n − 1, all non-trivial eigenvalues of A_n,i on X_Γ lie in the spectrum Ω_n,i of A_n,i on its universal cover ${\mathcal{B}}_{n,F}$. In other words, the non-trivial zeros of the polynomial

$$det(I-A_{n,1}u+\cdots +{(-1)}^i{q}^{i(i-1)/2}{A}_{n,i}{u}^i+\cdots +{(-1)}^n{q}^{n(n-1)/2}I{u}^n)$$

have the same absolute value q^−(n−1)/2. In terms of representations, this is equivalent to all non-trivial spherical representations in L²(Γ∖PGL_n(F)) being tempered. When n = 2, Ramanujan complexes are Ramanujan graphs. It follows from theorem 4.1 that Ramanujan complexes are spectrally extremal.

5. Various applications of the Ramanujan conjecture

(a). Explicit constructions of Ramanujan graphs and Ramanujan complexes

In real-world applications, it is time-consuming to compute the spectrum of a large graph. Hence explicitly constructed Ramanujan graphs ready for use are highly desirable. There are infinite families of explicitly constructed Ramanujan graphs and Ramanujan complexes based on the established cases of Ramanujan conjecture for generic cuspidal representations of certain inner forms of GL_n or unitary groups over $\mathbb{Q}$ or function fields discussed in §2c.

Theorem 5.1 —

For (i) $F={\mathbb{Q}}_p$ and n = 2, 3, and (ii) $F={\mathbb{F}}_q((t))$ and n≥ 2, there are infinitely many discrete torsion-free cocompact subgroups Γ of PGL_n(F) such that $X_{\mathit{\Gamma }}:=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{n,F}$ are finite Ramanujan complexes. When n = 2, they are Ramanujan graphs.

For case (i), this was done by Lubotzky–Phillips–Sarnak [3] and independently by Margulis [7] for n = 2 using the Ramanujan conjecture for classical weight-2 cusp forms by Eichler–Shimura [22,27]. The case n = 3 was done in [5] by Evra and Parzanchevski using the Ramanujan conjecture established for PGU₃ over $\mathbb{Q}$ with a special choice of the unitary group GU₃. (See §5b.) For case (ii), results on the Ramanujan conjecture for PGL_n and its inner forms over function fields were used. More precisely, this was done for n = 2 by Morgenstern [8] using Drinfeld [21], and for n≥2 by Li [4] using Laumon–Rapoport–Stuhler [25], Lubotzky-Samuels-Vishne [6] and Sarveniazi [9] using L. Lafforgue [24]. First [40] also constructed Ramanujan complexes by following the same method but using different inner forms of PGL_n. It remains to check whether different Ramanujan complexes are obtained by varying the inner forms.

We sketch below the main steps of the constructions originated from Lubotzky–Phillips–Sarnak [3]. Regard F as the completion K_v of a global field K at a nonarchimedean place v. For case (i), the field $K=\mathbb{Q}$ and v = p; denote by ∞ the archimedean place of $\mathbb{Q}$. For case (ii), $K={\mathbb{F}}_q(t)$ and v is a degree-1 place of K not equal to ∞, the place at infinity with 1/t as a uniformizer. Let H be a central simple division algebra of dimension n² over K, unramified at v and totally ramified at ∞. Write G for the group PH^×. The restricted product over all places w of K of local points G(K_w) of G at w with respect to the subgroups of integral points $G({\mathcal{O}}_w)$ at non-archimedean places w is the group $G({\mathbb{A}}_K)$ of adelic points on G. Let $\mathcal{K}$ be an open compact subgroup of $\prod _{w\ne v,\mathrm{\infty }}G({\mathcal{O}}_w)$, and

$$\mathit{\Gamma }={\mathit{\Gamma }}_{\mathcal{K}}:=G(K)\cap G(K_v)G(K_{\mathrm{\infty }})\mathcal{K}.$$

