Group Based Combinatorial Zeta Functions Fourth International Workshop on Zeta Functions in Algebra and Geometry Bielefeld, June 1, 2017 Winnie Li Pennsylvania State University 1
Selberg zeta functions • SL 2 ( R ) acts on H by fractional linear transformation. • Γ: discrete torsion-free cocompact subgroup of SL 2 ( R ) • X Γ = Γ \ H = Γ \ PGL 2 ( R ) /PO 2 ( R ) is a compact Riemann surface with fundamental group π 1 ( X Γ , pt ) ∼ = Γ. • Count geodesic cycles on X Γ up to equivalence, i.e., ignoring the starting point • A cycle is primitive if it is not obtained by repeating a cycle (of shorter length) more than once. • Equivalence classes [ C ] of primitive geodesics C are the ”primes” of X Γ . 2
The Selberg zeta function (1956) counts equiv. classes of geodesic cycles in X Γ : (1 − e − l ( C )( s + k ) ) � � Z ( X Γ , s ) = k ≥ 0 [ C ] prime (1 − e − l ( γ )( s + k ) ) � � = for ℜ s > 1 k ≥ 0 [ γ ] primitive because there is a length preserving bijection between primes [ C ] and conjugacy classes [ γ ] of primitive elements γ in Γ. 3
The Laplacian operator ∆ acts on L 2 ( X Γ ). It has discrete spec- trum 0 = λ 0 < λ 1 ≤ λ 2 ≤ · · · . The determinant of the Laplacian det(∆ − s (1 − s )), defined using the spectral zeta function, is formally equal to � ( λ n − s (1 − s )) . n ≥ 0 Sarnak, Voros (1987 independently): det(∆ − s (1 − s )) = Z ( X Γ , s )( e E + s (1 − s ) Γ 2 ( s ) 2 Γ( s ) (2 π ) s ) 2 g − 2 , where g is the genus of X Γ , and E is a constant. RH: Selberg showed that the nontrivial zeros of Z ( X Γ , s ) are � 1 1 2 ± 4 − λ n , n ≥ 1. 4
The Ihara zeta function of a graph • Let F be a nonarch. local field with the ring of integers O F and q elements in the residue field. Eg. F = Q p , F q (( t )). • Let Γ be a discrete torsion-free cocompact subgroup of PGL 2 ( F ) and X Γ = Γ \ PGL 2 ( F ) / PGL 2 ( O F ) = Γ \T . • Ihara defined the zeta function 1 � Z ( X Γ , u ) = 1 − u l ( γ ) , | u | << 1 . [ γ ] primitive • Serre observed that X Γ is a finite ( q +1)-regular graph, Z ( X Γ , u ) counts geodesic cycles in X Γ , and the definition works for all finite graphs. 5
• X : connected undirected finite graph • Primes of X are equivalence classes [ C ] of primitive geodesic cycles C in X . The Ihara zeta function of X counts the number N n ( X ) of closed geodesic cycles of length n : � � � N n ( X ) 1 u n � Z ( X ; u ) = exp = 1 − u l ( C ) n n ≥ 1 [ C ] prime 1 � = 1 − u l ( γ ) [ γ ] primitive where γ lies in the fundamental group π 1 ( X, pt ) of X . 6
Properties of zeta functions of regular graphs Ihara (1968): Let X be a finite ( q +1)-regular graph on n vertices. Then its zeta function Z ( X, u ) is a rational function of the form (1 − u 2 ) χ ( X ) Z ( X ; u ) = det( I − Au + qu 2 I ) , where χ ( X ) = # V − # E = − n ( q − 1) / 2 is the Euler characteristic of X and A = A ( X ) is the adjacency matrix of X . The eigenvalues of A are real: q + 1 = λ 1 > λ 2 ≥ · · · λ n ≥ − ( q + 1) . Hence det( I − Au + qu 2 I ) = (1 − λ j u + qu 2 ) . � 1 ≤ j ≤ n 7
Connection between zeta functions of graphs and curves The zeta function of a smooth irred. proj. curve V over F q counts V ( F q n ). It is a rational function P ( V, u ) Z ( V, u ) = (1 − u )(1 − qu ) . The modular curve X 0 ( N ) = Γ 0 ( N ) \ H ∗ is defined over Q , and has good reduction at p ∤ N . When N is a prime ≡ 1 mod 12, for each p � = N , there is a subgroup Γ N of PGL 2 ( Q p ) such that the zeta of the graph X Γ N is closely related to the zeta of X 0 ( N ) / F p , namely det( I − A ( X Γ N ) u + pu 2 I ) = P ( X 0 ( N ) / F p , u ) . 1 − ( p + 1) u + pu 2 8
Riemann Hypothesis and Ramanujan graphs • The trivial eigenvalues of X are ± ( q + 1), of multiplicity ≤ 1. The nontrivial eigenvalues λ satisfy q + 1 > λ > − ( q + 1). • Z ( X, u ) satisfies RH if the poles of Z ( X, u ) from nontrivial eigenvalues of X all have the same absolute value q − 1 / 2 iff all nontrivial eigenvalues λ satisfy the bound | λ | ≤ 2 √ q. Such a graph is called a Ramanujan graph . • Z ( X, u ) satisfies RH if and only if X is a Ramanujan graph. • [ − 2 √ q, 2 √ q ] is the spectrum of the ( q + 1)-regular tree, the universal cover of X . • Ramanujan graphs are spectrally extremal by Alon-Boppana. 9
