The Pennsylvania State University The Graduate School Eberly College of Science

PRIME GEODESIC THEOREMS FOR COMPLEXES OF PGL3(F ) AND

PGSp4(F )

A Dissertation in Mathematics by João Correia Matias

© 2018 João Correia Matias

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

December 2018 The dissertation of João Correia Matias was reviewed and approved∗ by the following:

Wen-Ching Winnie Li Distinguished Professor of Mathematics Dissertation Advisor, Chair of Committee

A. Kirsten Eisenträger Professor of Mathematics

Yuri Zarhin Professor of Mathematics

Martin Fürer Professor of Computer Science and Engineering

Carina Curto Associate Professor of Mathematics Co-Director of Graduates Studies

∗Signatures are on file in the Graduate School.

ii Abstract

In all history of Number Theory, prime numbers have been one of the greatest sources of questions, displaying many interesting properties. The distributions of prime numbers, and prime ideals of number fields are given by two famous results in Number Theory: the Prime Number Theorem and the Chebotarev Density Theorem. Moreover, prime numbers exhibit deep connections with the Riemann zeta function and the Riemann Hypothesis. Starting in the twentieth century, mathematicians have studied many objects analogous to the ones mentioned above in other mathematical contexts, including: algebraic varieties, Riemann surfaces, graphs and simplicial complexes. In this dissertation, we focus on studying closed geodesic paths in combinatorial complexes. Now, let F be a nonarchimedean local field. The equivalence classes of primitive closed geodesic paths, under cyclic rotation, in surfaces or multicomplexes are called primes. We describe the distribution of the primes in finite multicomplexes covered by the buildings associated to PGL3(F ) or PGSp4(F ). In the case of Ramanujan complexes covered by the building associated to PGL3(F ), we give more precise estimates, including bounds for the error term. Additionally, we study finite coverings of these multicomplexes, in which case the groups of automorphisms are analogous to the Galois groups of extensions of number fields. As a consequence, we derive results analogous to the Chebotarev Density Theorem in various contexts. Lastly, and throughout the dissertation, we define Zeta functions and L-functions attached to such multicomplexes. We show that they are rational functions and relate them to certain adjacency operators.

iii Table of Contents

List of Figures viii

Acknowledgments ix

Chapter 1 Introduction 1 1.1 Zeta functions and the distribution of primes – a historical review . . . . 1 1.1.1 Prime numbers and the Riemann zeta function ...... 2 1.1.2 Rational points and Zeta functions of algebraic varieties ...... 2 1.1.3 Primitive closed geodesics and the Selberg zeta function ...... 3 1.1.4 Primitive geodesic cycles and the Ihara zeta function ...... 4 1.2 Ramanujan graphs ...... 5 1.3 Primes and Zeta functions for finite quotients of buildings ...... 6 1.4 Chebotarev Density Theorem ...... 8 1.4.1 In Number Theory ...... 8 1.4.2 For Riemann surfaces ...... 8 1.4.3 For graphs ...... 9 1.5 Main results ...... 10

Chapter 2 Preliminaries 12 2.1 Graphs ...... 12 2.2 Simplicial complexes and multicomplexes ...... 13 2.3 Primes in finite quotients of X ...... 14 2.3.1 Zeta functions and L-functions ...... 16 2.3.2 Distribution of primes of given dimension and type ...... 19 2.3.3 Finite coverings of multicomplexes ...... 21 2.3.4 Cumulative distribution of primes ...... 24

Chapter 3 Analysis of PGL3(F ) 26 3.1 The PGL3(F ) building ...... 26

iv 3.1.1 Vertices ...... 26 3.1.2 Edges ...... 27 3.1.3 Pointed chambers ...... 27 3.1.4 Apartments ...... 28 3.2 Finite quotients of X ...... 29 3.3 Type 1 edge zeta function ...... 29 3.3.1 Adjacency operator for type 1 edges ...... 29 3.3.2 Edge zeta function ...... 30 3.3.3 Edge L-functions ...... 31 3.3.4 Main results for 1-dimensional primes of type 1 ...... 31 3.4 Chamber zeta function ...... 32 3.4.1 Adjacency of chambers ...... 32 3.4.2 Adjacency operator for directed chambers ...... 33 3.4.3 Chamber zeta function ...... 34 3.4.4 Chamber L-functions ...... 35 3.4.5 Main results for 2-dimensional primes of type 1 ...... 35 3.5 Proofs of Theorems 11 and 12 ...... 36 3.5.1 Edge representatives ...... 36 3.5.2 Successor map ...... 38 3.5.3 Properties of AE,Γ ...... 39 3.5.4 Proof of Theorem 11 ...... 41 3.5.5 Proof of Theorem 12 ...... 41 3.6 Proofs of Theorems 14 and 15 ...... 42 3.6.1 Pointed chamber representatives ...... 42 3.6.2 Successor map ...... 43 3.6.3 Properties of AB,Γ ...... 44 3.6.4 Proof of Theorem 14 ...... 46 3.6.5 Proof of Theorem 15 ...... 46 3.7 Error terms of Ramanujan complexes ...... 47

Chapter 4 Analysis of PGSp4(F ) 49 4.1 The symplectic group ...... 49 4.1.1 Definitions ...... 49 4.1.2 Lattices ...... 50 4.1.3 Arithmetic of GSp4(F ) ...... 52 4.2 The Bruhat-Tits building associated to Sp4(F ) ...... 53 4.2.1 Vertices, edges and chambers ...... 53 4.2.2 Types of vertices and edges ...... 53 4.2.3 Types of chambers ...... 55 4.2.4 Apartments ...... 57 4.3 Finite quotients of X ...... 58

v 4.4 Type 1 edge zeta functions ...... 59 4.4.1 Adjacency operator for type 1 edges ...... 59 4.4.2 Type 1 edge zeta function ...... 60 4.4.3 Type 1 edge L-functions ...... 60 4.4.4 Main results for 1-dimensional primes of type 1 ...... 60 4.5 Type 2 edge zeta function ...... 61 4.5.1 Adjacency of type 2 edges ...... 61 4.5.2 Type 2 edge zeta functions ...... 63 4.5.3 Long type 2 edges ...... 63 4.5.4 Type 2 edge L-functions ...... 65 4.5.5 Main results for 1-dimensional primes of type 2 ...... 65 4.6 Spin type chamber zeta function ...... 66 4.6.1 Adjacency operator ...... 66 4.6.2 Spin type chamber zeta function ...... 68 4.6.3 Spin type chamber L-functions ...... 69 4.6.4 Main results for 2-dimensional primes of spin type ...... 69 4.7 Standard type chamber zeta function ...... 70 4.7.1 Translation of chambers ...... 70 4.7.2 Standard type chamber zeta function ...... 72 4.7.3 Standard type chamber L-functions ...... 73 4.7.4 Main results for 2-dimensional primes of standard type ...... 73 4.8 Proofs of Theorems 19 and 20 ...... 74 4.8.1 Representatives of type 1 edges ...... 75 4.8.2 Successor map ...... 76

4.8.3 Properties of AP1,Γ ...... 78 4.8.4 Proof of Theorem 19 ...... 80 4.8.5 Proof of Theorem 20 ...... 80 4.9 Proofs of Theorems 22 and 23 ...... 81 4.9.1 Representatives of long type 2 edges ...... 81 s 0 4.9.2 Successor map P2 ...... 82 A 0 4.9.3 Properties of P2,Γ ...... 85 4.9.4 Proof of Theorem 22 ...... 88 4.9.5 Proof of Theorem 23 ...... 88 4.10 Proofs of Theorems 26 and 27 ...... 89 4.10.1 Representatives of spin type chambers ...... 89

4.10.2 Successor map sI1 ...... 90

4.10.3 Properties of AI1,Γ ...... 91 4.10.4 Proof of Theorem 26 ...... 93 4.10.5 Proof of Theorem 27 ...... 94 4.11 Proofs of Theorems 30 and 31 ...... 94 4.11.1 Standard type chamber representatives ...... 94

4.11.2 Successor map sI2 ...... 95

vi 4.11.3 Properties of AI2,Γ ...... 97 4.11.4 Proof of Theorem 30 ...... 99 4.11.5 Proof of Theorem 31 ...... 100

Appendix Perron-Frobenius Theorem for nonnegative matrices 101

Bibliography 103

vii List of Figures

3.1 Part of the standard apartment, with labelled vertices...... 28

3.2 Some edges in the standard apartment...... 30

3.3 A gallery in the standard apartment, containing F0...... 34

4.1 Some of the chambers in the standard apartment...... 56

4.2 Part of the standard apartment, with labelled vertices...... 58

4.3 Some edges in the standard apartment...... 62

4.4 Part of a spin type geodesic gallery in the standard apartment...... 68

4.5 Part of a geodesic gallery in the standard apartment...... 72

viii Acknowledgments

First and foremost, I would like to thank my advisor, Professor Winnie Li, for suggesting me this project and for her guidance throughout graduate school. Her support was crucial to complete this dissertation and I am most grateful for her endless patience during our regular discussions. She also questioned each of my results and listened to my explanations with interest, pushing me constantly to do better. Finally, I am also thankful to her for always giving me a great example both as a person and as a mathematician. I would also like to thank Professors Kirsten Eisenträger, Martin Fürer, Mihran Papikian and Yuri Zahrin, for serving in my Ph.D. committee, for the comments they provided to this dissertation, and for all that they taught me both in their classes and by their example. I was very fortunate for coming upon amazing people at Penn State that I will cherish for the rest of my life. These include, in particular, everyone in the Mathematics department that contributed to one of the best communities in Penn State. I am also very thankful to all my friends and peers for their camaraderie, and for all the good moments that we shared. Last, but definitely not least, I am forever indebted to my parents Ilda and Jorge for their unending support and love, for understanding me and for their genuine care when I was feeling low.

ix Dedication

Para o meu pai e a minha mãe.

x Chapter 1 | Introduction

In this dissertation we study the distribution of primitive closed geodesic paths in finite 2-dimensional simplicial complexes or, in general, multicomplexes (as in [LLR18]) covered by the buildings associated to PGL3(F ) or PGSp4(F ), where F is a local nonarchimedean field. We analyze paths in the 1-skeleton or 2-skeleton of multicomplexes, respectively. We shall refer to equivalence classes of primitive closed geodesic paths, under cyclic rotations, as primes. From this set of primes we define Zeta functions and L-functions whose analytic properties can be used to describe the distribution of primes, just like what happens to prime numbers and the Riemann zeta function. These Zeta functions are, in fact, rational functions and have a simple formula in terms of adjacency operators acting on the finite dimensional vector spaces of functions on the 1-skeleton or 2-skeleton of multicommplexes, respectively. This connection is useful, as it allows us to write the poles of the associated Zeta functions in terms of the eigenvalues of a linear transformation. The main contribution to the number of primes of given length comes from the largest eigenvalues, in absolute value, of the adjacency operators, and determining them is a central problem in the dissertation. This gives rise to the first main result of this dissertation: versions of the Prime Number Theorem for primes on finite complexes. The second main result is a collection of different versions of the Chebotarev Density Theorem, and gives a finer distribution of the primes, when they are partitioned by the associated Frobenius conjugacy class in the Galois group of a finite unramified Galois cover.

1.1 Zeta functions and the distribution of primes – a historical review

Before we present our main results, in this and the next sections, we give a brief historical review of the zeta functions, Prime Number Theorem and the Chebotarev Density

1 Theorem concerning the distribution of primes in different settings.

1.1.1 Prime numbers and the Riemann zeta function

Prime numbers have fascinated mathematicians for centuries and they have inspired many questions in all fields of mathematics. Moreover, there are still many open problems in Number Theory related to prime numbers. One of the most notorious problems, that is now a theorem, is the Prime Number Theorem, which roughly states that the number of prime numbers in the interval [1, x], denoted by π(x), is approximately x/ log x. We draw attention to the fact that the Prime Number Theorem may be derived from the analytic properties of the Riemann zeta function defined as

∞ X 1 Y ζ(s) = = (1 − p−s)−1 , 1 . ns n=1 p prime number

Here, the second equality is a consequence of the unique prime decomposition of natural numbers. A more precise version of the Prime Number Theorem states (cf. [MV07]) that the prime counting function, π(x), satisfies

π(x) = Li(x) + O(xβ log x) , where Z x 1 Li(x) = dt , and β = sup{

1.1.2 Rational points and Zeta functions of algebraic varieties

In his 1923 dissertation, Emil Artin defined and studied Zeta functions associated to algebraic curves over finite fields. These functions count the number of rational points of a given algebraic curve over different finite extensions of a given finite field. His definition is easily extended to Zeta functions of algebraic varieties, which we explain succinctly.

Let V be an algebraic variety over a finite field k, and let Nν be the number of rational points of V over the field extension of k with degree ν, kν. Then, the Zeta function

2 attached to V is defined by   X Nν Y Z(V, u) = exp uν = (1 − udeg v)−1 .  ν  ν≥1 closed point v

Here, a closed point is the orbit of a point x in V (k¯) under Gal(k/k¯ ) and deg v is the number of points in this orbit. The closed points play the role of primes for V . Part of the relevance of this function is due to four significant results known as the Weil Conjectures. These are four properties of the Zeta functions associated to algebraic varieties, conjectured by Weil [Wei49], and that had already been proved in the case of algebraic curves. The general case was proved by Deligne [Del74], Dwork [Dwo60] and Grothendieck [Gro65]. Two of those properties are the facts that these Zeta functions are rational and satisfy a version of the Riemann Hypothesis, which are in the same theme as this dissertation. 1 When V is the projective line P over k, the affine closed points are in bijective correspondence with the monic irreducible polynomials over k. Assume that k has size q.

One can verify that the number Mν of irreducible monic polynomials over k with degree 1 ν, in other words, the number of primes of P of degree ν, satisfies

qν qν/2  M = + O , for ν large . ν ν ν

This is the Prime Number Theorem for the projective line over a finite field with q elements.

1.1.3 Primitive closed geodesics and the Selberg zeta function

Recall that PSL2(R) acts transitively on the hyperbolic upper half-plane H by fractional linear transformations. These are also all the orientation preserving isometries of H.

In [Sel56], Selberg considered the compact Riemann surfaces of the form XΓ = Γ\H, where Γ is a discrete, torsion-free, cocompact subgroup of PSL2(R). The equivalence classes of primitive closed geodesics in this compact surface play the role of primes of XΓ.

Using these primes, Selberg defined a Zeta function attached to XΓ as

∞ Y Y  −(s+j)l(C) Z(XΓ, s) = 1 − e , [C] j=1 where [C] runs over all primes of XΓ and l(C) is the length of C under the hyperbolic metric. He proved that this function and the Riemann zeta function have similar analytic

3 behaviors, but, unlike ζ(s), all except finitely many real zeros of Z(XΓ, s), in the strip 0 <

Theorem (Prime Geodesic Theorem for compact Riemann surfaces). Let

π(XΓ, x) = |{[C]: C is a primitive geodesic with length ≤ x}|,

1 and let s1, . . . , sk be the real zeros of Z(XΓ, s) in the interval ( 2 , 1). Then

3 s1 sk 2 π(XΓ, x) = Li(x) + Li(x ) + ··· + Li(x ) + O(x 4 (log x) ) .

1.1.4 Primitive geodesic cycles and the Ihara zeta function

Let F be a nonarchimedean local field with uniformizer π, ring of integers OF and q elements in its residue field. Now, observe that H = SL2(R)/ SO2(R). In [Iha66], Ihara defined an analog of Selberg’s zeta function over p-adic fields. For that purpose, let Γ be a discrete, torsion-free, cocompact subgroup of PGL2(F ). We say that an element γ of Γ is primitive if it generates its own centralizer in Γ. Likewise, a conjugacy class of Γ is primitive if its elements are primitive. Furthermore, for a conjugacy class C of Γ, we define the number deg C as the minimum value of ordπ(det g), when g is a matrix with entries in OF that represents an element of C. Ihara defined the following Zeta function

−1 Y  deg P  Z(XΓ, u) = 1 − u , P where P runs over all primitive conjugacy classes of Γ. He also proved that the Ihara zeta function is a rational function.

Theorem (Ihara [Iha66]). Let Γ be a discrete, cocompact, torsion-free subgroup of

PGL2(F ). Then, the Ihara zeta function of XΓ = Γ\ PGL2(F )/ PGL2(OF ) satisfies

(1 − u2)χ(XΓ) Z(XΓ, u) = 2 , det(I − AΓu + qu ) where AΓ is a Hecke operator on the space of functions on Γ\ PGL2(F )/ PGL2(OF ), and

χ(XΓ) is the Euler characteristic of XΓ.

Observe that rationality is a feature that Ihara zeta functions share with the Zeta

4 functions of algebraic varieties over finite fields, as is known from the Weil conjectures, alluded to before.

Serre [Ser80] realized that the double-quotient Γ\ PGL2(F )/ PGL2(OF ) may be in- terpreted as a finite (q + 1)-regular graph, and generalized Ihara’s zeta function to any finite regular graph H, using the fact that primitive conjugacy classes of Γ correspond bijectively to equivalence classes of primitive geodesic cycles, called primes of H, of the same length. Bass [Bas92] extended Ihara’s Theorem to irregular graphs. In [Has89], Hashimoto also showed that the Ihara zeta function, as suggested by Serre, is a rational function by relating it with the directed edge adjacency matrix.

Theorem (Hashimoto [Has89]). Let H be a finite graph with directed edge adjacency T . Then, the Ihara zeta function of H satisfies

1 Z(H, u) = . det(I − T u)

In [Has92], Hashimoto showed a version of the Prime Geodesic Theorem counting the number of primes of a finite graph.

Theorem (Prime Geodesic Theorem for graphs, Hashimoto [Has92]). Let H be a con- nected graph with directed edge adjacency matrix T . Let λ be the largest eigenvalue, in absolute value, of T and let δH be the greatest common divisor of the lengths of the geodesic cycles of H. Then,

λδH l |{[C]: C is a primitive geodesic cycle of H with length δ l}| ∼ , (1.1) H l as l → ∞ .

1.2 Ramanujan graphs

It is known that the expansion properties of a regular graph are related to the size of the gap between the two largest eigenvalues of its vertex adjacency matrix. As a result, this gives a relation between the expansion properties of a graph and the poles of its Ihara zeta function. This fact has applications to the construction of expanders, that is, families of graphs where the distance between any two vertices is, on average, small. Examples of these are given by Ramanujan graphs, that is, (q + 1)-regular graphs for which all √ nontrivial eigenvalues satisfy |λ| ≤ 2 q. This class of graphs was first mentioned in papers of Lubotzky-Phillips-Sarnak [LPS88] and Margulis [Mar88] that, independently, constructed families of (q + 1)-regular Ramanujan graphs, with q prime, using Hecke

5 operators on quaternion algebras. In their paper, the authors verified that all nontrivial √ eigenvalues of their examples satisfy |λ| ≤ 2 q using the Ramanujan-Petersson conjecture, proved by Deligne [Del74]. This is equivalent to the Zeta function attached to the graph satisfying the Riemann Hypothesis. A theorem of Alon-Boppana [LPS88] shows that this bound cannot be smaller for (q + 1)-regular graphs with arbitrarily many vertices. Lastly, we observe that if H is a (q + 1)-regular Ramanujan graph, then λ = q, in the equation (1.1). Furthermore, we have a good control of the error term of the estimate in  δH l  λ 2 (1.1), which is given by O l .