Here G(K), the group of global K-points on G, is diagonally imbedded in $G({\mathbb{A}}_K)$. The group ${\mathit{\Gamma }}_{\mathcal{K}}$ consists of those global points which, outside v and ∞, fall in $\mathcal{K}$. It imbeds in G(K_v) as a discrete cocompact subgroup. Shrinking $\mathcal{K}$ if necessary, we may assume ${\mathit{\Gamma }}_{\mathcal{K}}$ is also torsion-free. Since H is unramified at v, we have ${\mathcal{B}}_{n,F}=G(K_v)/G({\mathcal{O}}_v)$, and its quotient by ${\mathit{\Gamma }}_{\mathcal{K}}$ is a finite (n − 1)-dimensional simplicial complex. By strong approximation theorem, we can write

$$\begin{array}{rl}{X}_{{\mathit{\Gamma }}_{\mathcal{K}}}& ={\mathit{\Gamma }}_{\mathcal{K}}\mathrm{\setminus }{\mathcal{B}}_{n,F}={\mathit{\Gamma }}_{\mathcal{K}}\mathrm{\setminus }G(K_v)/G({\mathcal{O}}_v)\\ \hspace{0.17em}& =G(K)\mathrm{\setminus }G({\mathbb{A}}_K)/G(K_{\mathrm{\infty }})G({\mathcal{O}}_v)\mathcal{K},\end{array}$$

so that the functions on $X_{{\mathit{\Gamma }}_{\mathcal{K}}}$ can be interpreted as automorphic forms on $G({\mathbb{A}}_K)$. By choosing $\mathcal{K}$ carefully, especially when n is a composite (cf. [57]), we can ensure that those automorphic forms orthogonal to the constant functions and their twists by nth roots of unity, under the Jacquet–Langlands correspondence established in [31,32], correspond to generic cuspidal representations of ${\mathrm{GL}}_n({\mathbb{A}}_K)$ so that the Ramanujan conjecture holds and, consequently, the complex $X_{{\mathit{\Gamma }}_{\mathcal{K}}}$ is Ramanujan. We obtain an infinite family by varying the group $\mathcal{K}$.

The infinite families of Ramanujan graphs constructed this way have valency k = q + 1 for a prime power q. It has been an open question to know whether the same holds for any valency k≥3. Using methods in combinatorics and analysis, in 2015 Marcus–Spielman–Srivastava [58] proved the existence of infinitely many k-regular bipartite Ramanujan graphs for any k≥3 by showing that a k-regular bipartite Ramanujan graph has a twofold unramified cover which is also Ramanujan. The question still remains open for non-bipartite k-regular Ramanujan graphs.

(b). Other applications of the Ramanujan conjecture

The group ${\mathit{\Gamma }}_{\mathcal{K}}$ defined in §5a consists of certain points of G(K) integral outside two places v and ∞ of K, and the Ramanujan graph or complex constructed from it arises from imbedding ${\mathit{\Gamma }}_{\mathcal{K}}$ in the local group G(K_v) at a nonarchimedean place v. When the field $K=\mathbb{Q}$, the place ∞ is archimedean and the group $G(K_{\mathrm{\infty }})=G(\mathbb{R})$ has very different properties than its non-archimedean counterparts. There are interesting applications obtained from imbedding ${\mathit{\Gamma }}_{\mathcal{K}}$ in $G(\mathbb{R})$ which we now explain. Assume that v = p for an odd prime p so that we may choose H to be the Hamiltonian quaternion algebra over $\mathbb{Q}$ and G = PH^×. The norm 1 elements in $H^{\times }(\mathbb{R})$ is SU(2), which projects onto SO(3) and acts on the sphere S².