Hashimoto’s expression Endow two orientations on each edge of a finite graph X . The neighbors of u → v are the directed edges v → w with w � = u . Associate the (directed) edge adjacency matrix T . Hashimoto (1989): N n ( X ) = Tr T n so that 1 Z ( X, u ) = det( I − Tu ) . Combine both expressions for Z ( X, u ) to get the identity (1 − u 2 ) χ ( X ) 1 Z ( X, u ) = det( I − Au + qu 2 I ) = det( I − Tu ) . This is the discrete analog of the relation between quantum and classical resonances. 10
Artin L -functions for graphs For a finite-dimensional unitary representation ρ of π 1 ( X, pt ), we associate the Artin L -function 1 � L ( X, ρ, u ) = . det( I − ρ ( γ ) u l ( γ ) ) [ γ ] primitive Ihara, Hashimoto, Stark-Terras: L ( X, ρ, u ) − 1 is a polynomial. Further, there are matrices A ρ and T ρ such that (1 − u 2 ) χ ( X ) deg ρ 1 L ( X, ρ, u ) = det( I − A ρ u + qu 2 I ) = det( I − T ρ u ) . 11
Artin L -functions for Riemann surfaces Back to the Riemann surface X Γ = Γ \ H considered by Selberg. For a finite-dim’l unitary representation ρ of Γ, Selberg defined det(1 − ρ ( γ ) e − l ( γ )( s + k ) ) . � � Z ( X Γ , ρ, s ) = k ≥ 0 [ γ ] primitive This is an Artin L -function. As Pohl explained, for Γ cofinite, it can be expressed in terms of a transfer operator Z ( X Γ , ρ, s ) = det( I − L s,ρ ) , which is parallel to the Hashimoto expression. The Ihara (spectral) expression for Z ( X Γ , ρ, s ) is known only for ρ trivial. 12
Distribution and density of primes in a graph Let α : Y → X be a finite unramified Galois cover of X with Ga- lois group G . Each prime [ C ] of X → primitive [ γ C ] in π 1 ( X, pt ) → Frob [ C ] := [ γ C ] in G . Given a conjugacy class C of G , let S C = { primes [ C ] of X : [ γ C ] = C} . Assume π 1 ( X, pt ) has rank ≥ 2. Cebotarev Density Theorem [Hashimoto, Stark-Terras] The Frob [ C ] are uniformly distributed in G w.r.t. Dirichlet density, i.e., for each conj. class C of G , [ C ] ∈ S C u l ( C ) � − . [ C ] p rime u l ( C ) → |C| 1 as u → | G | λ ( X ) � 1 Here λ ( X ) is the radius of convergence of Z ( X, u ). 13
Theorem [Huang-L] The natural density for S C given by |{ [ C ] ∈ S C : l ( C ) ≤ x }| lim x →∞ |{ primes [ C ] : l ( C ) ≤ x }| exists (and hence equal to |C| | G | ) if and only if δ ( Y ) = δ ( X ) . Here δ ( X ) = gcd primes [C] l ( C ), which corresponds to topolog- ical entropy for manifolds. Stark-Terras: δ ( Y ) = δ ( X ) or 2 δ ( X ) and both cases do occur. The zeta functions of Y and X are related by L ( X, ρ, s ) deg ρ . � Z ( Y, u ) = Z ( X, u ) n ontrivial ρ ∈ I rr ( G ) The proofs of both theorems use the analytic behavior of L ( X, ρ, u ). 14
Hashimoto showed: (1) If ρ has degree ≥ 2, then L ( X, ρ, u ) is holo. on the closed disk | u | ≤ 1 /λ ( X ); (2) If ρ has degree 1, then L ( X, ρ, u ) is holo. on the open disk | u | < 1 /λ ( X ), Z ( X, u ) has a simple pole at u = 1 /λ ( X ), and other L ( X, ρ, u ) are holo. there. (1) & (2) imply Cebotarev in Dirichlet density. Have λ ( X ) = λ ( Y ). By rescaling, may assume δ ( X ) = 1. Cebotarev in natural density holds iff L ( X, ρ, u ) are holo. on the circle | u | = 1 /λ ( X ) for all nontrivial ρ , which is equiv. to δ ( Y ) = δ ( X ) by using Prime Geodesic Theorem for graphs and Tauberian theorem. 15
The Bruhat-Tits building B 3 attached to SL 3 ( F ) • The vertices of the building B 3 of SL 3 ( F ) are equivalence classes of rank 3 lattices (i.e. O F -modules) in F 3 . • 3 vertices form a chamber (i.e., a triangle) if they are repre- sented by lattices L 1 , L 2 , L 3 satisfying L 1 � L 2 � L 3 � ̟L 1 . • The group PGL 3 ( F ) acts transitively on vertices and preserves adjacency. • Each vertex has two types of neighbors/out edges, described by two Hecke operators A 1 , A 2 . • Each edge has a direction of type 1, its opposite has type 2; 16
adjacency of type 1 (resp. type 2) edges given by the parahoric operator L E (resp. L t E ). • The Iwahori-Hecke operator L B describes adjacency of directed chambers. β α type 1 edges type 2 edges 17
Zeta functions for finite quotients of B 3 Take a discrete torsion-free cocompact subgroup Γ of PGL 3 ( F ) with ord det Γ ⊂ 3 Z . The quotient X Γ = Γ \ PGL 3 ( F ) /PGL 3 ( O F ) = Γ \B 3 is a finite 2-dimensional complex. The zeta function of X Γ counts the number N n ( X Γ ) of geodesic cycles of length n contained in the 1 -skeleton of X Γ : N n ( X Γ ) u n 1 � � � Z ( X Γ , u ) = exp( ) = 1 − u l ( C i ) i , n n ≥ 1 1 ≤ i ≤ 2 [ C i ] where [ C i ] runs through primes using only type- i edges. 18
Recommend
More recommend