1.3 Primes and Zeta functions for finite quotients of buildings

Graphs are 1-dimensional simplicial complexes. So, it is natural to ask the same questions for higher dimensional complexes. In simplicial complexes with dimension greater than 1 we may construct paths over adjacent cells with dimension 2. We call such paths galleries. Recall that in Serre’s interpretation of Ihara’s zeta function, he uses graphs given by

finite quotients of the building associated to PGL2(F ). Here we study 2-dimensional complexes, that we will denote by XΓ, covered by the buildings associated to PGL3(F ) or PGSp4(F ). In Deitmar-Hoffman [DH06] and Kang-Li-Wang [KLW10], the authors study thor- oughly finite quotients, XΓ, of the 2-dimensional building associated to PGL3(F ), by discrete, torsion-free, cocompact subgroups Γ of PGL3(F ). They also study the Zeta functions counting the number of closed geodesic cycles and galleries in XΓ. There are two types for geodesic cycles and galleries, respectively. Then, in [KL14], Kang-Li prove formulas for the Zeta functions associated to the closed geodesic cycles of types 1 and 2 of XΓ, and denoted by Z1,1(XΓ, u) and Z1,2(XΓ, u), respectively. They also give formulas for these Zeta functions in terms of linear operators. If we let AE,Γ be the adjacency operator on type 1 edges, then

1 1 Z1,1(XΓ, u) = , and Z1,2(XΓ, u) = T 2 . det(I − AE,Γu) det(I − (AE,Γ) u )

T Here, (AE,Γ) is the transpose of AE,Γ. A similar formula is given for the Zeta function associated to the closed geodesic galleries of type 1. If we let AB,Γ be the adjacency operator on type 1 chambers, then

1 Z2,1(XΓ, u) = . det(I − AB,Γu)

6 Furthermore, they show an identity relating the various adjacency operators acting on the different i-skeletons of XΓ, analogous to Ihara’s Theorem for graphs.

Theorem (Kang-Li [KL14]). The adjacency operators of XΓ satisfy the identity

3 χ(X ) (1 − u ) Γ det(I + AB,Γu) 2 3 3 = T 2 , det(I − A1u + A2u − q u I) det(I − AE,Γu) det(I − (AE,Γ) u ) where χ(XΓ) is the Euler characteristic of XΓ, and A1 and A2 are Hecke operators 2 associated to the vertices of XΓ, acting on L (Γ\ PGL3(F )/ PGL3(OF )).

In Fang-Li-Wang [FLW13], the authors study another kind of 2-dimensional complexes, this time given by finite quotients, XΓ, of the building associated to PGSp4(F ), by discrete, torsion-free, cocompact subgroups Γ of PGSp4(F ). They prove formulas for the Zeta functions attached to XΓ in terms of adjacency operators. Like the case of PGL3(F ), there are two types for paths of directed edges and two types for paths of directed chambers. The edge zeta functions of types 1 and 2 are reciprocal of the characteristic polynomials of adjacency matrices AP1,Γ (resp. AP2,Γ) of type 1 (resp. type 2) edges:

1 1 Z1,1(XΓ, u) = , and Z1,2(XΓ, u) = , det(I − AP1,Γu) det(I − AP2,Γu) and the Zeta function attached to the closed geodesic galleries of chambers of spin type satisfies 1 Z2,spin(XΓ, u) = , det(I − AI1,Γu) where AI1,Γ is the adjacency matrix of chambers of spin type. Lastly, they show an identity relating the adjacency operators on different i-skeletons

2 χ(XΓ) 2 2 2Np−Nns (1 − u ) (1 − q u ) det(I − AI1,Γu) 2 3 2 6 4 = 2 , det(I − A1u + qA2u − q A1u + q Iu ) det(I − AP1,Γu) det(I − AP2,Γu ) where χ(XΓ) is the Euler characteristic of XΓ, 2Np is the number of special vertices in

XΓ, Nns is the number of nonspecial vertices in XΓ, and A1 and A2 are Hecke operators 2 on L (Γ\ PGSp4(F )/ PGSp4(OF )). More recently, Kang-Li-Wang [KLW] defined Zeta functions attached to the closed geodesic galleries of chambers of standard type and obtained identities relating the aforementioned with other Zeta functions of the building associated to PGSp4(F ).

7 1.4 Chebotarev Density Theorem

1.4.1 In Number Theory

Whenever L/E is a finite Galois extension of number fields, the group of automorphisms of L that fix every element of E is called the Galois group of the extension, and is denoted by Gal(L/E). As the rings of integers, OL in L and OE in E, are Dedekind domains, we know that each ideal I of OL or OE admits a decomposition into prime ideals of OL or

OE, respectively. In particular, if p is a prime ideal of OE, unramified in L, then pOL admits a decomposition

pOL = P1 ··· Pr into prime ideals of OL. In this case, we say that the prime ideals Pj, j = 1, . . . , r lie over p.

For each prime p in OE, we associate a specific conjugacy class Frobp of Gal(L/E), called the Frobenius conjugacy class at p. It consists of the elements of Gal(L/E) that induce the map x 7→ xq on OL/P, for some prime ideal P of OL that lies over p and with q = |OE/p|. Denote pr by SC (E) the set of unramified prime ideals p of OE such that Frobp = C. We can now state the famous Chebotarev Density Theorem.

Theorem (Chebotarev Density Theorem). Let L/E be a finite Galois extension of pr number fields with Galois group G. Then, for any conjugacy class C of G, the set SC (E) has natural density |C|/|G|.

1.4.2 For Riemann surfaces

In his thesis (cf. [Sar80]), Sarnak also studied a version of the Chebotarev Density Theorem for compact Riemann surfaces covered by the hyperbolic upper half-plane. Let 0 0 Γ and Γ be two discrete, torsion-free, cocompact subgroups of PSL2(R) such that Γ is a normal subgroup of Γ of finite index. This inclusion induces a finite covering map from 0 XΓ0 = Γ \H to XΓ = Γ\H, denoted by ϕ : XΓ0 → XΓ. If p is a primitive closed geodesic cycle in XΓ, then there are finitely many primitive closed geodesics P1,..., Pr, such that

ϕ(Pj) = p, j = 1, . . . , r.

We denote by G the group of automorphisms g of XΓ0 such that ϕ ◦ g = ϕ. One can ∼ 0 easily see that G = Γ/Γ , so that the cover ϕ is Galois. Each closed geodesic p in XΓ is associated to a conjugacy class in G that we denote by Frobp, which consists of the

8 elements in G that map the initial point of a lift of p in some Pj to the terminal point of pr that lift. Similarly to before, for a conjugacy class C of G, we denote by SC (Γ) the set of primitive closed geodesics p in XΓ such that Frobp = C. We now state the following theorem due to Sarnak.

Theorem (Sarnak [Sar80]). Let Γ0 and Γ be discrete, torsion-free, cocompact subgroups 0 of PSL2(R) such that Γ is a normal subgroup of Γ. Then, this inclusion induces a Galois ∼ 0 pr covering ϕ : XΓ0 → XΓ with group of automorphisms G = Γ/Γ , and the set SC (Γ) has natural density |C|/|G| for any conjugacy class C of G.

1.4.3 For graphs

We now present the case for coverings of graphs. Such coverings were studied thoroughly in Stark-Terras [ST00]. Let ϕ : X → Y be an unramified covering of finite graphs with d sheets. We further assume that the covering is normal, meaning that the group of automorphisms of ϕ acts transitively on each fiber, hence it has cardinality d. Call this group the Galois group G. We remark that Hashimoto [Has90] gives some insight to the case of ramified graph coverings. For each primitive geodesic cycle p in Y there are a number of primitive geodesic cycles P1,..., Pr in X, such that ϕ(Pj) = p, j = 1, . . . , r. We say that each prime

Pj lies over p. This was shown in the work of Terras [Ter11], Somodi [Som15], and Huang-Li [HL]. Furthermore, to each prime p in Y we associate a conjugacy class of G, that we denote by Frobp, in the same way as in section 1.4.2. Given a conjugacy class C of G, denote by pr SC (Y ) the set of primes p such that Frobp = C.

Theorem (Terras [Ter11]). Let ϕ : X → Y be a d-sheeted normal unramified covering of pr finite graphs with Galois group G. Then, for each conjugacy class C of G, the set SC (Y ) has Dirichlet density |C|/|G|.

Hashimoto [Has90] considered the above result in natural density. However, his argument was incomplete. Later, Huang-Li [HL] gave a necessary and sufficient condition for which the Chebotarev Density Theorem holds for natural density. To state their result, denote by δX and δY the greatest common divisors of all lengths of primes in X and Y , respectively.

pr Theorem (Huang-Li [HL]). In the same context as in the previous theorem, SC (Y ) has natural density if and only if δX = δY , in which case its value agrees with the Dirichlet density.

9 1.5 Main results

As before, let F be a nonarchimedean local field with uniformizer π, ring of integers

OF and q elements in its residue field. Our objective is to study versions of the Prime Geodesic Theorem and Chebotarev Density Theorem for equivalence classes of primitive closed geodesic cycles and galleries of given type for finite quotients of PGL3(F ) and

PGSp4(F ). Note that the asymptotic distribution of the 1 and 2-dimensional primes of given type is determined by the largest eigenvalues, in absolute value, of the corresponding adjacency operator. Moreover, as each cell in these buildings has the same number of adjacent cells, it is straightforward to determine the largest eigenvalue in absolute value. However, the asymptotic distribution depends on the multiplicity of such eigenvalues, which is not trivial to determine. The first main result of this dissertation delves with this question and gives a version of the Prime Geodesic Theorem for finite quotients of the buildings associated to PGL3(F ) and PGSp4(F ).

Theorem 1 (Prime Geodesic Theorem for 2-dimensional complexes). Let G be either the group PGL3(F ) or PGSp4(F ) and X the associated building. Let Γ be a discrete, cocompact, torsion-free subgroup of G, such that ordπ(det Γ) is divisible by 3 or 4, respec- tively. Given the dimension i and the type θ, denote by δ the greatest common divisor of the lengths of the i-dimensional primes with type θ. Then,

ηqκδl |{p | p is a type θ prime of dimension i in X with length δl}| ∼ , (1.2) Γ l as l → ∞, and

η qκδl |{p | p is a type θ prime of dimension i in XΓ with length < δl}| ∼ , qκδ − 1 l as l → ∞, where the value of each variable is given in the table below.

Group G PGL3(F ) PGSp4(F ) Dimension i 1 2 1 2 Type θ 1 1 2 spin standard η 1 1 2 1 2 δ 3 2 1 2 1 κ 2 1 3 4 2 3

For G = PGL3(F ), we can make Theorem 1 more precise when XΓ is Ramanujan. Ramanujan complexes were introduced in [Li04] and [LSV05b], and are the simplicial

10 complexes given by finite quotients of the building associated to PGLn(F ) whose spectra are contained in that of their universal cover. In particular, when n = 2, these are the (q + 1)-regular Ramanujan graphs. Furthermore, Kang-Li-Wang [KLW10] (Theorem 2) give equivalent conditions for Ramanujan complexes, in terms of the poles of the associated Zeta functions. Using that result, we can verify that the error term of (1.2) is κδl  q 2  O l . 0 Now, let Γ and Γ be discrete, cocompact, torsion-free subgroups of PGL3(F ) or 0 PGSp4(F ) such that Γ is a normal subgroup of Γ of finite index. We further assume 0 that ordπ(det Γ ) and ordπ(det Γ) are divisible by 3 or 4, depending on whether they are subgroups of PGL3(F ) or PGSp4(F ), respectively. We denote by XΓ0 and XΓ the 0 quotients of the building associated to either PGL3(F ) or PGSp4(F ) by the groups Γ and Γ, respectively. The inclusion of subgroups induces a covering map ϕ : XΓ0 → XΓ with automorphism group Γ/Γ0. Similarly to what we did in the previous sections, for each i-dimensional prime p with type θ in XΓ we associate a conjugacy class from the 0 pr,θ group Γ/Γ , denoted by Frobp. We denote by Si,C (Γ) the set of i-dimensional primes p with type θ such that Frobp = C. We then have the following result.

pr,θ 0 Theorem 2. With notation as above, the set Si,C (Γ) has natural density |C|/|Γ/Γ |.

In particular, we remark that, unlike the graph case, we do not distinguish between the cases of Dirichlet or natural densities, since we know that the greatest common divisor of the lengths of primes with given dimension i and type θ is the same for both quotients

XΓ and XΓ0 . Hence the Chebotarev Density Theorem holds for the stronger natural density. This dissertation is organized as follows. Our main results are Theorems 1 and 2, that we further divide into Theorems 11, 12, 14, 15, 19, 20, 22, 23, 26, 27, 30 and 31. In chapter 2, we introduce the necessary concepts to understand the structure of simplicial complexes and multicomplexes, and we prove some preliminary results that we use multiple times in chapters 3 and 4, in various situations. Chapters 3 and 4 have a similar structure, and both begin with an explanation of the structure of the buildings attached to PGL3(F ) and PGSp4(F ), respectively. Then, we state the main results of the chapter, and give proofs for them in the end.

11 Chapter 2 | Preliminaries

2.1 Graphs

A graph is the classic abstract representation of a network. We define a directed graph −→ X as a set of vertices, V , together with a set of directed edges given by a subset E of −→ V × V − {(v, v): v ∈ V }. For each e = (u, v) ∈ E , we write e−1 = (v, u) to denote the edge with opposite direction from e, and we call an unordered pair of vertices {u, v} undirected edge. The set of unordered edges will be denoted by E. We will often refer to these just by edges, whenever it is clear from the context whether they are directed or undirected edges.

A path is a sequence of adjacent edges C = (e1, e2, . . . , el) where the terminal vertex of ei coincides with the initial vertex of ei+1, i = 1, . . . , l − 1. The length of C is l(C) = l.

Moreover, if the initial vertex of e1 is the same as the terminal vertex of el, we say that C is a closed path or that it is a cycle, and we denote by [C] the equivalence class containing 0 0 0 all cyclic rotations of C. If C = (e1, . . . , el) and C = (e1, . . . , el0 ) are two paths in X, 0 such that the terminal vertex of el is the same as the initial vertex of e1, we define their 0 0 0 concatenation as the path CC = (e1, . . . , el, e1, . . . , el0 ). We say that a cycle C is primitive if there does not exist another cycle C0 so that k C0 = C, for some integer k ≥ 2, and in this case [C] is also called primitive. Whenever −1 every pair of consecutives edges in C = (e1, . . . , el) satisfies ei 6= ei+1 we say that C is −1 backtrackless. Additionally, if e1 =6 el, we say that C is tailless. If V is finite and has size n, we define the adjacency matrix of X as the n × n matrix

AX = A = (auv) with indices in V , whose entries are given by ( −→ 1 , if (u, v) ∈ E auv = −→ . 0 , if (u, v) 6∈ E

12 −→ Moreover, if E is finite and has size 2m, we define the edge adjacency matrix of X as 0 −1 the 2m × 2m matrix TX = T = (tee0 ), where tee0 = 1 if e =6 e and the terminal vertex 0 of e coincides with the initial vertex of e , and tee0 = 0 otherwise. We wish to introduce a useful generalization of graphs. Multigraphs are similar to graphs except that they can have parallel edges. As a consequence, in this case, edges are not uniquely described by their endpoints. We define a multigraph Xe over a graph X as a set of vertices, V , together with a pair (E, m), where m : E → N is the multiplicity function defined on the undirected edges of X. The pairs (e, i) with e ∈ E and 1 ≤ i ≤ m(e) are the multiedges of Xe, and they form the set Ee. We also denote by ι : Xe → X the natural projection map that satisfies

ι(v) = v , v ∈ V and ι(e, i) = e , (e, i) ∈ E.e

2.2 Simplicial complexes and multicomplexes

Simplicial complexes are a generalization of graphs for higher dimensions. Again, use V to denote a set of vertices. We define a simplicial complex X with vertices in V as a collection of subsets of V , called cells, with the extra condition that if a = {v0, v1, . . . , vn} is a cell of X and b ⊂ a, then b is also a cell of X . The dimension of a cell a = {v0, v1, . . . , vn} is dim a = |a| − 1 = n. We also define the boundary of a cell a = {v0, . . . , vn}, denoted by ∂a, as the set of cells that result from removing a single vertex from a. For each i ∈ N we denote by Xi the i-skeleton of X, that is, the set of cells with dimension i in X or i-cells, for short. Finally, we define directed cells as the cells of X equipped with an orientation, that is, a tuple where each of its vertices appears exactly once. We present a further generalization of the concept of simplicial complex. Multi- complexes are to simplicial complexes like multigraphs are to graphs. This means that each cell is not uniquely determined by the vertices that it contains. Here we show the definition from [LLR18].

Definition (Multicomplex). Let X be a simplicial complex with set of vertices V , and m : X → N be a function that we will call multiplicity function. We define the set of pairs Xm = {(a, r): a ∈ X, r ∈ {1,..., m(a)}}, whose elements are multicells, and the forgetful map ι : Xm → X, by ι(a) = a, whenever a = (a, r). We also assume that m(v) = 1 if v ∈ V , and we define the dimension of a multicell as dim a = |ι(a)| − 1.

A gluing map g : {(a, σ) ∈ Xm × X : σ ∈ ∂ι(a)} → Xm is a map that satisfies (ι ◦ g)(a, σ) = σ for (a, σ) in the domain of g. In particular, we will see that g tells us how the multicells of a given dimension are glued together. We define the multiboundary

13 of a cell as the following set ( {g(a, σ): σ ∈ ∂ι(a)} if dim a ≥ 1 ∂ma = . ∅ if dim a = 0

The multicells obtained by taking multiboundaries of a multicell, one or more times, are called submulticells. Finally, we say that the triple Xe = (X, m, g) is a multicomplex if it satisfies all the conditions above and the following consistency property: if a = (a, r) is a multicell with two submulticells b = (b, r) and b0 = (b0, r0) such that dim b = dim b0 and the intersection of their projections onto X is ρ = σ∩σ0 ∈ Xdim b−1, then g(b, ρ) = g(b0, ρ). For a multicomplex Xe, we denote by Xei the set of multicells with dimension i or i-multicells, for short. We will call directed multicell to any multicell equipped with an orientation, that is, a tuple where each of its vertices appears exactly once.