In their paper [3], Lubotzky–Phillips–Sarnak showed that the vertices of the (p + 1)-regular tree ${\mathcal{B}}_{2,{\mathbb{Q}}_p}$ are represented by a group Λ_p generated by a set S_p consisting of p + 1 integral quaternions with norm p which give rise to the Hecke operator at p. Normalize the elements in Λ_p so that all elements have norm 1. Imbed it in PU(2) then project it to SO(3). Lubotzky–Phillips–Sarnak [11] proved that the image of Λ_p is dense in SO(3). Moreover, the orbit of any point x in S² under Λ_p is uniformly distributed and can be used to approximate integrals over S². Similar results hold for ${\mathit{\Gamma }}_{\mathcal{K}}$ [12].

As shown by Parzanchevski & Sarnak [14], the image of Λ_p in PU(2) distributes uniformly, and S_p is a Golden Gates Set in quantum computation. For a survey of this topic, the reader is referred to [13].

The Ramanujan conjecture for holomorphic cusp forms of weight 2 is behind the above applications. There are other choices for $\mathcal{K}$. For instance, we may choose $\mathcal{K}$ to be an open subgroup of $\prod _{w\ne v,2}G({\mathcal{O}}_w)$ and ${\mathit{\Gamma }}_{\mathcal{K}}=G(\mathbb{Q})\cap G({\mathbb{Q}}_2)G({\mathbb{Q}}_v)\mathcal{K}$, as in [11,12]. In this case, the result of Deligne on Ramanujan conjecture for holomorphic cusp forms of weight greater than or equal to 2 is used.

The applications below rely on the validity of Ramanujan conjecture for certain unitary groups.

Clozel [10] extended the result in [11] from S² to S²ⁿ⁻¹ using the unitary group U_n over $\mathbb{Q}$ obtained from a central simple division algebra of dimension n² over an imaginary quadratic extension of $\mathbb{Q}$. Very recently, Evra & Parzanchevski [5] took G = PGU₃ over $\mathbb{Q}$, where GU₃ preserves the usual Hermitian form on three-dimensional vector space over $\mathbb{Q}(\sqrt{-1})$ up to scalars in ${\mathbb{Q}}^{\times }$. At ∞, the group $G(\mathbb{R})$ is compact. They extended the results in [3,14] to G. More precisely, they constructed an infinite family of ${\mathit{\Gamma }}_{\mathcal{K}}$ such that the finite quotients ${\mathit{\Gamma }}_{\mathcal{K}}\mathrm{\setminus }G({\mathbb{Q}}_p)/G({\mathbb{Z}}_p)$ are two-dimensional Ramanujan complexes at unramified primes p≡1 mod 4, and finite biregular bipartite Ramanujan graphs at unramified primes p≡3 mod 4. At ∞ they obtain Golden Gate Sets on PU(3) from suitable global integral points of G.

6. Ramanujan, strongly Ramanujan and the Riemann hypothesis

We saw in §3d that Ramanujan graphs can be characterized by the associated zeta functions satisfying the Riemann hypothesis. In this section, we address the corresponding question for higher-dimensional Ramanujan complexes.

(a). Zeta functions of finite quotients of ${\mathcal{B}}_{n,F}$

Let $X_{\mathit{\Gamma }}=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{n,F}$ be a finite quotient of ${\mathcal{B}}_{n,F}$. In the building ${\mathcal{B}}_{n,F}$, there are many apartments, which are (n − 1)-dimensional Euclidean spaces tiled by (n − 1)-dimensional simplicies, called chambers; and the building is obtained by gluing together the apartments along certain chambers. The Euclidean metric on the apartments induces a metric on the building since any two points in the building are contained in a common apartment. The quotient X_Γ inherits the metric on the building so that geodesic cycles on X_Γ as well as primes can be defined similar to graphs, except that we restrict ourselves to those contained in the 1-skeleton of X_Γ. Therefore, geodesic cycles in X_Γ can be lifted to straight lines contained in the 1-skeleton of ${\mathcal{B}}_{n,F}$ regardless of the starting points. Like the zeta function of graphs, the zeta function of X_Γ counts the number N_m(X_Γ) of closed geodesic cycles of length m in X_Γ, defined by

$$Z(X_{\mathit{\Gamma }},u)=\exp \left(\sum _{m\ge 1}\frac{N_m(X_{\mathit{\Gamma }})u^m}{m}\right)=\prod _{\mathrm{prime}\hspace{0.17em}[C]}\frac{1}{1-u^{l(C)}}.$$