2.3 Primes in finite quotients of X

Let X be an infinite simply connected simplicial complex. The neighborhood of a cell a in X consists of the cells b, with the same dimension as a, such that a ∩ b is a subcell of a and b with dimension dim a − 1. In the sequel, the central object of our study will be backtrackless and tailless paths in Xi, i = 1, 2. However, we want to focus on paths that minimize the distance between cells, that is, geodesic paths. To help us in that pursuit, we will use directed cells and we introduce the concept of out-neighbors of a directed cell. −→ Furthermore, we shall denote by Xi the set of directed cells contained in Xi. The out-neighbors of a directed cell c are the directed cells with the same dimension, that share a given face with c. The orientation of c determines which of its faces it has in common with the out-neighbors and the orientation of the out-neighbors. For each directed cell c, we will also consider a subset R(c) of its out-neighbors, that we will call proper out-neighbors (R stands for “right multiplication”, which should become natural from the concrete examples considered later on). To give an idea, these adjacent cells give rise to straight lines in the apartments of buildings. We will describe these in more detail in each of the cases considered in the next chapters.

A geodesic path of i-cells is a sequence of directed i-cells C = (c1, c2, . . . , cl), such that cj+1 is a proper out-neighbor of cj for any 1 ≤ j ≤ l − 1, or equivalently, cj+1 ∈ R(cj), and its length is l(C) = l. We call a path of 1-cells path and a path of 2-cells gallery.

Whenever C is closed, that is c1 ∈ R(cl), we denote by [C] the equivalence class of all rotation shifts of C. If such C is composed of 1-cells then we say that C is a geodesic cycle, and if it is composed of 2-cells we say that C is a closed geodesic gallery. Furthermore,

14 we say that [C] is a 1 or 2-dimensional prime. We shall study different kinds of geodesic paths, that may be characterized by their shape or direction and the dimension of their directed cells. Likewise, we shall assign a type θ to each directed cell or variations of it, so that each geodesic contains cells of a given dimension and type. We denote by Xi,θ the set of directed cells with dimension i and type θ in X. We assume that a directed cell and its out-neighbors have the same type. We denote by Aut(X) the group of automorphisms of X and let Γ be a subgroup of Aut(X), such that Γ has no torsion, the quotient space Γ\X is finite, the stabilizer of any vertex v of X is trivial, and Γv does not contain two adjacent vertices for 0 any v ∈ X . In this situation, XΓ = Γ\X has a structure of simplicial complex or multicomplex induced by the orbits of cells in X under the action of Γ. In more detail, 0 XΓ is defined over the simplicial complex ι(XΓ), which has Γ\X as set of vertices and whose cells are given by the subsets {Γv0,..., Γvn} such that {v0, . . . , vn} is a cell of X.

Then, XΓ is a multicomplex over ι(XΓ) where the multiplicity function at a given cell

{Γv0,..., Γvn} is equal to the number of orbits of cells Γ{w0, . . . , wn} of X such that

{Γw0,..., Γwn} = {Γv0,..., Γvn}. The boundary of a multicell Γ{v0, . . . , vn} is given −→ i,θ by the orbits of the cells in ∂{v , . . . , v }. Finally, we define the sets Xi , Xi and X 0 n −→ Γ Γ Γ similarly to before, by taking the orbits of Xi, Xi and Xi,θ, respectively, under Γ. The proper out-neighbors and geodesic paths in XΓ are given by the projection of proper out-neighbors or geodesic paths in X under the projection map ϕ : X → XΓ.

Additionally, in order to guarantee that each path of i-multicells of XΓ has a unique lift, we assume that Γ satisfies the following condition.

(C) for any cells σ, a and b of X, with σ ∈ ∂a and σ ∈ ∂b, then Γ(a, σ) = Γ(b, σ) if and only if (a, σ) = (b, σ).

In this context, Γ is isomorphic to the fundamental group of XΓ. θ θ 1,θ 2,θ Let S1,Γ and S2,Γ be sets of representatives of the orbits in Γ\X and Γ\X , respectively. Consider C a closed geodesic path of i-multicells with i = 1 or 2, in XΓ. C θ has a unique lift to X starting from an i-multicell in Si,Γ, like the following

s0 → γ1s1 → γ2s2 → · · · → γl−1sl−1 → γlsl = γls0 ,

θ where sj ∈ Si,Γ, j = 0, . . . , l, and we take s0 = sl, because C is closed. Thus, the initial and terminal cells of the lift of C differ by the action of γl ∈ Γ. Note that if we replace

15 the initial cell by γs0 ∈ Γs0, the lift of C becomes

−1 −1 −1 −1 γs0 → (γ · γ1 · γ )γs1 → · · · → (γ · γl−1 · γ )γsl−1 → (γ · γl · γ )γsl = (γ · γl · γ )γs0 , so the initial and terminal cells differ by the action of a conjugate of γl. On the other hand, if we apply a rotation shift to C, the resulting path or gallery has the following lift

γ1s1 → γ2s2 → · · · → γl−1sl−1 → γlsl = γls0 → γl(γ1s1) .

Consequently, the conjugacy class [γl] is well-defined in terms of the equivalence class [C].

In this situation, we define the Frobenius conjugacy class of [C] as Frob[C] = [γl]. From here on, the equivalence classes, under cyclic rotation, of primitive closed geodesic cycles or galleries in XΓ of type θ will be called i-dimensional primes of type θ.

2.3.1 Zeta functions and L-functions

We will define counting functions of primes of given dimension and type, known as Zeta functions. These functions have many interesting features, and their analytic properties have consequences to the distribution of closed geodesic cycles and galleries in XΓ. We start by introducing adjacency operators in the i-skeleton of X, using the proper out-neighbors. We define the following operator on C(Xi,θ), the set of complex valued functions on Xi,θ, θ X 0 (Ai f)(c) = f(c ) . c→c0 proper

θ Then, we may define a similar operator, Ai,Γ, on the complex valued functions on directed θ cells of the finite quotient XΓ, by taking the restriction of Ai to the finite dimensional space θ i,θ i,θ Vi,Γ = {f : X → C | f(γc) = f(c), ∀c ∈ X , γ ∈ Γ}.

A Zeta function is given by a product over i-dimensional primes of type θ. Given the set i,θ of primes in XΓ , the corresponding Zeta function is given by

−1 Y  l(p) Zi,θ(XΓ, u) = 1 − u . p type θ prime of dimension i

In the next section, we will prove that this Zeta function is a rational function in u.

16 Theorem 3. The type θ Zeta function on the i-skeleton of XΓ satisfies

1 Zi,θ(XΓ, u) = θ . (2.1) det(I − Ai,Γu)

Let ρ be a d-dimensional unitary representation of Γ, with representation space Vρ. i,θ For dim X ≥ i ≥ 1, we define the following space of Vρ-valued functions in X

θ i,θ i,θ Vi,ρ = {f : X → Vρ : f(γc) = ρ(γ)f(c), ∀c ∈ X , γ ∈ Γ} ,

i,θ which is a subspace of C(X ) ⊗ Vρ equipped with the following Γ action

−1 i,θ i,θ (γf)(c) = ρ(γ)f(γ c) , ∀c ∈ X , f ∈ C(X ) ⊗ Vρ . (2.2)

θ i,θ Using the action of Γ we may also define Vi,ρ as the subspace of C(X ) ⊗ Vρ fixed by Γ, that is, θ i,θ Γ Vi,ρ = (C(X ) ⊗ Vρ) .

Denote by idρ : Vρ → Vρ the identity endomorphism on Vρ. Then, we introduce the θ θ θ adjacency operator Ai,ρ as the restriction of Ai ⊗ idρ to the subspace Vi,ρ. This definition θ θ is equivalent to saying that Ai,ρ is an operator that sends f ∈ Vi,ρ to

X 0 (Ai,ρf)(c) = f(c ) . c→c0 proper

We define the L-function associated to ρ as

−1 Y  l(p) Li,θ(XΓ, ρ, u) = det I − ρ(Frobp)u . p type θ prime of dimension i

It turns out that this function is also a rational function, as shown below.

Theorem 4. The type θ L-function associated to XΓ and ρ satisfies

1 Li,θ(XΓ, ρ, u) = θ . (2.3) det(I − Ai,ρu)

Remark. When ρ = 1 is the trivial representation, this L-function becomes the type θ

Zeta function on the i-skeleton of XΓ.

Proof. The proof follows the same structure as the one for Theorem 1 in [Has89] and

17 Theorem 1.1 in [KL15]. We use the identity tr(log(I − A)) = log(det(I − A)). In order to prove the desired equality we apply this identity to both sides of (2.3). θ We first analyze the right hand side of (2.3). Consider Si,Γ = {s1, . . . , sn} a set of i,θ i,θ representatives of Γ\X and fix a basis v1, . . . , vd of Vρ. The functions fjk : X → Vρ, j = 1, . . . , d, k = 1, . . . , n, given by ( ρ(γ)vj if k = t fjk(γst) = , for γ ∈ Γ , 0 otherwise

θ θ l form a basis for Vi,ρ with size dn. Note that the operator (Ai,ρ) is a sum over geodesic paths of i-cells in X with length l + 1, like the following

st = st0 → γ1st1 → · · · → γlstl .

Denote by fk = a1f1k + ··· + adfdk an arbitrary linear combination of elements in the θ θ l basis of Vi,ρ that are zero outside Γsk. The components f•k of (Ai,ρ) fk are determined by

 θ l  X (Ai,ρ) fk (st) = fk(γst) st→···→γst proper, with length l X = ρ(γ)fk(st) , st→···→γst proper, with length l when t = k. As a result, we obtain the following lemma.

θ l Lemma 1. The trace of (Ai,ρ) may be given in terms of the i-dimensional primes of type θ in XΓ, as follows

 θ l X X  l/k tr (Ai,ρ) = k tr ρ(Frobp) . (2.4) k|l p type θ prime of dimension i l(p)=k

Using Lemma 1, we see that the logarithm of the right hand side of (2.3) can also be written as   θ l l   X (Ai,ρ) u − tr log(I − Aθ u) = tr i,ρ  l  l≥1

18     X 1 X X  l/k l l = tr  k tr ρ(Frobp) u  , t =  l  k  l≥1 k|l p type θ prime  of dimension i l(p)=k t
l(p)t X X tr ρ(Frobp) u = . t p type θ prime t≥1 of dimension i

On the other hand, taking the logarithm of the left hand side of (2.3) and expanding, we obtain

X  l(p)  X  l(p)  − log det(I − ρ(Frobp)u ) = − tr log(I − ρ(Frobp)u ) p type θ prime p type θ prime of dimension i of dimension i   k l(p)k X X ρ(Frobp) u = tr  k  p type θ prime k≥1 of dimension i k
l(p)k X X tr ρ(Frobp) u = . k p type θ prime k≥1 of dimension i

This concludes the proof.

2.3.2 Distribution of primes of given dimension and type

Recall that in the previous section we saw that the Zeta functions of XΓ may be given by rational functions of the form

1 Zi,θ(XΓ, u) = θ . det(I − Ai,Γu)

θ θ i,θ Let λi,Γ = max{|λ| : λ ∈ C is an eigenvalue of Ai,Γ} . Let n = ni,θ = |XΓ | and let θ α1, . . . , αt, β1, . . . , βn−t be all the eigenvalues of Ai,Γ counted with multiplicity, such that θ θ |αr| = λi,Γ, r = 1, . . . , t, and |βj| < λi,Γ, j = 1, . . . , n − t . We can write,

1 Z X , u . i,θ( Γ ) = Qt Qn−t (2.5) r=1(1 − αru) j=1(1 − βju)

19 Taking the logarithmic derivative of the left hand side of (2.5), we obtain

d −u log Z (X , u) du i,θ Γ   d Y  −1 X ul(p) = −u log  1 − ul(p)  = − l(p) du   − ul(p) p type θ prime p type θ prime 1 of dimension i of dimension i

    X X l(p)k X  X  l = − l(p)u = −  l(p) u   p type θ prime k≥1 l≥1 p type θ prime  of dimension i of dimension i l(p) | l

And calculating the Möbius inversion for the coefficients in the last expression, we have

X X − l(p)ul . (2.6) l≥1 p type θ prime of dimension i l(p)=l

On the other hand, we expect to get an equivalent expression if we apply the same operations to the right hand side of (2.5). Taking the logarithmic derivative, we obtain

! t n−t d 1 X αru X βju −u log = − − du Qt Qn−t 1 − α u 1 − β u r=1(1 − αru) j=1(1 − βju) r=1 r j=1 j

t n−t  t n−t  X X l X X l X X l X l l = − (αru) − (βju) = − αr + βj u . r=1 l≥1 j=1 l≥1 l≥1 r=1 j=1

And calculating the Möbius inversion for the coefficients in the last expression, we have

t n−t X X  l  X X  − µ αk+ βk ul k r j l≥1 k|l r=1 j=1   t n−t  t n−t  X X l X l X  l  X k X k  l = −  α + β + µ α + β  u  r j k r j  l≥1 r=1 j=1 k|l r=1 j=1 k

20  t  X X l θ l
l = − αr + o (λi,Γ) u . (2.7) l≥1 r=1

Finally, combining (2.6) and (2.7) we deduce a general version of the Prime Geodesic Theorem. This result will play a central role in the examples that will be presented in the next chapters.

Theorem 5. Let X be an infinite simply connected simplicial complex and Γ a subgroup i,θ of Aut(X) as given above. Then, for l large, the number of primes of XΓ satisfies

t θ l ! X 1 X (λi,Γ) 1 = αl + o . l r l p type θ prime r=1 of dimension i l(p)=l

2.3.3 Finite coverings of multicomplexes

Consider the subgroups Γ0 and Γ of Aut(X), such that both of them satisfy the conditions 0 assumed above for the finite quotients XΓ0 and XΓ, and so that Γ is a normal subgroup of Γ. Geometrically, this inclusion induces a natural covering map ϕ : XΓ0 → XΓ, with 0 0 0 automorphism group Γ/Γ in XΓ0 which commutes with ϕ. When |Γ/Γ | is finite, |Γ/Γ | is the degree of the covering map ϕ, and we call ϕ a Galois covering with Galois group Γ/Γ0. We will prove the following functoriality property for L-functions.

Theorem 6. Let X, Γ0 and Γ be as above. In particular, we assume that Γ0 is a normal subgroup of Γ, of finite index, so that there is a natural covering map ϕ : XΓ0 → XΓ. Let 0 (ρ, Vρ) be a finite dimensional unitary representation of Γ . Then, we have

Γ Li,θ(XΓ0 , ρ, u) = Li,θ(XΓ, IndΓ0 ρ, u) . (2.8)

Γ Proof. We shall denote by ψ the representation IndΓ0 ρ. Recall that the L-functions θ θ in (2.8) may be written in terms of the adjacency operators Ai,ρ and Ai,ψ, and the Γ representation space of ψ = IndΓ0 ρ may be identified with

|Γ/Γ0| M Vψ = hrVρ , h1 = e , (2.9) r=1

0 0 where hr, 1 ≤ r ≤ |Γ/Γ |, are representatives of the cosets in Γ/Γ . We claim that the vector spaces (cf. (2.2))

θ i,θ Γ0 Vi,ρ = (C(X ) ⊗ Vρ) , and

21 θ i,θ Γ Vi,ψ = (C(X ) ⊗ Vψ) ,

θ are isomorphic. Similarly to the proof of Theorem 4 we fix a set Si,Γ = {s1, . . . , sn} of i,θ θ 0 representatives of Γ\X . The set Si,Γ0 = {hrsk : 1 ≤ r ≤ |Γ/Γ |, 1 ≤ k ≤ n} is a set of 0 i,θ 0 θ θ representatives of Γ \X . We give bases {fjkr} and {fjkr} of Vi,ρ and Vi,ψ, respectively, that are described below.

( 0 0 0 0 0 ρ(γ )vj if r = m and k = t fjkr(γ hmst) = ρ(γ )fjkr(hmst) = , 0 otherwise ( 0 −1 0 0 ψ(γ hmhr )(e · vj) if k = t fjkr(γ hmst) = ψ(γ hm)fjkr(st) = , 0 otherwise 0 1 ≤ j ≤ dim Vρ , 1 ≤ k, t ≤ n , 1 ≤ r, m ≤ |Γ/Γ | .

0 i,θ Notice, that the term γ hmst runs through all cells of X . Furthermore, since Vψ can be described as multiple copies of Vρ (cf. (2.9)), we write e · vj to denote the image of vj in the copy corresponding to e ∈ Γ/Γ0. Now, we observe that the map

Q : V θ → V θ i,ψ i,ρ , 0 fjkr 7→ fjkr

θ is an isomorphism of vector spaces. In particular, since the image of the functions in Vi,ψ, θ corresponds to multiple copies of Vρ, we can decompose each f ∈ Vi,ψ as

θ 0 f = (e · f1, h2 · f2, . . . , h|Γ/Γ0| · f|Γ/Γ0|) , fj ∈ Vi,ρ , j = 1,..., |Γ/Γ | ,

Furthermore, since the component e · f1 in fjkr is nonzero only in the case r = m, in its definition, we have 0 Q(fjkr) = fjkr = f1 .

Additionally, as f is determined uniquely by f1, one can see that

−1 θ θ Q Ai,ρQ = Ai,ψ ,

θ
θ
and we conclude that det I − Ai,ρu = det I − Ai,ψu , which ends the proof.

The next result follows immediately from the definition of L-function.

Theorem 7. Let X and Γ be as above. Let (ρ1,Vρ1 ) and (ρ2,Vρ2 ) be two unitary

22 representations of Γ. Then,

Li,θ(XΓ, ρ1, u)Li,θ(XΓ, ρ2, u) = Li,θ(XΓ, ρ1 ⊕ ρ2, u) .

Γ Γ/Γ0 L 0 1 1 ρ ρ Now, we use the fact that IndΓ = Ind{e} = ρ∈Γ[/Γ0 deg( ) to decompose Zi,θ(XΓ0 , u). From Theorems 6 and 7, we obtain

Y deg(ρ) Zi,θ(XΓ0 , u) = Zi,θ(XΓ, u) Li,θ(XΓ, ρ, u) . (2.10) ρ∈Γ[/Γ0 ρ6=1

We also make the assumption that the smallest poles, in absolute value, of Zi,θ(XΓ0 , u) and Zi,θ(XΓ0 , u) are the same (including multiplicities). In particular, this implies that θ θ 0 θ λi,Γ0 = λi,Γ and that for any ρ ∈ Γ[/Γ , with ρ =6 1, the eigenvalues of Ai,ρ have absolute θ value smaller than λi,Γ. The following result is used to give the distribution of primes whose Frobenius conjugacy class is a given conjugacy class of Γ/Γ0. Therefore, it provides an analogous version of the celebrated Chebotarev Density Theorem from Number Theory, in the context of finite quotients of simplicial complexes.

Theorem 8. Let X be an infinite simply connected simplicial complex, Γ, Γ0 subgroups of Aut(X) as above and αr, r = 1, . . . , t as in (2.5). Then, for any conjugacy class C of 0 i,θ Γ/Γ the distribution of the primes of XΓ with Frobp = C satisfies

t |C| X X αl = l 1 + o(λθ )l
. |Γ/Γ0| r i,Γ r=1 p type θ prime of dimension i l(p)=l Frobp=C

Proof. From Lemma 1 we have

 θ l X X  l/k tr (Ai,ρ) = k tr ρ(Frobp) . (2.11) k|l p type θ prime of dimension i l(p)=k

Let C be a conjugacy class of Γ/Γ0 and τ ∈ C. If we multiply both sides of (2.11) by tr(ρ(τ)) and sum over ρ ∈ Γ[/Γ0, we obtain

X  θ l X X X  l/k tr(ρ(τ)) tr (Ai,ρ) = k tr(ρ(τ)) tr ρ(Frobp) ρ∈Γ[/Γ0 ρ∈Γ[/Γ0 k|l p type θ prime of dimension i l(p)=k

23 X X X  l/k = k tr(ρ(τ)) tr ρ(Frobp) , k|l p type θ prime ρ∈Γ[/Γ0 of dimension i l(p)=k and using Schur’s orthogonality relations, we have

X   X X |Γ/Γ0| tr(ρ(τ)) tr (Aθ )l = k . i,ρ |C| ρ∈Γ[/Γ0 k|l p type θ prime of dimension i l(p)=k l/k Frobp =C

This implies

|C|   tr (Aθ )l = |Γ/Γ0| i,Γ ! X X X X  θ l = l 1 + k 1 − tr(ρ(τ)) tr (Ai,ρ) . p type θ prime k|l p type θ prime ρ∈Γ[/Γ0 of dimension i k

θ l
We claim that the second term in the right hand side of this equation is o (λi,Γ) . This θ θ results from the facts that the eigenvalues of Ai,ρ are smaller than λi,Γ, in absolute θ l
value, as pointed out after (2.10), and the part summing over primes is o (λi,Γ) , by the  θ l Prime Geodesic Theorem (Theorem 5). Furthermore, if we write tr (Ai,Γ) in terms of αr, r = 1, . . . , t, we obtain the expression we wanted to prove.

2.3.4 Cumulative distribution of primes

We end the chapter with a result on sequences that, later on, will be used to describe the cumulative distributions of primes of given dimension and type over finite quotients of the buildings PGL3(F ) or PGSp4(F ). Recall the following well known lemma. ∞ ∞ Lemma 2. Let (an)k=1 and (bn)k=1 be two sequences of real numbers satisfying the following conditions

1. bk > 0, k ∈ N ; Pl 2. lim k=1 bk = ∞ ; and l→∞ 3. lim al = c . l→∞ bl Then, Pl ak lim k=1 = c . l→∞ Pl k=1 bk

24 Hashimoto [Has92] showed the following result.

Theorem 9 (Hashimoto [Has92]). Let a > 1 be a real number. Then,

l−1 X ak 1 al ∼ , l → ∞ . (2.12) k a − 1 l k=1

Proof. Partly motivated by the left hand side of (2.12), it will be useful to consider the following integral Z l ax I(k, l) := dx , 0 < k < l . k x Using integration by parts twice, we obtain

 ax l  ax l 2 Z l ax I(k, l) = + 2 2 + 3 3 dx . (2.13) x log a k x (log a) k (log a) k x

Taking into account the sizes of the numerators and denominators in each summand of (2.13), we easily see that

al I(1, l) ∼ , l → ∞ (2.14) l log a  ax k ak−1 a 1  ak−1 (a − 1)k − a I(k − 1, k) ∼ = − = x log a k−1 log a k k − 1 log a k(k − 1) ak−1 a − 1 ∼ , k → ∞ . (2.15) log a k − 1

By Lemma 2 and (2.15), we have

l l l−1 X a − 1 X ak−1 a − 1 X ak I(1, l) = I(k − 1, k) ∼ = , l → ∞ . (2.16) log a k − 1 log a k k=2 k=2 k=1

Combining (2.14) with (2.16) we obtain the desired theorem.

25 Chapter 3 | Analysis of PGL3(F )

3.1 The PGL3(F ) building

Let F be a nonarchimedean local field with uniformizer π, ring of integers OF and residue

field with size q. In this chapter, we denote by G the general linear group GL3(F ) and by K its standard maximal subgroup GL3(OF ). The center of G, denoted by Z, is the set of all nonzero scalar multiples of the identity matrix.

The structure of the Bruhat-Tits building X, associated to PGL3(F ) as a 2-dimensional contractible simplicial complex is explained in detail in [KL14]. In this chapter, we shall use the same notation almost every time, and for the sake of completion, we will introduce the necessary concepts again here.

3.1.1 Vertices

Let V be a 3-dimensional vector space over F . The set of vertices of X is given by the equivalence classes of rank 3 lattices over OF contained in V , with the following equivalence relation L ∼ L0 ⇔ L = zL0, z ∈ F × .

Unless otherwise stated, all lattices considered will have rank 3. We also write [L] to refer to the equivalence class of L. Note that selecting a basis of such a lattice and constructing the 3 × 3 matrix whose columns are the vectors from that basis, we obtain a parametrization of X0, the vertices of X, as the cosets of G/KZ, and we will also use this notation to refer to the vertices whenever it is convenient.

26 3.1.2 Edges

We say that two vertices [L1] and [L2] of X are adjacent if they have representatives L1 and L2, respectively, that satisfy

πL1 ( L2 ( L1 .

3 i Given that [L1 : πL1] = q , then [L1 : L2] = q , with i = 1 or 2. In each case, we say that the directed edge ([L1], [L2]) has type i, and [L2] is a type i out-neighbor of [L1].

Likewise, the reverse edge ([L2], [L1]) has type 3 − i and we say that the [L1] is a type 3 − i out-neighbor of [L ]. We will denote the set of directed edges of type i by X1,i. We −→ 2 also denote X1 = X1,1 ∪ X1,2 and the set of (undirected) edges of X by X1. Throughout the sections in this chapter we use the notation

    1 1     σ =  1  , and α =  1  . π π

Let E0 = (KZ, σKZ) be a type 1 edge in X. The stabilizer of E0, under left multiplication by K ⊂ G, is the standard parahoric subgroup of K, which is defined as

  ∗ ∗ ∗     −1   E = K ∩ σKσ = g ∈ K : g ≡  ∗ ∗ ∗  (mod πOF ), ∗ ∈ OF .    ∗ 

Since G acts transitively on X1,1 by left multiplication, the type 1 edges may be parametrized by the cosets of G/EZ, where gEZ represents the type 1 edge (gKZ, gσKZ).

3.1.3 Pointed chambers

Pointed chambers are ordered triples of mutually adjacent vertices. In specific, we say that ([L1], [L2], [L3]) is a pointed chamber if there are representatives L1, L2 and L3 of

[L1], [L2] and [L3], respectively, that satisfy

πL1 ( L3 ( L2 ( L1 .

In this case, the unordered set {[L1], [L2], [L3]} is called a chamber. Thus, each chamber shares the same vertices with three pointed chambers. We shall denote by X2 the set of −→ chambers in X, and by X2,1 or X2 the set of pointed chambers in X.

27 2 Let F0 = (KZ, σKZ, σ KZ) be a pointed chamber in X. The stabilizer of F0 under left multiplication by K ⊂ G is the Iwahori subgroup of K, which is defined as

  ∗ ∗ ∗     −1 2 −2   B = K ∩ σKσ ∩ σ Kσ = g ∈ K : g ≡  ∗ ∗  (mod πOF ), ∗ ∈ OF .    ∗ 

Again, note that the action of G is transitive, thus X2,1 can be parametrized by the cosets of G/BZ, so that gBZ represents the pointed chamber (gKZ,gσKZ,gσ2KZ). Moreover, we denote by Be the subgroup of G generated by B, Z, and hσi, and note that the set of chambers of X is parametrized by G/Be.

3.1.4 Apartments

The building associated to PGL3(F ) can also be realized as a union of Euclidean planes, called apartments. For any two chambers in X, there is an apartment containing both of them and, in particular, that apartment contains a geodesic path between the mentioned chambers. Conveniently, any apartment can be described as the set of vertices a b c [OF π u1 + OF π u2 + OF π u3], a, b, c ∈ Z, for a given choice of linearly independent a b c vectors u1, u2, u3 ∈ V . To abbreviate some notation, we will use [π , π , π ] to refer a b c to the vertex [OF π e1 + OF π e2 + OF π e3], where {e1, e2, e3} is the standard basis of V =∼ F 3 . We consider the standard aparment to be the one containing the vertices a b c [π , π , π ], a, b, c ∈ Z.

[1, π2, π] [1, π2, π2]

, π, 1] [1, π, π][1 [1, π, π2]

[π, π, 1] [1, 1, 1] [1, 1, π]

[π, 1, 1][ π, 1, π]

Figure 3.1. Part of the standard apartment, with labelled vertices.

28 3.2 Finite quotients of X

Let Γ be a discrete, cocompact, torsion-free subgroup of PGL3(F ). We will assume that ordπ(det Γ) ⊂ 3Z. The quotient XΓ = Γ\X is a finite connected 2-dimensional simplicial complex or multicomplex inheriting its structure from X. Thus, its set of vertices is 0 0 1 1 2 2 XΓ = Γ\X , its set of edges is XΓ = Γ\X and its set of chambers is XΓ = Γ\X . The 1,1 1,1 2,1 2,1 set of type 1 edges is XΓ = Γ\X and the set of pointed chambers is XΓ = Γ\X . By abuse of notation, we will refer to the multicells in XΓ using representatives from X.

Finally, we say that two cells in XΓ are adjacent if they have representatives in X that are adjacent.

3.3 Type 1 edge zeta function

We will see different kinds of Zeta functions in the sequel. The first such kind is the type

1 edge Zeta function, which is associated to geodesic cycles of type 1 edges in XΓ. Since a type 2 edge is the opposite of a type 1 edge, all results in this section can be directly adapted to type 2 edges.

3.3.1 Adjacency operator for type 1 edges

Recall that E0 = (KZ, σKZ) may be represented by EZ. We define the out-neighbors of gEZ as the type 1 edges with initial vertex gKZ. Out-neighbors are discussed again in detail in section 3.5.1. The proper out-neighbors of gEZ are a subset of its out-neighbors, given by the edges that belong to an apartment containing gEZ and that have the same direction as gEZ. As an example, αEZ is a proper out-neighbor of EZ (cf. figure 3.2). Then, since E is the stabilizer of EZ, the set of all proper out-neighbors of EZ is obtained from multiplying αEZ on the left by an element of E, and is represented by the cosets below:

  1 G   G EαEZ =  1  EZ = hEZ . x,y,∈OF /πOF xπ yπ π h

By transitivity of the G action, each type 1 edge gEZ has q2 proper out-neighbors, given 1,1 by ghEZ. Hence, the adjacency operator AE on the space C(X ) of complex-valued functions on X1,1 can be expressed as

29 X (AEf)(gEZ) = f(ghEZ) , hEZ where hEZ runs through all the proper out-neighbors of EZ.

The induced operator AE,Γ on functions on the type 1 edges of XΓ can be viewed as the restriction of AE to the following finite dimensional vector space,

1,1 1,1 VE,Γ = {f : X → C | f(γe) = f(e), ∀e ∈ X , γ ∈ Γ} .

σEZ

2 EZ σ

E0 αEZ

Figure 3.2. Some edges in the standard apartment.

These observations imply the following result.

2 Proposition 1. Each type 1 edge of X or XΓ has q proper out-neighbors.

We have a similar statement for out-neighbors, which is proved in Section 3.5.1.

2 Proposition 2. Each type 1 edge of X or XΓ has q + q + 1 out-neighbors.

3.3.2 Edge zeta function

Cycles in XΓ arising from closed paths of type 1 edges where any two consecutive edges are proper out-neighbors are called geodesic. Moreover, we call the equivalence classes of such geodesic cycles 1-dimensional primes of type 1. We now introduce a counting function of 1-dimensional primes of type 1. The type 1 edge zeta function is defined as

−1 Y  l(p) Z1,1(XΓ, u) = 1 − u . p type 1 prime of dimension 1

30 It was proven in [KL14] that this function is rational and satisfies

∞ ! X Nn(XΓ) 1 Z (X , u) = exp un = , 1,1 Γ n I − A u n=1 det( E,Γ ) where Nn(XΓ) is the number of type 1 geodesic cycles with length n in XΓ.

3.3.3 Edge L-functions

Let ρ be a finite d-dimensional unitary representation of Γ on Vρ, and define VE,ρ as the 1,1 set of of Vρ-valued functions on X compatible with ρ, that is,

1,1 1,1 VE,ρ = {f : X → Vρ f(γe) = ρ(γ)f(e), ∀e ∈ X , γ ∈ Γ} .

Then, we denote by AE,ρ the restriction of AE ⊗ idρ to VE,ρ. For a prime of dimension

1, p, Frobp is a conjugacy class of Γ defined in the same manner as in section 2.3. We define the type 1 edge L-function associated to ρ by

−1 Y  l(p) L1,1(XΓ, ρ, u) = det I − ρ(Frobp)u . p type 1 prime of dimension 1

These L-functions were studied to a great extent in [KL15] and, in particular, they were proved to be rational and given by the following formula.

Theorem 10. The type 1 edge L-function associated to XΓ and ρ satisfies

1 L1,1(XΓ, ρ, u) = . det(I − AE,ρu)

3.3.4 Main results for 1-dimensional primes of type 1

Our main results for type 1 edges concern the distribution of 1-dimensional primes of type

1 in XΓ. The next theorem is analogous to the Prime Number Theorem from Number Theory.

Theorem 11 (Prime Geodesic Theorem for 1-dimensional primes of type 1). The number of 1-dimensional primes of type 1 in XΓ satisfies

q6l |{p : p is a type 1 prime of dimension 1 in X with l(p) = 3l}| ∼ , Γ l as l → ∞.

31 And using Theorem 9, we obtain the following corollary.

Corollary 1. The number of 1-dimensional primes of type 1 in XΓ satisfies

1 q6l |{p : p is a type 1 prime of dimension 1 in XΓ with l(p) < 3l}| ∼ , q6 − 1 l as l → ∞.

We now wish to show a version of the Cheboratev Density Theorem for 1-dimensional primes of type 1. Consider Γ0 and Γ two discrete, cocompact, torsion-free subgroups of 0 PGL3(F ), so that Γ is a normal subgroup of Γ. Geometrically, this means that there is 0 a natural covering map ϕ : XΓ0 → XΓ, with Galois group Γ/Γ .

Theorem 12 (Chebotarev Density Theorem for cycles in natural density). Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

6l p is a type 1 prime of dimension 1 in X |C| q |{p : Γ }| ∼ , with l(p) = 3l and Frobp = C |Γ/Γ0| l as l → ∞.

Corollary 2. Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

6l p is a type 1 prime of dimension 1 in X |C| 1 q |{p : Γ }| ∼ , with l(p) < 3l and Frobp = C |Γ/Γ0| q6 − 1 l as l → ∞.

3.4 Chamber zeta function

Now, we analyze the paths obtained from combining adjacent chambers in XΓ.

3.4.1 Adjacency of chambers

We start by defining adjacency. Two chambers are adjacent if they share an edge. As each chamber has three edges, we can group its adjacent chambers according to the edge that is shared. We will use the following matrices

      π−1 1 1       t1 =  1  , t2 =  1  , and t3 =  1  . π 1 1

32 −1 i Note that σtiσ = ti+1, with indices taken modulo 3, and the edges σ EZ are fixed by left multiplication by ti. One can see that the chambers tiBe, i = 1, 2, 3, are the chambers from the standard apartment that are adjacent of F0 (which may be represented by Be), i along the type 1 edges σ EZ, i = 1, 2, 3. Furthermore, left multiplication by ti, i = 1, 2, 3, gives three different reflexions on the standard apartment. We call the paths formed by connecting adjacent chambers galleries. Each step of a gallery can be described by multiplying a chamber gBe by an element in Bte iB/e Be, i = 1, 2, 3. Amongst the galleries connecting two given chambers, those with the smallest number of intermediate chambers are called geodesic. To determine a geodesic gallery between g1Be and g2Be, we just need to find an element w = ti1 ti2 ··· tik , with the minimum −1 number of factors, and such that g1 g2 ∈ Bwe Be. Furthermore, such a geodesic follows a straight path in an apartment if and only if ij+1 − ij, j = 1, . . . , k, is constant modulo 3. In that case, we say that the geodesic has either type 1 or 2 depending on whether the difference of consecutive indices is 1 or 2 modulo 3.

3.4.2 Adjacency operator for directed chambers

In the case of pointed chambers, we give a slightly more specific definition. We say that 2 the pointed chamber g2BZ or (g2KZ, g2σKZ, g2σ KZ) is an out-neighbor of g1BZ or 2 2 (g1KZ, g1σKZ, g1σ KZ) if g1σKZ = g2KZ and g1σ KZ = g2σKZ. Moreover, we say that g2BZ is a proper out-neighbor of g1BZ if it is an out-neighbor with the added 2 condition that g1KZ =6 g2σ KZ. A path formed by consecutive proper out-neighbors of pointed chambers has the shape of a straight strip in an apartment. We will use the matrix   1   σt3 =  1 . π

Note that σt3BZ is an out-neighbor of BZ, which does not overlap with the latter and is also contained in the standard apartment (cf. figure 3.3). For that reason, we see that

σt3BZ is a proper out-neighbor of BZ. That said, by applying a transformation that fixes BZ, we see that all proper out-neighbors of BZ are represented by the following cosets,

  1 G   G Bσt3BZ =  1  BZ = hBZ . x∈OF /πOF xπ π h

Also, by transitivity of the G action, the proper out-neighbors of any given pointed

33 2,1 chamber gBZ in X are ghBZ. The corresponding adjacency operator AB on C(X ) sends f ∈ C(X2,1) to X (ABf)(gBZ) = f(ghBZ) . hBZ where hBZ runs through all proper out-neighbors of BZ.

We also have a similar operator, AB,Γ, on the finite quotient XΓ defined as the restriction of AB to the finite dimensional vector space

2 2,1 VB,Γ = {f : X → C | f(γc) = f(c), ∀c ∈ X , γ ∈ Γ} .

σt3σEZ

σt3BZ σEZ

F0 EZ

Figure 3.3. A gallery in the standard apartment, containing F0.

As a remark, the following result is an easy implication from the observations above.

Proposition 3. Each pointed chamber in X and XΓ has q proper out-neighbors.

In section 3.6.1, we prove the following result.

Proposition 4. Each pointed chamber in X and XΓ has q + 1 out-neighbors.

3.4.3 Chamber zeta function

A closed geodesic gallery in XΓ is said to have type 1 if its boundary is a union of type 1 edges. The equivalence classes of primitive closed geodesic galleries will be called 2-dimensional primes of type 1. The chamber zeta function collects information about the 2-dimensional primes of type 1 in XΓ. It is defined as

−1 Y  l(p) Z2,1(XΓ, u) = 1 − u . p type 1 prime of dimension 2

34 It was proven in [KL14] that this is a rational function and satisfies

∞ ! X Mn(XΓ) 1 Z (X , u) = exp un = , 2,1 Γ n I − A u n=1 det( B,Γ ) where Mn(XΓ) is the number of closed geodesic galleries in XΓ of type 1 and length n.