This is the zeta function for one-dimensional closed geodesic cycles in X_Γ. Starting from these, we recursively define higher dimensional closed geodesic cycles and their associated zeta functions. More precisely, for 2≤r≤n − 1, an r-dimensional geodesic cycle in X_Γ is obtained by taking a directed geodesic closed path of adjacent r-dimensional simplicies in X_Γ and gluing together each adjacent pair of r-dimensional simplicies along the (r − 1)-dimensional simplex they share such that the boundary of the r-dimensional cycle consists of closed (r − 1)-dimensional geodesic cycle(s) in X_Γ. We similarly define r-dimensional primes and the zeta function Z_r(X_Γ, u) counting closed r-dimensional geodesic cycles in X_Γ. See [39, Ch. 8] for more detail.

(b). A zeta identity

Hashimoto [59] showed that the zeta function of a graph can be expressed in terms of the adjacency operator of directed edges. This can be extended to higher dimensional complexes. For a finite quotient X_Γ of the building ${\mathcal{B}}_{n,F}$ and 1≤r≤n − 1, the zeta function Z_r(X_Γ, u) introduced in §6a can be expressed in terms of finitely many operators describing a more refined adjacency relations among r-dimensional simplicies. We explain this for the case n = 3. See [38] for details. As explained in §4a, vertices in ${\mathcal{B}}_{3,F}$ have 3 types. Hence there are two types of directed edges in ${\mathcal{B}}_{3,F}$: for j = 1, 2, those of type j are from type i vertices to type i + j vertices for all $i\in \mathbb{Z}/3\mathbb{Z}$. There is a parahoric operator L_E describing adjacent type 1 edges which do not share a chamber. Type 2 edges are the opposite of the type 1 edges, described by the adjoint of L_E, denoted L^t_E. Using them one can show that the zeta function Z(X_Γ, u) is a rational function in u, with a closed form expression

$$Z(X_{\mathit{\Gamma }},u)=\frac{1}{det(I-L_{E}u)det(I-L_E^t{u}^2)}.$$

Likewise, there is an Iwahori–Hecke operator L_B describing the adjacency of directed chambers in the building ${\mathcal{B}}_{3,F}$ such that

$$Z_2(X_{\mathit{\Gamma }},u)=\frac{1}{det(I-L_{B}u)}.$$

Analogous to theorem 3.1, the zeta function Z(X_Γ, u) is related to the Hecke operators A_3,1 and A_3,2 of PGL₃(F) as follows.

Theorem 6.1 —

For n = 3 and a finite quotient $X_{\mathit{\Gamma }}=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{3,F}$ of ${\mathcal{B}}_{3,F},$ the zeta function Z(X_Γ, u) is a rational function with two closed form expressions

$$\begin{array}{rl}Z(X_{\mathit{\Gamma }},u)& =\frac{1}{det(I-L_{E}u)det(I-L_E^t{u}^2)}\\ \hspace{0.17em}& =\frac{{(1-u^3)}^{\chi (X_{\mathit{\Gamma }})}}{det(I-A_{3,1}u+q{A}_{3,2}{u}^2-q^3{u}^{3}I)det(I+L_{B}u)},\end{array}$$

where χ(X_Γ) is the Euler characteristic of X_Γ, L_E is a parahoric operator, and L_B is an Iwahori–Hecke operator as above.

There are three proofs for this identity, representing different interpretations of the operators involved: [38] is algebraic and combinatorial, [60] is representation-theoretic, and [61] is cohomological.

In their unpublished work, Kang and J-K Yu obtained a similar identity for n≥4 involving Z_r(X_, u) for 1≤r≤n − 1.