3.4.4 Chamber L-functions

Let ρ be a d-dimensional unitary representation of Γ on Vρ. We define the space of 2,1 Vρ-valued functions on X compatible with ρ as

2,1 2,1 VB,ρ = {f : X → Vρ f(γc) = ρ(γ)f(c), ∀c ∈ X , γ ∈ Γ } .

Furthermore, we define the adjacency operator AB,ρ as the restriction of AB ⊗ idρ to

VB,ρ. We associate to each 2-dimensional prime, p, a conjugacy class Frobp of Γ, just as in section 2.3. We define the type 1 chamber L-function attached to ρ as

−1 Y  l(p) L2,1(XΓ, ρ, u) = det I − ρ(Frobp)u . p type 1 prime of dimension 2

Yet again, the above L-functions were studied to great extent in [KL15] and it was proved that they are rational functions.

Theorem 13. The type 1 chamber L-function over XΓ attached to ρ has the following expression 1 L2,1(XΓ, ρ, u) = . det(I − AB,ρu)

3.4.5 Main results for 2-dimensional primes of type 1

Our main results for pointed chambers give the distribution of 2-dimensional primes of type 1 in XΓ. The next theorem is analogous to the Prime Number Theorem from Number Theory.

Theorem 14 (Prime Geodesic Theorem for 2-dimensional primes of type 1). The number of 2-dimensional primes of type 1 in XΓ satisfies

q3l |{p : p is a type 1 prime of dimension 2 in X with l(p) = 3l}| ∼ , Γ l as l → ∞.

35 And as a corollary, we obtain

Corollary 3. The number of 2-dimensional primes of type 1 in XΓ satisfies

1 q3l |{p : p is a type 1 prime of dimension 2 in XΓ with l(p) < 3l}| ∼ , q3 − 1 l as l → ∞.

Now, we state a version of the Chebotarev Density Theorem for 2-dimensional primes 0 0 of type 1. Consider Γ and Γ two discrete, cocompact subgroups of PGL3(F ), so that Γ is a normal subgroup of Γ and there is a natural covering map ϕ : XΓ0 → XΓ.

Theorem 15 (Chebotarev Density Theorem for galleries in natural density). Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

3l p is a type 1 prime of dimension 2 in X |C| q |{p : Γ }| ∼ , with l(p) = 3l and Frobp = C |Γ/Γ0| l as l → ∞.

Corollary 4. Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

3l p is a type 1 prime of dimension 2 in X |C| 1 q |{p : Γ}| ∼ , with l(p) < 3l and Frobp = C |Γ/Γ0| q3 − 1 l as l → ∞.

3.5 Proofs of Theorems 11 and 12

In this section we introduce a number of ideas applied to the 1-skeleton of X, in preparation to give a combinatorial proof of the Prime Geodesic Theorem and the Cheboratev Density Theorem for 1-dimensional primes of type 1. In the next two subsections we describe a way to construct geodesic cycles starting from the neighborhood of a vertex in XΓ and progressively extending to the whole XΓ.

Then, we will be able to prove some relevant properties of AE,Γ. Finally, we use our knowledge of the eigenvalues of AE,Γ to prove Theorems 11 and 12.

3.5.1 Edge representatives

Next, we will list the edges in the neighborhood of the vertex σKZ.

36 Type 1 edges with initial vertex σKZ

The edge E0 = (KZ, σKZ), which is represented by the coset EZ, has terminal vertex σKZ = αKZ, so the type 1 edges with initial vertex σKZ are the out-neighbors of EZ. One can easily check that such edges are given by gEZ, where g ∈ G satisfies gKZ = αKZ. So, we conclude that the type 1 edges with initial vertex σKZ = αKZ are parametrized by the cosets in

EαKZ/EZ .

By the definition of E, we have Eα ⊂ αK, thus we may simplify the quotient above as αKZ/EZ, and we may use the class representatives shown below

αKZ =       1 1 1 G   G      1  EZ ∪  x 1  EZ ∪  1  EZ. x,y∈OF /πOF xπ yπ π x∈OF /πOF π π

Additionally, this enumeration proves Proposition 2.

Type 1 edges with terminal vertex σKZ

In order to describe the type 1 edges with terminal vertex σKZ, we note that they may be represented by the cosets gEZ such that gαKZ = gσKZ = σKZ. For this reason, they are parametrized by the cosets in

EαKα−1EZ/EZ .

From the definition of E we have Eα ⊂ αK and α−1E ⊂ Kα−1, so we may simplify the quotient above and use the class representatives shown below

    1 xπ−1 π−1 −1 G  −1  G  −1  αKZα =  1 yπ  EZ ∪  1 xπ  EZ x,y∈OF /πOF 1 x∈OF /πOF π   1  −1  ∪  π  EZ. π

37 Naturally, a type 1 edge with initial vertex σKZ is an out-neighbor of each type 1 edge with terminal vertex σKZ. We describe a useful map between these two sets of edges below.

3.5.2 Successor map

Using all the coset representatives computed above, we now wish to define a map sE that assigns to each type 1 edge in αKZα−1/EZ one of its proper out-neighbors in αKZ/EZ. Furthermore, we will require this map to be bijective, and we use the fact that proper out-neighbors of a type 1 edge are obtained from right multiplication by an element in

EαE/EZ. Later on, we construct the map for all type 1 edges of XΓ. We begin by constructing sE for the edges with terminal point αKZ:

      1 xπ−1 1 xπ−1 1  −1  sE  −1    (1)  1 yπ  EZ −→  1 yπ   1  EZ 1 1 yπ −xπ π       1 + xy −x2 x 1 −x 1 − xy x2  2     2  =  y 1 − xy y   1 −y   −y 1 + xy  EZ yπ −xπ π 1 1   1   =  1  EZ, for x, y ∈ OF /πOF yπ −xπ π

        1 1 1 1  −1  sE  −1      (2)  π  EZ −→  π   1  EZ =  1  EZ π π π π

      π−1 π−1 1  −1  sE  −1    (3)  1 −π  EZ −→  1 −π   1  EZ π π π       1 1 1       =  1 −1   1 1  EZ =  1  EZ π 1 π

38       π−1 π−1 1  −1  sE  −1    (4)  1 xπ  EZ −→  1 xπ   1  EZ π π π π       1 1 1 1       =  1 + x x   1 1  EZ =  1 + x 1  EZ π −1 π

for x ∈ OF /πOF and x 6≡ −1 (mod πOF ) .

In order to extend sE to all type 1 edges of XΓ, we consider SΓ(K) a set of representatives of the orbits in Γ\X0. Then, we define

−1 −1 −1 sE(aα · gEZ) := aα · sE(gEZ) , ∀a ∈ SΓ(K), gEZ ∈ αKZα /EZ .

Note that the expression above does indeed give a bijection, because for any given a ∈ −1 −1 SΓ(K), aα gEZ runs through all the edges with terminal vertex aKZ and aα sE(gEZ) runs through all the edges with initial vertex aα−1αKZ = aKZ, as we vary gEZ ∈ αKZα−1/EZ.

3.5.3 Properties of AE,Γ

Here we show two relevant properties of AE,Γ.

Proposition 5. The adjacency operator AE,Γ is irreducible.

Proof. To prove that AE,Γ is irreducible for any XΓ we shall see that any two distinct type

1 edges in XΓ can be connected using a type 1 path that only uses proper out-neighbors in every step. Note that as X is connected we can join any two edges using a path of type 1 and type 2 edges. Moreover, any type 2 edge has the same initial and terminal vertices as a path composed by two type 1 edges that form a chamber together with the former. Thus, we conclude that we can join any pair of type 1 edges in X and XΓ using a type 1 path. The only thing that remains to prove is that two type 1 edges belonging to a common chamber of XΓ can be joined using a type 1 geodesic path.

Suppose that g0EZ and g1EZ are two type 1 edges of XΓ, such that the terminal vertex of g0EZ is the same as the initial vertex of g1EZ, and yet g1EZ is not a proper 0 out-neighbor of g0EZ. We wish to prove that there is an edge g0EZ with the same terminal vertex as g0EZ and that satisfies:

0 (i) sE(g0EZ) is a proper out-neighbor of g0EZ ;

39 0 (ii) g1EZ is a proper out-neighbor of g0EZ .

0 We start by noting that there are at most q + 1 exceptions sE(g0EZ) to (i) and at most 0 q + 1 exceptions g0EZ to (ii), by Propositions 1 and 2. So, the number of the remaining 0 possibilities for g0EZ is

q2 + q + 1 − 2(q + 1) = q2 − q − 1 > 0 .

Hence, our claim holds. 0 Now, starting from g0EZ, we see that sE(g0EZ) is a proper out-neighbor. The k 0 1,1 1,1 sequence (sE(g0EZ))k≥1 gives an infinite geodesic path in XΓ . However, since XΓ is finite it must have a repeated edge which, together with the fact that sE is a bijection, implies that the sequence is periodic. Therefore, there is a geodesic path like the following

0 2 0 0 g0EZ → sE(g0EZ) → sE(g0EZ) → ... → g0EZ → g1EZ.

In particular, note that this geodesic path starts at g0EZ and ends at g1EZ, and this concludes the proof that AE,Γ is irreducible.

Proposition 6. AE,Γ has period 3.

Proof. First of all, recall that any proper out-neighbor of a type 1 edge gEZ can be written as ghEZ, where h ∈ EαEZ. In this case, we see that ordπ(det h) = 1. Consequently, since ordπ(det Γ) ⊂ 3Z, the length of a type 1 geodesic cycle in XΓ must be divisible by 3. 2 Now, consider the pointed chamber (KZ, σKZ, σ KZ) in XΓ. Using results and notation from the previous proposition, one can see that there is a geodesic cycle of type

1 edges like the one represented below. We use g0EZ for EZ, g1EZ for σEZ and g2EZ for σ2EZ.

EZ 0 2 0 0 → sE(g0EZ) → sE(g0EZ) → ... → g0EZ

→ g1EZ 0 2 0 0 → sE(g1EZ) → sE(g1EZ) → ... → g1EZ

→ g2EZ 0 2 0 0 → sE(g2EZ) → sE(g2EZ) → ... → g2EZ → EZ

40 Notice that this geodesic contains three intermediate geodesic loops that may be repeated more than once. Now, call d to the sum of the lengths of the intermediate loops. Consequently, the total length of the above geodesic is 3 + d. However, if instead we consider that each loop is repeated 3 + d times, the resulting cycle is still geodesic and will have length 3 + (3 + d)d. Therefore, we have constructed two geodesic cycles of type 1 with lengths 3 + d and 3 + (3 + d)d, and gcd 3 + d, 3 + (3 + d)d
= gcd(3 + d, 3) ≤ 3, which concludes the proof of the desired result.

3.5.4 Proof of Theorem 11

By Propositions 5 and 6 we know that the adjacency operator AE,Γ, acting on type 1 edges, is irreducible and has period 3. Furthermore, taking into account the fact that each edge has q2 proper out-neighbors, we deduce by the Perron-Frobenius Theorem that 2 2 2 2 q , ζ3q and ζ3 q are the eigenvalues of AE,Γ with largest absolute value. Together with the general Prime Geodesic Theorem (Theorem 5) we conclude that

X 1 q6l  1 = 3q6l + o . l l p type 1 prime 3 of dimension 1 l(p)=3l

This proves the theorem.

3.5.5 Proof of Theorem 12

Γ Γ/Γ0 L 0 1 1 ρ ρ We use the fact that IndΓ = Ind{e} = ρ∈Γ[/Γ0 deg( ) to see the decomposition

Y deg(ρ) Z1,1(XΓ0 , u) = Z1,1(XΓ, u) L1,1(XΓ, ρ, u) . ρ∈Γ[/Γ0 ρ6=1

Moreover, in the previous proof we saw that the smallest poles of Z1,1(XΓ0 , u) and 2 2 2 2 Z1,1(XΓ, u), in absolute value, are 1/q , 1/ζ3q and 1/ζ3 q for both Zeta functions. Consequently, the remaining L-functions in the decomposition have poles with absolute 2 values larger than 1/q , since each L1,1(XΓ, ρ, u) is the reciprocal of a polynomial.

As seen above, the eigenvalues of AE,Γ0 and AE,Γ with the largest absolute value 2 2 2 2 are q , ζ3q and ζ3 q for both operators. In this situation, we can apply the general

41 Chebotarev Density Theorem (Theorem 8), which implies

|C| X 3q6l = 3l 1 + o(q6l) . | / 0| Γ Γ p type 1 prime of dimension 1 l(p)=3l Frobp=C

As a result X |C| q6l q6l  1 = + o | / 0| l l p type 1 prime Γ Γ of dimension 1 l(p)=3l Frobp=C and the proof is finished.

3.6 Proofs of Theorems 14 and 15

In this section we wish to give proofs of the Prime Geodesic Theorem and the Chebotarev Density Theorem for closed geodesic galleries, similarly to what was done for type 1 geodesic cycles. In the first two sections, we describe the pointed chambers in the neighborhood of an edge in XΓ. Then, we show some properties of AB,Γ that we use to characterize its eigenvalues and prove the desired results.

3.6.1 Pointed chamber representatives

Pointed chambers containing the edge σEZ

We wish to define a map from the set D1 that contains the pointed chambers g1BZ or 2 2 2 (g1KZ, g1σKZ, g1σ KZ) that satisfy g1σKZ = σKZ and g1σ KZ = σ KZ to the set 2 D2 of pointed chambers g2BZ or (g2KZ, g2σKZ, g2σ KZ) that satisfy g2KZ = σKZ 2 and g2σKZ = σ KZ. The first set D1 may be given by the elements g1 ∈ G that fix the type 1 edge σEZ, so it may be parametrized by the cosets of BσEZσ−1B/BZ or equivalently σEZσ−1/BZ as, by definition, σ−1Bσ ⊂ E.

−1 −1 D1 = {g1BZ : g1σEZ = σEZ} = {g1BZ : g1 ∈ σEZσ } = σEZσ /BZ

Hence the pointed chambers in D1 are represented by the cosets

42     1 xπ−1 π−1 −1 G     σEZσ =  1  BZ ∪  1  BZ. x∈OF /πOF 1 π

As for the second set D2, we note that it corresponds to the elements g2 ∈ G that map EZ to σEZ.

D2 = {g2BZ : g2EZ = σEZ} = {g2BZ : g2 ∈ σEZ} = σEZ/BZ

Thus, we can parametrize these pointed chambers with the cosets of BσEZ/BZ, or just σEZ/BZ, described below:

    x 1 1 G     σEZ =  1  BZ ∪  1  BZ. x∈OF /πOF π π

Additionally, Proposition 4 follows immediately from this partition.

3.6.2 Successor map

−1 We now define a map sB from D1 = σEZσ /BZ to D2 = σEZ/BZ, so that it is bijective and it maps each chamber to one of its proper out-neighbors. Recall that the proper out-neighbors of a pointed chamber are obtained by right multiplication by an element in Bσt3BZ. The following sB has the desired properties:

        π−1 π−1 1 1   sB       (1)  1  BZ −→  1   1 BZ =  1 BZ π π π π

        1 −π−1 1 −π−1 1 1 −1   sB       (2)  1  BZ −→  1   1 BZ =  1 BZ 1 1 π π   1   =  1 BZ π

43       1 xπ−1 1 xπ−1 1   sB     (3)  1  BZ −→  1   1 BZ 1 1 π π       x + 1 x 1 1 x + 1 1       =  1  −1  BZ =  1 BZ, π π 1 π

for x ∈ OF /πOF and x 6≡ −1 (mod πOF )

We shall extend this map to any pointed chamber in XΓ. In order to do so, let SΓ(E) be a set of representatives of the orbits in Γ\X1,1. Then, we define

−1 −1 −1 sB(aσ gBZ) := aσ sB(gσBZ) , ∀a ∈ SΓ(E), gBZ ∈ σEZσ /BZ .

0 0 0 0 2 Note that each pointed chamber g BZ = (g KZ, g σKZ, g σ KZ) of XΓ lies in a unique set −1 0 0 2 of the form aσ D1, where a ∈ SΓ(E) and (g σKZ, g σ KZ) = aEZ. Similarly, a pointed 0 0 0 0 2 −1 chamber g BZ = (g KZ, g σKZ, g σ KZ) of XΓ lies in a unique set of the form aσ D2, 0 0 −1 −1 where a ∈ SΓ(E) and (g KZ, g σKZ) = aEZ. Moreover, we have sB(aσ D1) = aσ D2.

This way, the map becomes defined over all pointed chambers of XΓ and is indeed bijective.

3.6.3 Properties of AB,Γ

We now show two useful properties of AB,Γ.

Proposition 7. AB,Γ is irreducible.

Proof. The building X is a connected simplicial complex, so there is a gallery joining any pair of pointed chambers of X. Likewise, the same statement should be true for XΓ.

However, AB,Γ corresponds to sums solely over proper out-neighbors, so it is not obvious that this operator is irreducible.

It suffices to show that it is always possible to find a geodesic gallery in XΓ connecting any given pointed chamber gBZ to one of its rotations, obtained from multiplication by

σ, that is, gσBZ. We claim that given the directed chambers g0BZ and g1BZ = g0σBZ 0 in XΓ, there is another directed chamber g0BZ that satisfies:

0 (i) g0σEZ = g0σEZ ;

0 (ii) g1BZ is a proper out-neighbor of g0KZ ;

0 (iii) sB(g0BZ) is a proper out-neighbor of g0BZ .

44 We first note that there are q +1 pointed chambers satisfying condition (i), by Proposition 4. From those pointed chambers there is 1 that does not satisfy condition (ii), and 1 that 0 does not satisfy (iii), by Proposition 3. Thus, we can pick g0BZ from a non-empty set with at least q + 1 − 2 = q − 1 > 0 pointed chambers.

As XΓ is finite, we can find a closed geodesic gallery that runs through the following pointed chambers in the order displayed:

g0BZ 0 2 0 0 → sB(g0BZ) → sB(g0BZ) → ... → g0BZ → g0σBZ = g1BZ.

We conclude that AB,Γ is irreducible as we desired.

Proposition 8. AB,Γ has period 3.

Proof. We start by seeing that any proper out-neighbor of the pointed chamber gBZ can be given by right multiplication by an element in Bσt3B and that ordπ(det Bσt3B) = {1}. As ordπ(det Γ) ⊂ 3Z, we conclude that the length of a closed geodesic gallery in XΓ is divisible by 3. We now verify that 3 is the exact value of the period. Using the directed 2 chambers g0BZ, g1BZ = g0σBZ and g2BZ = g0σ BZ, we can use the same strategy as in the previous proof to find the following closed geodesic gallery:

g0BZ 0 2 0 0 → sB(g0BZ) → sB(g0BZ) → ... → g0BZ → g1BZ = g0σBZ 0 2 0 0 → sB(g1BZ) → sB(g1BZ) → ... → g1BZ 2 → g2BZ = g0σ BZ 0 2 0 0 → sB(g2BZ) → sB(g2BZ) → ... → g2BZ → g0BZ.

Now, denote by d the sum of the lengths of the three intermediate loops in the diagram above. Then, the total length of this geodesic gallery is d + 3. If instead, we repeat each loop d + 3 times, we still obtain a geodesic gallery, but this time with length 3 + (d + 3)d. One can easily see that the greatest common divisor of 3 + d and 3 + (d + 3)d is at most

3. Therefore, it follows that the period of AB,Γ is 3.

45 3.6.4 Proof of Theorem 14

From Propositions 7 and 8 we determined that AB,Γ is irreducible and has period 3. Moreover, keeping in mind that each directed chamber has q proper out-neighbors, we 2 conclude by the Perron-Frobenius Theorem that q, ζ3q, and ζ3 q are the eigenvalues of AB,Γ with the largest absolute value. Together with the general Prime Geodesic Theorem (Theorem 5) we obtain

X 1 q3l  1 = 3q3l + o . l l p type 1 prime 3 of dimension 2 l(p)=3l

This proves the desired result.

3.6.5 Proof of Theorem 15

Similarly to what we showed for the edge L-functions, we have a decomposition

Y deg(ρ) Z2,1(XΓ0 , u) = Z2,1(XΓ, u) L2,1(XΓ, ρ, u) . ρ∈Γ[/Γ0 ρ6=1

Observe that the poles of Z2,1(XΓ0 , u) and Z2,1(XΓ, u) with the smallest absolute value 2 are 1/q, 1/(ζ3q) and 1/(ζ3 q), for both of them, thus the remaining L-functions in the decomposition have poles with larger absolute value.

We already know that the eigenvalues of AB,Γ0 and AB,Γ with the largest absolute 2 value are q, ζ3q and ζ3 q for both operators. For that reason, we can apply the Chebotarev Density Theorem (Theorem 8) and we obtain

|C| X 3q3l = 3l 1 + o(q3l) , | / 0| Γ Γ p type 1 prime of dimension 2 l(p)=3l Frobp=C which implies X |C| q3l q3l  1 = + o . | / 0| l l p type 1 prime Γ Γ of dimension 2 l(p)=3l Frobp=C This concludes the proof.

46 3.7 Error terms of Ramanujan complexes

In the case of finite quotients of the building associated to PGL3(F ), we can give a better estimate of the error term associated to the estimates in Theorems 11 and 14, if we know that XΓ is Ramanujan. Ramanujan complexes were introduced in [Li04] and [LSV05b], and they generalize the Ramanujan property, that was known for graphs, to higher dimensions. In [KLW10], Kang-Li-Wang showed that XΓ being Ramanujan is −1 − 1 equivalent to all nontrivial poles of Z1,1(XΓ, u) having absolute value q or q 2 , and is − 1 − 1 equivalent to all nontrivial poles of Z2,1(XΓ, u) having absolute value 1, q 4 , or q 2 .

As a consequence, when XΓ is Ramanujan, we have clear bounds on the poles of

Z1,1(XΓ, u) and Z2,1(XΓ, u), and we can improve the error term of the Prime Geodesic Theorem for 1 and 2-dimensional primes of type 1.

Theorem 16. Let Γ be a discrete, torsion-free, cocompact subgroup of PGL3(F ), such that XΓ is Ramanujan. Then, the number of 1-dimensional primes of type 1 onXΓ satisfies

X q6l q3l  1 = + O . l l p type 1 prime of dimension 1 l(p)=3l

Proof. In section 2.3.2, we can combine equations (2.5) and (2.7) to obtain

t n−t X 1 X 3l X X  1 = µ αk + βk 3l k r j p type 1 prime k|3l r=1 j=1 of dimension 1 l(p)=3l t n−t t n−t 1 X 1 X 1 X 3l X X  = α3l + β3l + µ αk + βk , 3l r 3l j 3l k r j r=1 j=1 k|3l r=1 j=1 k<3l where 1/αr are the trivial poles of Z1,1(XΓ, u), and 1/βj are the nontrivial poles of 3 6 1 Z1,1(XΓ, u). Using the facts that αr = q , t = 3 and |βj| = q or q 2 , the value of the last expression becomes

q6l  q3l  q6l q3l  = + O n = + O , l l l l

1,1 where n = XΓ . This concludes the proof of the theorem.

Theorem 17. Let Γ be a discrete, torsion-free, cocompact subgroup of PGL3(F ), such

47 that XΓ is Ramanujan. Then, the number of 2-dimensional primes of type 1 in XΓ satisfies 3l  3 l  X q q 2 1 = + O . l l p type 1 prime of dimension 2 l(p)=3l

Proof. We use the same procedure as in the previous proof. Recall that the trival poles 3 3 1/αr of Z2,1(XΓ, u) satisfy αr = q , and the nontrivial poles 1/βj of Z2,1(XΓ, u) satisfy 1 1 |βj| = 1, q 4 or q 2 . Then, we have

t n−t X 1 X 3l X X  1 = µ αk + βk 3l k r j p type 1 prime k|3l r=1 j=1 of dimension 2 l(p)=3l t n−t t n−t 1 X 1 X 1 X 3l X X  = α3l + β3l + µ αk + βk 3l r 3l j 3l k r j r=1 j=1 k|3l r=1 j=1 k<3l 3l 3 l 3l 3 l q  q 2  q q 2  = + O n = + O , l l l l

2,1 where n = XΓ . This concludes the proof of the theorem.

48 Chapter 4 |

Analysis of PGSp4(F )

Consider a nonarchimedean local field F with a residue field of size q. In this chapter we describe some geometric properties of simplicial complexes arising as finite quotients of the Bruhat Tits building associated to the group Sp4(F ). These simplicial complexes were studied before by Fang-Li-Wang [FLW13] and Kang-Li-Wang [KLW10]. In particular, the results that we show are also a consequence of the analysis carried out in [KLW10]. However, whilst the mentioned papers use the perspective of representation theory, we use the combinatorics point of view and our approach is self-contained.

4.1 The symplectic group

4.1.1 Definitions

Let F be a nonarchimedean local field with uniformizer π, ring of integers OF and residue field with size q. Consider V a 4-dimensional vector space over F equipped with an alternating bilinear form h·, ·i, and a standard basis {e1, e2, f1, f2} satisfying

he1, f2i = he2, f1i = 1 ;

hek, fki = 0 , k = 1, 2 ;

hei, eji = hfi, fji = 0 , 1 ≤ i, j ≤ 2 .

Any basis {u1, u2, w1, w2} satisfying these conditions is called a symplectic basis. The symplectic group Sp4(F ) is the set of matrices preserving h·, ·i, that is, g ∈ Sp4(F ) if and

49 only if   1   T  1  g Jg = J , where J =   .  −   1  −1

Additionally, define the group GSp4(F ) as the set of matrices g ∈ GL4(F ) such that

gT Jg = λ(g)J, for some scalar λ(g) ∈ F × , and the group PGSp4(F ) as GSp4(F )/Z, where Z is the center of GSp4(F ), which consists of the non-zero scalar multiples of the identity matrix.

We will denote by G the group GSp4(F ), and by G0 the index 2 subgroup of G consisting of the elements g ∈ G such that ordπ(det g) ≡ 0 (mod 4). We will use

  1    1  τ =   ,  π    π and one can easily see that G = G0 ∪ G0τ .

We will also call K to the standard maximal subgroup of G, that is, GSp4(OF ).

4.1.2 Lattices

Denote by L a rank 4 lattice over OF contained in V . The group GL4(F ) acts tran- sitively on the set of lattices of V . In specific, selecting a basis for any given lattice L and constructing the 4 × 4 matrix whose columns are the vectors in that basis, we obtain a parametrization of all rank 4 lattices of V over OF given by the cosets of

GL4(F )/ GL4(OF ).

We say that a lattice L is primitive if hL, Li ⊂ OF and h·, ·i induces a nondegenerate alternating bilinear form h·, ·iL/πL on the vector space L/πL over OF /πOF . Note that, by definition, Sp4(F ) acts transitively on the set of primitive lattices. Moreover, we 0 0 0 call a lattice L incident if hL ,L i ⊂ πOF and there is a primitive lattice L, such that πL ⊂ L0 ⊂ L . For any lattice L in V , we denote by [L] its equivalence class under the following equivalence relation

L ∼ L0 ⇔ L = αL0 , α ∈ F × .

50 Let L0, L1, L2 and L3 be the lattices defined by

L0 = OF e1 + OF e2 + OF f1 + OF f2 ;   1    1  L1 = OF e1 + OF e2 + OF f1 + πOF f2 =   L0 ;    1  π   1    1  L2 = OF e1 + OF e2 + πOF f1 + πOF f2 =   L0 ;  π    π   1    π  L3 = OF e1 + πOF e2 + πOF f1 + πOF f2 =   L0 .  π    π

Out of these four, L0 is primitive, and L2 and L3 are incident. Furthermore, [τ·Li] = [Li+2], where the indices are taken modulo 4.

Dual lattice

We define the dual lattice of L as

∗ L = {v ∈ V | hv, wi ∈ OF , ∀w ∈ L} .

One can verify that for a lattice L = g · L0, where g ∈ GL4(F ), the dual operation can be calculated as follows ∗ −1 T L = (Jg J) · L0 .

As a result of this, we see that [L0] and [L2] are fixed by the dual operator and the same operator interchanges [L1] with [L3]. Using basic algebra, one can also verify that for

L = gh · L0 = g · (hL0) we have

∗ −1 T −1 −1 T −1 T −1 T L = (J(gh) J) · L0 = −(Jh JJg J) · L0 = −(Jg J) (Jh J) · L0 −1 T ∗ = −(Jg J) · (hL0) .

Therefore, if g ∈ G, then −(Jg−1J)T = λ(g)−1g ∈ gZ, and we see that, up to scalar multiplication, left multiplication by g commutes with the dual operator. Moreover, for

51 hL0 = Li, i = 1, 2, 3 we have ( ∗ ∗ [gLi] , if i is even [(gLi) ] = [gLi ] = . [gL4−i] , if i is odd

In the continuation, we will further identify dual lattices as the same vertex, thus the orbits of L0, L2 and L3 (or L1) will contain all primitive and incident lattices.

4.1.3 Arithmetic of GSp4(F )

As mentioned above, the group GSp4(F ) is given by the matrices g ∈ GL4(F ) that satisfy

gT Jg = λ(g)J, for some scalar λ(g) ∈ F ×.

For the matrix g below , this condition becomes   a1d4 + a2d3 − a3d2 − a4d1 = λ(g)    a b c d  b c + b c − b c − b c = λ(g) 1 1 1 1  1 4 2 3 3 2 4 1    a2 b2 c2 d2  a1b4 + a2b3 − a3b2 − a4b1 = 0 g =   , . a3 b3 c3 d3  a1c4 + a2c3 − a3c2 − a4c1 = 0     a4 b4 c4 d4  d1b4 + d2b3 − d3b2 − d4b1 = 0   d1c4 + d2c3 − d3c2 − d4c1 = 0

These 6 equations are useful while doing elementary operations with GSp4(F ). Further- more, it is useful to keep in mind that GSp4(F ) is closed under transpose.

We can easily verify that GSp4(F ) contains the matrices     1 1     1   1    ,   ,    −   1  1  1 1 that permute rows or columns. It also contains the diagonal matrices     1 s      1   s  ×   ,   , r, s ∈ F .  r   s      r s

52 And finally, it contains the following unipotent matrices         1 a 1 1 c 1 d          1   1 b   1 c  1    ,   ,   ,   , a, b, c, d ∈ OF .  −a        1   1   1   1  1 1 1 1

If we choose pivots appropriately, for example, the entries of g with smaller order, we conclude through a proccess of elimination, that the elementary matrices used above generate the whole group GSp4(F ). This fact will be especially useful later on to list coset representatives.

4.2 The Bruhat-Tits building associated to Sp4(F )

4.2.1 Vertices, edges and chambers

Denote by X the building associated to the simplicial group Sp4(F ) . The vertices of X are given by the equivalence classes of lattices [L] that contain a primitive or an incident lattice. If L is a primitive lattice, we say that the vertices [L] and [L0] are connected by an edge if there is a representative L0 of [L0], such that πL ⊂ L0 ⊂ L. Otherwise, we say that two vertices [L0] and [L00] are connected by an edge if they have representatives L0 and L00, respectively, such that either L0 ⊂ L00 or L00 ⊂ L0, and there is a primitive lattice L such that πL ⊂ L0,L00 ⊂ L. Furthermore, in this situation we say that the vertices [L], [L0] and [L00] form a chamber. X can be given as a union of planes like the one shown in figure 4.2. In the figure, the vertices on the intersection of vertical and horizontal lines are called special and the remaining ones are called nonspecial.

4.2.2 Types of vertices and edges

For any lattice L, there is an element g ∈ GL4(F ) such that L = gL0. We define the type of [L] as the congruence class of ordπ(det g) modulo 4, except for when ordπ(det g) ≡ 1 (mod 4), in which case we say that it has type 3, since that is the type of its dual. Thus, the vertices of X have either types 0, 2, or 3. Type 0 vertices correspond to the equivalence classes containing primitive lattices. Vertices with types 0 and 2 are the special vertices and vertices with type 3 are the nonspecial vertices. G acts transitively by left multiplication on the set of special vertices of X and the stabilizer of [L0] is K = GSp4(OF ), so we may parametrize the special vertices by G/KZ.

53 G also acts transitively on the set of nonspecial vertices. We also observe that, right multiplication by elements of the group K4 = GL4(OF )Z corresponds to changing the basis of a lattice, while preserving it as a set. Under such transformation the new basis is not necessarily a symplectic basis. That said, we can also refer to the vertices of X using cosets of the form gK4 where g represents a lattice in G · [L0] ∪ G · [L3] . We say that a directed edge has type 1 if it connects two special vertices, and has type 2 if it connects one special and one nonspecial vertices. Note that there are no edges between two nonspecial vertices. We give another description for the edges of X below.

Let E1 be the type 1 edge from [L0] to [L2]. The stabilizer of E1 in K is the Siegel congruence subgroup and is given by

−1 P1 = K ∩ diag(1, 1, π, π)K diag(1, 1, π, π)     ∗ ∗ ∗ ∗         ∗ ∗ ∗ ∗   = g ∈ K : g ≡   (mod πOF ) .   ∗ ∗        ∗ ∗ 

As G acts transitively on the type 1 edges of X, we conclude that these are parametrized by the cosets from the quotient G/P1Z. The set of type 1 edges of X will be denoted by X1,1.

Now, consider the directed edge E2 from [L0] to [L3]. The stabilizer of E2 in K is the Klingen congruence subgroup given by

−1 P2 = K ∩ diag(1, π, π, π)K4 diag(1, π, π, π)     ∗ ∗ ∗ ∗         ∗ ∗ ∗   = g ∈ K : g ≡   (mod πOF ) .   ∗ ∗ ∗        ∗ 

By transitivity of the G action on the type 2 edges of X, we conclude that these are parametrized by the cosets from the quotient G/P2Z. The set of type 2 edges of X will be denoted by X1,2. Later on, we will see that in order to create paths, it is useful to use a variation of the type 2 edges, that we call long type 2 edges. These are edges contained in some apartment (cf. section 4.2.4) that are obtained from extending a type 2 edge by a factor 2 of 2. Consider E3 the long type 2 edge from KZ to diag(1, π, π, π )KZ. The stabilizer of E3 in K is the following group

54 0 2 2 −1 P2 = K ∩ diag(1, π, π, π )K diag(1, π, π, π )     ∗ ∗ ∗ ∗         π∗ ∗ ∗ ∗   = g ∈ K : g =   , ∗ ∈ OF .   π∗ ∗ ∗ ∗        π2∗ π∗ π∗ ∗ 

The set of long type 2 edges is also given by the orbit G·E3, and thus may be parametrized 0 1,3 by G/P2Z. We denote the set of long type 2 edges by X . Lastly, we denote by X1 the set of undirected type 1 and 2 edges of X, and we use −→ X1 to denote X1,1 ∪ X1,2.

4.2.3 Types of chambers

As mentioned before, chambers are the 2-dimensional cells of X. Chambers equipped with an orientation, that is, where the order of their vertices is not arbitrary, are called pointed chambers. As an example, let C be the pointed chamber ([L0], [L2], [L3]). The stabilizer of C is the Iwahori group given by     ∗ ∗ ∗ ∗         ∗ ∗ ∗  I = P1 ∩ P2 = g ∈ K : g ≡   (mod πOF ) .   ∗ ∗       ∗ 

As a result, we see that the pointed chambers of X are described by the cosets in G/IZ. −→ The set of pointed chambers will be denoted by X2,1 or X2 and we use X2 to denote the set of chambers. In what follows, we will use the following matrices     −π−1 1      1   1  s0 =   , and s2 =   .    −   1   1  π 1

Left multiplication by these two matrices gives two different reflexions on the standard apartment (cf. section 4.2.4) with respect to lines parallel to type 2 edges. We will also

55 use the matrices     1 1      1   1  s1 =   , and τ =   ,   π   1    1 π that represent two reflexions on the standard apartment with respect to lines parallel to type 1 edges.

s0IZ IZ, s1IZ τIZ

s2IZ

Figure 4.1. Some of the chambers in the standard apartment.

Paths of adjacent chambers are called galleries, and depending on their direction we study two different kinds of geodesic galleries. The first kind is associated to the spin representation of the Langlands dual of GSp4(F ). In this case, it is convenient to use a variation of a pointed chamber, that is called spin type chamber. Spin type chambers are given by the union of two chambers in some apartment, that intersect along a type

1 edge. As an example, consider C1 the union of the pointed chambers IZ and s1IZ, which can also be identified with the square

C1 = (KZ, diag(π, 1, π, π)K4Z, diag(1, 1, π, π)KZ, diag(1, π, π, π)K4Z) .

56 The stabilizer of C1 is     ∗ ∗ ∗       −1   ∗ ∗ ∗  I1 = I ∩ s1Is1 = g ∈ K : g ≡   (mod πOF ) .   ∗        ∗ 

The set of spin type chambers is given by the orbit G · C1, so we conclude that the spin type chambers are parametrized by G/I1Z. We denote the set of spin type chambers by X2,spin. The second kind of geodesic galleries we would like to study are associated to the standard representation of the Langlands dual of GSp4(F ). In this case, it is convenient to use a variation of the pointed chamber, that is called standard type chamber. A standard type chamber is given by the union of four pointed chambers in some apartment, that intersect along type 2 edges. For example, take C2 the union of the pointed chambers

IZ, s0IZ, s2IZ and s0s2IZ, which can also be identified with the square

−1 C2 = (KZ, diag(1, 1, π, π)KZ, diag(π , 1, 1, π)KZ, diag(1, π, 1, π)KZ) .