(c). Ramanujan and strongly Ramanujan complexes

Motivated by the Ramanujan conjecture, a Ramanujan complex arising as a finite quotient of the building ${\mathcal{B}}_{n,F}$ is defined in §4b in terms of the Hecke operators, or equivalently, adjacency operators on vertices. To see this from the viewpoint of zeta functions as we did in §3d for graphs, as shown in theorem 6.1, for n≥3 other operators also appear in the zeta function. In fact, adjacency operators on r-dimensional simplicies for each 0≤r≤n − 1 would appear. It is natural to ask whether the temperedness condition holds for each r and how they affect each other.

For n = 3, Kang–Li–Wang proved in [60] that a finite quotient X_Γ of ${\mathcal{B}}_{3,F}$ is a Ramanujan complex if and only if the non-trivial spherical representations in L²(Γ∖PGL₃(F)) are tempered (by definition), which is equivalent to the non-trivial parahoric representations in L²(Γ∖PGL₃(F)) being tempered, which in turn is equivalent to the non-trivial Iwahori-spherical representations in L²(Γ∖PGL₃(F)) being tempered. We note that a spherical representation is parahoric, and a parahoric representation is Iwahori-spherical. This holds true in general: among the representations in L²(Γ∖PGL_n(F)) which occur in the zeta function of the finite quotient $X_{\mathit{\Gamma }}=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{n,F}$, spherical representations contain non-zero vectors fixed by the maximal compact subgroup of PGL_n(F), Iwahori-spherical representations contain non-zero vectors fixed by the Iwahori subgroup of the maximal compact subgroup, and other relevant representations contain non-zero vectors fixed by a subgroup sandwiched by the maximal compact and the Iwahori subgroups.

Since Ramanujan complexes are defined in terms of spherical representations, call a finite quotient X_Γ of ${\mathcal{B}}_{n,F}$ strongly Ramanujan if the non-trivial Iwahori-spherical representations in L²(Γ∖PGL_n(F)) are tempered.

First [40] has verified that all known Ramanujan complexes constructed from local fields F with positive characteristic discussed before are strongly Ramanujan.

To date, we know that Ramanujan and strongly Ramanujan are equivalent for n = 2 by Hashimoto [59], and n = 3 by [60]; it is unknown for n≥4.

(d). Strongly Ramanujan and the Riemann hypothesis

Given a finite quotient $X_{\mathit{\Gamma }}=\mathit{\Gamma }\mathrm{\setminus }{\mathcal{B}}_{n,F}$ and 1≤r≤n − 1, the zeta function Z_r(X_Γ, u) is said to satisfy the Riemann hypothesis if all non-trivial poles have specifically prescribed absolute values, as given in [41].

In [41] Kang showed that if X_Γ is strongly Ramanujan, then all Z_r(X_Γ, u) satisfy the Riemann hypothesis. The converse is not always true. He gave a counterexample for n = 4. On the other hand, if all representations in L²(Γ∖PGL_n(F)) are generic, then he proved that the validity of the Riemann hypothesis for all Z_r(X_Γ, u) and strongly Ramanujan are equivalent.

Selberg [45] pioneered the viewpoint that equivalent classes of geodesic cycles in a compact Riemann surface play the role of prime numbers in $\mathbb{Q}$. Their distribution, described by the prime geodesic theorem similar to the celebrated prime number theorem for primes in $\mathbb{Q}$, was proved by Sarnak in his thesis [62]. The combinatorial counterpart of the prime geodesic theorem is expected to hold for the primes of X_Γ in each dimension 1≤r≤n − 1. If, furthermore, X_Γ is strongly Ramanujan, then one would also have a good estimate of the error term in the prime geodesic theorem for r-dimensional primes, just like the role of the original Riemann hypothesis in the distribution of prime numbers. To-date, this is proved for n = 2 by Hashimoto [63], and n = 3 by Li-Matias [64] (cf. [39, Theorem 8.8]). The reader is referred to [39] for the development and current status of the prime geodesic theorems.

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