The stabilizer of C2 fixes the chambers IZ and s0s2IZ, thus it is given by     ∗ ∗ ∗ ∗       −1   π∗ ∗ π∗ ∗  I2 = I ∩ s0s2I(s0s2) = g ∈ K : g =   , ∗ ∈ OF .   π∗ π∗ ∗ ∗       π2∗ π∗ π∗ ∗ 

The set of standard type chambers is given by the orbit G · C2, so we conclude that the standard type chambers of X are parametrized by G/I2Z. The set of standard type chambers will be denoted by X2,st.

4.2.4 Apartments

The building X can also be realized as a union of Euclidean planes called apartments. a Each apartment can be described as the union of all vertices of the form [OF π 1 u1 + a b b OF π 2 u2 + OF π 1 w1 + OF π 2 w2] for some ai, bi ∈ Z, i = 1, 2 (cf. [Set08]), and where {u1, u2, w1, w2} is a simplicial basis of V . In order to simplify some notation later, we will a a b b a a b b use [π 1 , π 2 , π 1 , π 2 ] to refer to the vertex [OF π 1 e1 + OF π 2 e2 + OF π 1 f1 + OF π 2 f2].

57 [1, π−2, π2, 1] [π, 1, π3, π2] [1, 1, π2, π2] [1, π, π2, π3] [π−2, 1, 1, π2]

[1, π−2, π, 1] [1, 1, π2, π] [1, 1, π, π2] [π−2, 1, 1, π]

[π2, 1, π3, π] [1, π−1, π, 1] [1, 1, π, π] [π−1, 1, 1, π] [1, π2, π, π3]

[π, 1, π2, 1] [π, 1, π, π] [1, π, π, π] [1, π, 1, π2]

π2, 1, π2, 1] [π, 1, π, 1][ [1, 1, 1, 1] [1, π, 1, π] [1, π2, 1, π2]

[π2, 1, π, 1] [π, π, π, 1] [π, π, 1, π] [1, π2, 1, π]

[π3, π, π2, 1] [π, 1, 1, π−1] [π, π, 1, 1] [1, π, π−1, 1] [π, π3, 1, π2]

Figure 4.2. Part of the standard apartment, with labelled vertices.

4.3 Finite quotients of X

In previous sections we mentioned the concepts required to describe X, the Bruhat-Tits building associated to Sp4(F ). We now wish to describe finite multicomplexes that are locally the same as X.

Let Γ be a discrete, cocompact, torsion-free subgroup of PGSp4(F ) such that ordπ(det Γ) ⊂ 4Z. Then, the quotient multicomplex XΓ = Γ\X inherits its struc- 0 0 1 1 ture from X, having set of vertices XΓ = Γ\X , set of edges XΓ = Γ\X and set of 2 2 chambers XΓ = Γ\X . In particular, observe that the multicells of XΓ are well-defined, since every cell in X has vertices with different types, and multiplication by Γ preserves 1,1 1,1 the type of vertices. Moreover, XΓ has set of type 1 edges XΓ = Γ\X , set of type 2 1,2 1,2 1,3 1,3 edges XΓ = Γ\X , set of long type 2 edges XΓ = Γ\X , set of pointed chambers 2,1 2,1 2,spin 2,spin XΓ = Γ\X , set of spin type chambers XΓ = Γ\X and set of standard type 2,st 2,st chambers XΓ = Γ\X . By abuse of notation, we will refer to the multicells in XΓ using representatives in X. Lastly, we say that two multicells in XΓ are adjacent if they have representatives in X that are adjacent.

58 4.4 Type 1 edge zeta functions

4.4.1 Adjacency operator for type 1 edges

Recall that E1 or P1Z is the type 1 edge from KZ to diag(1, 1, π, π)KZ. We define the out-neighbors of gP1Z as the type 1 edges with initial vertex g diag(1, 1, π, π)KZ. Also, starting from the edge P1Z, the type 1 edge diag(1, 1, π, π)P1Z, also contained in the standard apartment of X, continues from P1Z in a parallel direction (cf. figure 4.3). In this case, we say that diag(1, 1, π, π)P1Z is a proper out-neighbor of E1. Then, if we apply all transformations that fix P1Z, we define its proper out-neighbors to be the P1Z cosets occurring in

    1 1      1  G  1  P1   P1Z =   P1Z.  π   cπ bπ π    b,c,d∈OF /πOF   π dπ cπ π

3 In particular, the type 1 edge P1Z has q proper out-neighbors. By transitivity of the G action over the type 1 edges, we deduce that left translation by g ∈ G of these cosets parametrizes the proper out-neighbors of the type 1 edge gP1Z of X. Furthermore, this adjacency relation gives the following operator on C(X1,1), the space of complex valued functions on X1,1,

X (AP1 f)(gP1Z) = f(ghP1Z) . hP1Z where hP1Z runs over all proper out-neighbors of P1Z.

Similarly, for each finite quotient XΓ we define the operator AP1,Γ as the restriction of AP1 to the finite dimensional vector space

1,1 1,1 VP1,Γ = {f : X → C | f(γe) = f(e), ∀e ∈ X , γ ∈ Γ} .

We record the following result that is an easy consequence of the observations above.

3 Proposition 9. Each type 1 edge in X or XΓ has q proper out-neighbors.

We have a comparable result for out-neighbors, which is proved in section 4.8.1.

3 2 Proposition 10. Each type 1 edge in X or XΓ has q + q + q + 1 out-neighbors.

59 4.4.2 Type 1 edge zeta function

We study cycles of type 1 edges. Note that a cycle is geodesic if any two consecutive type 1 edges are proper out-neighbors. The equivalence classes, under cyclic rotation, of type 1 primitive geodesic cycles are called 1-dimensional primes of type 1. The counting function of such objects is the type 1 edge zeta function and is defined as

−1 Y  l(p) Z1,1(XΓ, u) = 1 − u . p type 1 prime of dimension 1

It was proven in [FLW13] that this is a rational function and satisfies

∞ ! X N1,n(XΓ) 1 Z (X , u) = exp un = , 1,1 Γ n I − A u n=1 det( P1,Γ ) where N1,n(XΓ) is the number of primitive type 1 geodesic cycles in XΓ with length n .

4.4.3 Type 1 edge L-functions

Let ρ be a d-dimensional unitary representation of Γ on Vρ. For any type 1 cycle p, we define Frobp just as in section 2.3. We define the type 1 edge L-function associated to ρ as −1 Y  l(p) L1,1(XΓ, ρ, u) = det I − ρ(Frobp)u . p type 1 prime of dimension 1 1,1 Furthermore, we define the space of Vρ-valued functions on the type 1 edges in X compatible with ρ as

1,1 1,1 VΓ,ρ = {f : X → Vρ | f(γe) = ρ(γ)f(e), ∀e ∈ X , γ ∈ Γ} .

Then, we denote by AP1,ρ the restriction of AP1 ⊗ idρ to VP1,ρ. One can verify that the L-functions defined above are rational of the form given in the next theorem.

Theorem 18. The type 1 L-function of XΓ associated to ρ satisfies

1 L1,1(XΓ, ρ, u) = . det(I − AP1,ρu)

4.4.4 Main results for 1-dimensional primes of type 1

In this section, our objective is to analyze the distribution of type 1 primitive geodesic cycles. The first result is a version of the Prime Geodesic Theorem.

60 Theorem 19. The number of 1-dimensional primes of type 1 in XΓ satisfies

q6l |{p : p is a type 1 prime of dimension 1 in X with l(p) = 2l}| ∼ , Γ l as l → ∞.

Which in turn implies the result below.

Corollary 5. The number of 1-dimensional primes of type 1 in XΓ satisfies

1 q6l |{p : p is a type 1 prime of dimension 1 in XΓ with l(p) < 2l}| ∼ , q6 − 1 l as l → ∞.

We now wish to give a version of the Chebotarev Density Theorem. Let Γ0 and Γ 0 be two discrete, cocompact, torsion-free subgroups of PGSp4(F ), so that Γ is a normal subgroup of Γ. This inclusion induces a covering map ϕ : XΓ0 → XΓ, with Galois group Γ/Γ0.

Theorem 20. Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

6l p is a type 1 prime of dimension 1 in X |C| q |{p : Γ}| ∼ , with l(p) = 2l and Frobp = C |Γ/Γ0| l as l → ∞.

As a corollary we have the following result.

Corollary 6. Let Γ0 and Γ be as above. Then, for any conjugacy class C of Γ/Γ0

6l p is a type 1 prime of dimension 1 in X |C| 1 q |{p : Γ}| ∼ , with l(p) < 2l and Frobp = C |Γ/Γ0| q6 − 1 l as l → ∞.

4.5 Type 2 edge zeta function

4.5.1 Adjacency of type 2 edges

Type 2 edges go from special vertices to nonspecial vertices. However, there are no edges that go from nonspecial vertices to special vertices. Due to this, the proper out-neighbors of a type 2 edge are given by the next type 2 edge in a straight line in an apartment of

61 2 X. For example, (1, π, π, π )P2Z is a proper out-neighbor of P2Z (cf. figure 4.3), and we may parametrize all proper out-neighbors of P2Z using the following cosets   1   2 G  aπ π  P2 diag(1, π, π, π )P2Z =   P2Z.  cπ π  a,c∈OF /πOF   2 2 2 2 d∈OF /π OF dπ cπ −aπ π

These adjacency relations induce an operator on C(X1,2), the space of complex valued functions on X1,2, given by

X (AP2 f)(gP2Z) = f(ghP2Z) , hP2Z where hP2Z runs over all proper out-neighbors of P2Z.

For each finite quotient XΓ, we define AP2,Γ as the restriction of AP2 to the finite dimensional vector space

1,2 1,2

VP2,Γ = {f : X → C | f(γe) = f(e), ∀e ∈ X , γ ∈ Γ} .

Z

1

P ) Z 2 )P

,π,π 2

1 ,

,π,π,π

diag(1 diag(1

1 E

2 E

Figure 4.3. Some edges in the standard apartment.

62 4.5.2 Type 2 edge zeta functions

A cycle of type 2 edges is geodesic if any two consecutive edges in it are proper out- neighbors. The equivalence classes of type 2 primitive geodesic cycles are called 1- dimensional primes of type 2. We now introduce a counting function of 1-dimensional primes of type 2 in XΓ. The type 2 edge zeta function is defined as

−1 Y  2l(p) Z1,2(XΓ, u) = 1 − u . p type 2 prime of dimension 1

It was proven in [FLW13] that this is a rational function and satisfies

∞ ! X N2,n(XΓ) 1 Z (X , u) = exp u2n = , 1,2 Γ n I − A u2 n=1 det( P2,Γ ) where N2,n(XΓ) is the number of primitive tailless type 2 geodesic cycles in XΓ with length n .

4.5.3 Long type 2 edges

As we mentioned before, consecutive edges in a type 2 path are not actually adjacent from a geometric perspective. Our solution for this issue is to take paths of long type 2 edges instead.

Adjacency of long type 2 edges

Similarly to what we did for type 1 edges, we describe the out-neighbors of the long 0 2 type 2 edge gP2Z as the long type 2 edges with initial point g diag(1, π, π, π )KZ.A proper out-neighbor of a long type 2 edge is an out-neighbor that is parallel to it and 2 0 such that both belong to the same apartment. For example, diag(1, π, π, π )P2Z is a 0 proper out-neighbor of E3 or P2Z, and consequently the set of proper out-neighbors of 0 P2Z is given by the following cosets     1 1     0  π  0 G  aπ π  0 P   P Z =   P Z. 2  π  2  cπ π  2   a,c∈OF /πOF   2 2 2 2 2 2 π d∈OF /π OF dπ cπ −aπ π

0 0 0 For any edge gP2Z its proper out-neighbors have the form ghP2Z where hP2Z is a proper 0 P Z A 0 A 0 out-neighbor of 2 . Then, we can also define the operators P2 and P2,Γ, along with

63 V 0 the vector space P2,Γ, in a similar way to what was done in other sections. Finally, the following remark can be verified easily from the observations above.

Proposition 11. Each type 2 and long type 2 edges have q4 proper out-neighbors.

We give a comparable result in the case of out-neighbors, and prove it in section 4.9.1.

Proposition 12. Each type 2 and long type 2 edges have q4 + q3 + q2 + q out-neighbors.

After introducing long type 2 edges we wish to verify that there is no difference between geodesic paths formed by type 2 edges or long type 2 edges.

Proposition 13. In X, the geodesic paths of type 2 edges are the same as those of long type 2 edges, in the sense that they contain the same sets of vertices.

Proof. To prove this statement we will reverse the direction of geodesic paths. Observe 0 that if the type 2 edge g P2Z is a proper out-neighbor of gP2Z then

0 2 g ∈ gP2 diag(1, π, π, π )P2Z, which is equivalent to 0 2 g ∈ g P2 diag(π , π, π, 1)P2Z, and a similar statement can be said for long type 2 edges. Therefore, describing each step of type 2 geodesic paths in the opposite direction corresponds to right multiplication by the elements in the following cosets     π2 π2 −aπ cπ d      π  G  π c  P2   P2Z =   P2Z.  π   π a    a,c∈OF /πOF   2 1 d∈OF /π OF 1

As for the case of geodesic paths of long type 2 edges, we obtain     π2 π2 −aπ cπ d     0  π  0 G  π c  0 P   P Z =   P Z. 2  π  2  π a  2   a,c∈OF /πOF   2 1 d∈OF /π OF 1

Since, we have the same sets of representatives in both the case with type 2 edges and long type 2 edges, we conclude that the sets of vertices in the geodesic paths must be the same as well.

64 In particular, as a consequence of this result we conclude that defining a Zeta function in terms of geodesic cycles of type 2 edges or of long type 2 edges, gives the same function.

4.5.4 Type 2 edge L-functions

Let ρ be a d-dimensional unitary representation of Γ on Vρ. For each 1-dimensional prime of type 2, p, Frobp is defined in the same way as in section 2.3. Note that by Proposition

13, Frobp is independent of the interpretation of geodesic cycles as containing type 2 edges or long type 2 edges. We define the type 2 edge L-function associated to ρ by

−1 Y  2l(p) L1,2(XΓ, ρ, u) = det I − ρ(Frobp)u , p type 2 prime of dimension 1

1,3 V 0 V X ρ Let P2,ρ be the space of ρ-valued functions on compatible with , that is

1,3 1,3 V 0 {f X → V | f γe ρ γ f e , ∀γ ∈ , e ∈ X } . P2,ρ = : ρ ( ) = ( ) ( ) Γ

A 0 A 0 ⊗ V 0 Define the adjacency operator P2,ρ as the restriction of P2 idρ to P2,ρ. The following result is a consequence of Theorem 4.

Theorem 21. The type 2 edge L-function of XΓ associated to ρ satisfies

1 L1,2(XΓ, ρ, u) = 2 . I − A 0 u det( P2,ρ )

4.5.5 Main results for 1-dimensional primes of type 2

We start by analyzing the distribution of 1-dimensional primes of type 2. The next result is a version of the Prime Geodesic Theorem.

Theorem 22. The number of 1-dimensional primes of type 2 satisfies

2q4l |{p : p is a type 2 prime of dimension 1 in X with l(p) = l}| ∼ , Γ l as l → ∞.

Additionally, this implies the next result.

Corollary 7. The number of 1-dimensional primes of type 2 satisfies

2 q4l |{p : p is a type 2 prime of dimension 1 in XΓ with l(p) < l}| ∼ , q4 − 1 l

65 as l → ∞.

Again, we give a version of the Chebotarev Density Theorem. Similarly to what was done in previous sections, consider Γ0 and Γ two discrete, cocompact, torsion-free 0 subgroups of PGSp4(F ), such that Γ is a normal subgroup of Γ. This inclusion induces 0 a covering map ϕ : XΓ0 → XΓ with Galois group Γ/Γ .

Theorem 23. Consider Γ0 and Γ as above. Then, we have

4l p is a type 2 prime of dimension 1 in X |C| 2q |{p : Γ }| ∼ , with l(p) = l and Frobp = C |Γ/Γ0| l as l → ∞.

As a consequence we obtain the following corollary.

Corollary 8. Consider Γ0 and Γ as above. Then, we have

4l p is a type 2 prime of dimension 1 in X |C| 2 q |{p : Γ }| ∼ , with l(p) < l and Frobp = C |Γ/Γ0| q4 − 1 l as l → ∞.

4.6 Spin type chamber zeta function

Now, we wish to describe the closed geodesic galleries in XΓ, which means studying the paths obtained from connecting adjacent chambers. The first kind we study are based on spin type chambers.

4.6.1 Adjacency operator

Recall that C1 or I1Z is the union of the chambers IZ and s1IZ, and is also the square with vertices KZ, diag(π, 1, π, π)K4Z, diag(1, 1, π, π)KZ and diag(1, π, π, π)K4Z. The sides of I1Z are given by the type 2 edges P2Z, s1P2Z, τs1P2Z and τP2Z. Moreover, we point out that that the union of any two distinct chambers that intersect along a type 1 edge is in the orbit G · C1. This claim can be verified by the fact that each type 1 edge is contained in [P1 : I] = q + 1 chambers, and the number of ordered pairs with two different such chambers is q(q + 1), which is equal to [P1 : I1].

66 We will use the following matrix   1    1  t =   ,  −π    π and we observe that the spin type chamber tI1Z corresponds to the translation and reflexion of I1Z in a diagonal direction (cf. figure 4.4). In particular, we have ts1P2Z =

τP2Z, so the two spin type chambers intersect along a type 2 edge. 0 0 We say that the spin type chamber g I1Z is an out-neighbors of gI1Z if g s1P2Z = gτP2Z, so that they intersect along a specific type 2 edge. A proper out-neighbor of a spin type chamber is an out-neighbor that results from a translation and reflexion of it, and such that both belong to a common apartment of X. As an example, tI1Z is a proper out-neighbor of I1Z, so we may describe all proper out-neighbors of I1Z using the following cosets   1   G  1  I1tI1Z =   I1Z. aπ −π  a,d∈OF /πOF   dπ aπ π

The proper out-neighbors of a spin type chamber gI1Z may be given in the form ghI1Z, where hI1Z is a proper out-neighbor of I1Z. We remark that in Fang-Li-Wang [FLW13], the authors chose to define proper out- neighbors in terms of the pointed chambers, and obtain the following decomposition   1   G  1  ItIZ =   IZ.  aπ −π  a,d∈OF /πOF   dπ aπ π

Since the representatives of cosets happen to be the same, we may conclude that the geodesic galleries in XΓ arising from both descriptions are also the same. Additionally, the Zeta functions arising from both descriptions are the same functions. The next result is an immediate consequence of the discussion above.

Proposition 14. Each spin type chamber has q2 proper out-neighbors.

We give a similar proposition for out-neighbors.

67 tτP

2 Z tIZ

τP 2 Z

s1IZ IZ

s 1 P 2 Z

Figure 4.4. Part of a spin type geodesic gallery in the standard apartment.

Proposition 15. Each spin type chamber has q2 + q out-neighbors.

2,spin 2,spin Now, we define the adjacency operator AI1 on C(X ) by sending f ∈ C(X ) to X (AI1 f)(gI1Z) = f(ghI1Z) , hI1Z where hI1Z runs over all the proper out-neighbors of I1Z.

There is a similar adjacency operator, AI1,Γ, for XΓ defined as the restriction of AI1 to the following finite dimensional space

2,spin 2,spin VI1,Γ = {f : X → C | f(γc) = f(c), ∀γ ∈ Γ, c ∈ X } .

4.6.2 Spin type chamber zeta function

A spin type gallery is geodesic if any two consecutive chambers in it are proper out- neighbors. The 2-dimensional primes of spin type will be the equivalence classes of spin type primitive closed geodesic galleries in XΓ. Then, we define the spin type chamber zeta function as −1 Y  l(p) Z2,spin(XΓ, u) = 1 − u . p spin type prime of dimension 2

68 Using the theorem on rationality of the Zeta function (Theorem 3) we obtain the following result.

Theorem 24. The spin type chamber zeta function satisfies the identity

1 Z2,spin(XΓ, u) = . det(I − AI1,Γu)

4.6.3 Spin type chamber L-functions

Let ρ be a d-dimensional unitary representation of Γ on Vρ. The Frobenius conjugacy class of a spin type closed gallery is given in the same way as the definition in section 2.3. We define the spin type chamber L-function associated to ρ as

−1 Y  l(p) L2,spin(XΓ, ρ, u) = det I − ρ(Frobp)u . p spin type prime of dimension 2

Yet again, this function is rational. To prove that, we use the adjacency operator

AI1,ρ, which is defined as the restriction of AI1 ⊗ idρ to the space of Vρ-valued functions on X2,spin that are compatible with the action of ρ, that is,

2,spin 2,spin VI1,ρ = {f : X → Vρ | f(γc) = ρ(γ)f(c), ∀γ ∈ Γ, c ∈ X } .

As a consequence of the theorem on rationality of L-functions (Theorem 4) we obtain

Theorem 25. The spin type chamber L-function associated to XΓ and ρ satisfies

1 L2,spin(XΓ, ρ, u) = . det(I − AI1,ρu)

4.6.4 Main results for 2-dimensional primes of spin type

The main theorem of this section gives the distribution of spin type primitive closed geodesic galleries. This is a version of the Prime Geodesic Theorem.

Theorem 26. The number of 2-dimensional primes of spin type satisfies

q4l |{p : p is a spin type prime of dimension 2 in X with l(p) = 2l}| ∼ , Γ l as l → ∞.

And as a corollary, we obtain

69 Corollary 9. The number of 2-dimensional primes of spin type satisfies

1 q4l |{p : p is a spin type prime of dimension 2 in XΓ with l(p) < 2l}| ∼ , q4 − 1 l as l → ∞.

Next, we give a version of the Chebotarev Density Theorem. Let Γ0 and Γ be discrete, 0 cocompact, torsion-free subgroups of PGSp4(F ) as considered above and so that Γ is a normal subgroup of Γ. This inclusion induces a covering map ϕ : XΓ0 → XΓ with Galois group Γ/Γ0.

Theorem 27. Consider Γ0 and Γ as above. Then, for any conjugacy class C of Γ/Γ0, we have 4l p is a spin type prime of dimension 2 in X |C| q |{p : Γ }| ∼ , with l(p) = 2l and Frobp = C |Γ/Γ0| l as l → ∞.

As a corollary of this theorem we obtain the following result.

Corollary 10. Consider Γ0 and Γ as above. Then, for any conjugacy class C of Γ/Γ0, we have 4l p is a spin type prime of dimension 2 in X |C| 1 q |{p : Γ }| ∼ , with l(p) < 2l and Frobp = C |Γ/Γ0| q4 − 1 l as l → ∞.

4.7 Standard type chamber zeta function

Following the analysis carried out in the previous section, we now wish to describe a different kind of galleries, this time using standard type chambers.

4.7.1 Translation of chambers

The following decomposition shows the cosets corresponding to the translation of the pointed chambers in an apartment in an horizontal or vertical direction (cf. figure 4.5).

    d 1 aπ−1 1  −1   −1 0 G  π  0  π  It IZ =   IZ , t =   . π −a b  π  a,b,d∈OF /πOF     1 1

70 Looking at the standard apartment of X we see that the chambers IZ and t0IZ are not adjacent in the natural sense of the word. However, we can recognize a pattern in the trajectory of the galleries in the standard apartment using this translation. In particular, 0 0 note that multiplication by t maps the type 1 edge s0s2P1Z to the edge P1Z = t s0s2P1Z.

This is the point where C2 and standard type chambers become useful. Recall that C2 or I2Z may be given as the union of the pointed chambers IZ, s0IZ, s2IZ and s0s2IZ, and is also the square with vertices KZ, diag(1, 1, π, π)KZ, diag(π−1, 1, 1, π)KZ, and diag(1, π, 1, π)KZ. The sides of I2Z are the type 1 edges P1Z, s0P1Z, s0s2P1Z and s2P1Z. On a side note we observe that the union of any four distinct pointed chambers that intersect along type 2 edges is in the orbit G · C2. Note that we are not requiring that the pointed chambers belong to a common apartment. To verify this statement, we note that each type 2 edge is contained in [P2 : I] = q + 1 pointed chambers. So, starting from gIZ and selecting two other chambers adjacent to it through each of its type 2 edges, we obtain q2 different collections of 3 chambers containing 4 vertices, that may or may not be part of a square in the conditions mentioned before. Moreover, we see that there 2 are [I : I2] = q elements in the orbit of C2 that take IZ to gIZ. Consequently, each such union of 3 chambers is part of an element in the orbit of C2, and there is a fourth chamber that completes a square together with the previous ones. 0 We remark that the standard type chambers I2Z and t I2Z intersect along the 0 0 type 1 edge P1Z = t s0s2P1Z. Then, we say that g I2Z is an out-neighbor of gI2Z if 0 gP1Z = g s0s2P1Z, so that they intersect along a type 1 edge. A proper out-neighbor of a standard type chamber is an out-neighbor that results from a translation and reflexion 0 of it, and such that both belong to the same apartment. For example, t I2Z is a proper out-neighbor of I2Z, so we may characterize all the proper out-neighbors of I2Z using the cosets in the following decomposition   d 1 aπ−1  −1  0 G  π  I2t I2Z =   I2Z. π −a b  a,b,d∈OF /πOF   1

For each standard type chamber gI2Z, its proper out-neighbors are given by ghI2Z, where hI2Z runs over the proper out-neighbors of I2Z. Next, we give an immediate consequence of the description of proper out-neighbors as cosets.

Proposition 16. Each standard type chamber has q3 proper out-neighbors.

71 We will also prove the following result in section 4.11.1.

Proposition 17. Each standard type chamber has q3 + q2 out-neighbors.

We define the following adjacency operator on C(X2,st), the set of complex valued functions on X2,st X (AI2 f)(gI2Z) = f(ghI2Z) , hI2Z where hI2Z runs over the proper out-neighbors of I2Z.

In a similar manner, we define AI2,Γ as the restriction of AI2 to the following finite dimensional vector space

2,st 2,st VI2,Γ = {f : X → C | f(γc) = f(c), ∀c ∈ X , γ ∈ Γ} . s 0 s Z s0IZ 2 1 P P 0 1 t

0 Z t IZ s1IZ IZ P 1 Z s2IZ

Figure 4.5. Part of a geodesic gallery in the standard apartment.

4.7.2 Standard type chamber zeta function

A standard type gallery where any two consecutive chambers are proper out-neighbors is a standard type geodesic gallery. One can also observe, that the boundary of a standard type geodesic gallery in X is given by two type 1 geodesic paths in an apartment of X.

72 The equivalence classes of standard type primitive closed geodesic galleries in XΓ will be called 2-dimensional primes of standard type. We define the standard type chamber zeta function, which counts the 2-dimensional primes of standard type, as

−1 Y  l(p) Z2,st(XΓ, u) = 1 − u . p standard type prime of dimension 2

As a consequence of Theorem 3, we conclude that this Zeta function is a rational function.

Theorem 28. The standard type chamber zeta function satisfies the identity

1 Z2,st(XΓ, u) = . det(I − AI2,Γu)

4.7.3 Standard type chamber L-functions

Let ρ be a d-dimensional unitary representation of Γ on Vρ. For each standard type closed geodesic gallery p, we define Frobp as in section 2.3. The standard type chamber L-function associated to ρ is defined as

−1 Y  l(p) L2,st(XΓ, ρ, u) = det I − ρ(Frobp)u . p standard type prime of dimension 2

In order to prove that this is a rational function, denote by VI2,ρ the set of Vρ-valued functions on X2,st that are compatible with the action of ρ, that is,

2,st 2,st VI2,ρ = {f : X → Vρ | f(γc) = ρ(γ)f(c), ∀γ ∈ Γ, c ∈ X } .

We define the adjacency operator AI2,ρ as the restriction of AI2 ⊗ idρ to VI2,ρ. The Theorem on rationality of L-functions (Theorem 4) implies the following result.

Theorem 29. The standard type chamber L-function associated to XΓ and ρ satisfies

1 L2,st(XΓ, ρ, u) = . det(I − AI2,ρu)

4.7.4 Main results for 2-dimensional primes of standard type

Now, we list some results giving the distribution of 2-dimensional primes of standard type. The first one is a version of the Prime Geodesic Theorem.

73 Theorem 30. The number of 2-dimensional primes of standard type satisfies

2q3l |{p : p is a standard type prime of dimension 2 in X with l(p) = l}| ∼ , Γ l as l → ∞.

As a consequence, this gives the next result. Corollary 11. The number of 2-dimensional primes of standard type satisfies

2 q3l |{p : p is a standard type prime of dimension 2 in XΓ with l(p) < l}| ∼ , q3 − 1 l as l → ∞.

Next, we present a version of the Chebotarev Density Theorem. Consider two discrete, 0 0 cocompact, torsion-free subgroups Γ and Γ of PGSp4(F ), such that Γ is a normal subgroup of Γ. In these conditions, there is a covering map ϕ : XΓ0 → XΓ with Galois group Γ/Γ0. Theorem 31. Consider Γ0 and Γ as above. Then, for any conjugacy class C of Γ/Γ0, we have: 3l p is a standard type prime of dimension 2 in X |C| 2q |{p : Γ}| ∼ , with l(p) = l and Frobp = C |Γ/Γ0| l as l → ∞.

This theorem implies the following result. Corollary 12. Consider Γ0 and Γ as above. Then, for any conjugacy class C of Γ/Γ0, we have:

3l p is a standard type prime of dimension 2 in X |C| 2 q |{p : Γ}| ∼ , with l(p) < l and Frobp = C |Γ/Γ0| q3 − 1 l as l → ∞.

4.8 Proofs of Theorems 19 and 20

In this section we prove versions of the Prime Geodesic Theorem and the Chebotarev Density Theorem for 1-dimensional primes of type 1. In the first two parts we analyze the type 1 edges in the neighborhood of a vertex. Then, we define a map that allows us to construct type 1 geodesic cycles over the whole XΓ. After that, we will be able to show some properties of AP1,Γ and characterize its eigenvalues. Lastly, we prove the desired theorems.

74 4.8.1 Representatives of type 1 edges

We now list the edges in the neighborhood of the vertex τKZ = diag(1, 1, π, π)KZ.

Type 1 edges with initial point τKZ

We are interested in determining the type 1 edges with initial vertex τKZ, which are given by gP1Z with gKZ = τKZ. Therefore, such edges may be parametrized by the cosets in P1τKZ/P1Z, that we list below.

      1 1 1        1  G  1   1    KZ =   P1Z ∪   P1Z  π   cπ bπ π   π    c,b,d∈OF /πOF     π dπ cπ π π     1 1     G  a 1  G  1  ∪   P1Z ∪   P1Z  −π   bπ π  a,d∈OF /πOF   b∈OF /πOF   dπ aπ π −π

Additionally, this proves Proposition 10.

Type 1 edges with terminal point τKZ

The type 1 edges with terminal point τKZ are represented by the edges gP1Z satisfying −1 gτKZ = τKZ. So, these may be parametrized by P1τKτ P1Z/P1Z. Furthermore, note that the type 1 edge τP1Z represents the opposite direction of P1Z. This implies that the type 1 edge gτP1Z represents the opposite direction of the edge gP1Z. So, we may multiply the representatives in the previous decomposition on the right by τ or τ −1 to obtain the desired decomposition.     π−1 1  −1    −1 G  π   1  τKZτ =   P1Z ∪   P1Z  π c b   1  c,b,d∈OF /πOF     π d c 1     π−1 1  −1   −1  G  1 aπ  G  π  ∪   P1Z ∪   P1Z  −1   π b  a,d∈OF /πOF   b∈OF /πOF   π d a −1

75 4.8.2 Successor map

−1 We now describe a bijective map sP1 from τKZτ /P1Z to τKZ/P1Z, which assigns to each type 1 edge one of its proper out-neighbors. Recall that the proper out- neighbors of a type 1 edge can be obtained by right multiplication by an element in P1 diag(1, 1, π, π)P1Z/P1Z. Later on, we construct sP1 for all type 1 edges of XΓ. We begin by describing sP1 for the edges with terminal vertex τKZ.

      π−1 π−1 1  −1  s  −1     1 aπ  P1  1 aπ   1  (1)   P1Z −−→     P1Z  −   −   π π   1   1    π d a π d a π π     1 1 1 1      a + 1 a   1 1  =     P1Z  −π −π   −     1  dπ (a + 1)π dπ aπ −1   1    a + 1 1  =   P1Z, for a, d ∈ OF /πOF  −π    dπ (a + 1)π π

    1 1   s    1  P1  1  (2)   P1Z −−→   P1Z    π π   1    1 π π

      π−1 π−1 1  −1  s  −1     π  P1  π   1  (3)   P1Z −−→     P1Z  π   π   π        π π π   1    1  =   P1Z  π    π

76       1 1 1  −1  s  −1     π  P1  π   1  (4)   P1Z −−→     P1Z  π b   π b   π π        −1 −1 π     1 1      1 1   1 1  =     P1Z  b π bπ   −   ( + 1)   1  −π −1   1    1  =   P1Z, for b ∈ OF /πOF  b π π   ( + 1)  π

      π−1 π−1 1  −1  s  −1     π  P1  π   1  (5)   P1Z −−→     P1Z  π − b   π − b   π   1   1    π −1 π −1 π π     1 1 1 1      1 1   1 −1  =     P1Z  b π −π bπ     ( + 1)   1  −π π −π −1 −1   1    1  =   P1Z, for b ∈ OF /πOF  b π π   ( + 1)  −π

      π−1 π−1 1  −1  s  −1     π  P1  π   1  (6)   P1Z −−→     P1Z  π c b   π c b   π π        π d c π d c π π

77     1 1 1 1      1 1   1 1  =     P1Z  c π bπ cπ bπ   −   ( + 1)   1  dπ (c + 1)π dπ cπ −1   1    1  =   , for b, c, d ∈ OF /πOF  c π bπ π   ( + 1)  dπ (c + 1)π π

except when c ≡ −1 and d ≡ 0 (mod πOF ), or b ≡ c ≡ d ≡ 0 (mod πOF )

0,sp Denote by X the set of special vertices of X. In order to extend sP1 to all type 1 0,sp edges of XΓ, we choose a set SΓ(K) of representatives of Γ\X , and we define

−1 −1 −1 sP1 (aτ gP1Z) = aτ sP1 (gP1Z) , ∀a ∈ SΓ(K), gP1Z ∈ τKZτ /P1Z.

−1 Note that this does indeed define a bijection, as aτ gP1Z covers all the type 1 edges −1 with terminal vertex aKZ and aτ sP1 (gP1Z) covers all the type 1 edges with initial vertex aKZ, when gP1Z varies.

4.8.3 Properties of AP1,Γ

Now, we show two properties of AP1,Γ.

Proposition 18. The adjacency operator AP1,Γ is irreducible.

Proof. To verify that AP1,Γ is irreducible we must prove that any two type 1 edges in XΓ can be joined using a type 1 geodesic path in XΓ. Since X is connected, XΓ is connected as well, and it is enough to see that a type 1 edge can be joined to any of its type 1 out-neighbors through a geodesic path.

Consider two type 1 edges g0P1Z and g1P1Z, such that the terminal vertex of g0P1Z is the same as the initial vertex of g1P1Z, and yet g1P1Z is not a proper out-neighbor 0 of g0P1Z. We wish to prove that there is a type 1 edge g0P1Z with the same terminal vertex as g0P1Z that satisfies:

0 (i) sP1 (g0P1Z) is a proper out-neighbor of g0P1Z ;

0 (ii) g1P1Z is a proper out-neighbor of g0P1Z .

2 0 We note that there are at most q + q + 1 exceptions sP1 (g0P1Z) to (i), and at most 2 0 q + q + 1 exceptions to g0P1Z (ii), by Propositions 9 and 10. Thus, removing these

78 exceptions we are left with

q3 + q2 + q + 1 − 2(q2 + q + 1) = q3 − q2 − q − 1 > 0

0 choices of type 1 edges g0P1Z satisfying (i) and (ii). Hence our claim holds. 0 Considering g0P1Z as above, we can construct a type 1 geodesic path that starts at g P Z sk g0 P Z X1,1 s 0 1 and then follows the sequence ( P1 ( 0 1 ))k≥1. Since Γ is finite and P1 is a bijection, this sequence must be periodic. This shows that

0 2 0 0 g0P1Z → sP1 (g0P1Z) → sP1 (g0P1Z) → ... → g0P1Z → g1P1Z is a geodesic path from g0P1Z to g1P1Z. This completes the proof of the proposition.

Proposition 19. The adjacency operator AP1,Γ has period 2.

Proof. Recall that any proper out-neighbor of a type 1 edge gP1Z can be written in the form ghP1Z, where h ∈ P1 diag(1, 1, π, π)P1Z. It is easy to see that ordπ(det h) ≡ 2 (mod 4). Furthermore, as ordπ(det Γ) ⊂ 4Z, we conclude that the length of any type 1 geodesic cycle is divisible by 2.

To verify that 2 is the actual period, take the edge g0P1Z = P1Z in XΓ and let g1P1Z be the same edge with the opposite direction. We see that g1P1Z is an out-neighbor of g0P1Z, which in turn is an out-neighbor of g1P1Z. However, neither of these edges is a proper out-neighbor of the other. Then, as shown in the proof of the previous proposition, there is a type 1 geodesic cycle passing through the following edges.

P1Z 0 2 0 0 → sP1 (g0P1Z) → sP1 (g0P1Z) → ... → g0P1Z

→ g1P1Z 0 2 0 0 → sP1 (g1P1Z) → sP1 (g1P1Z) → ... → g1P1Z

→ P1Z

This geodesic contains two geodesic loops that we can repeat a number of times, with the result remaining a geodesic cycle. Denote by d the sum of the lengths of the geodesic loops. Then, the cycle above has length 2 + d. Now, consider the modified geodesic cycle obtained from repeating the geodesic loops 2 + d times. The total length of this last cycle is 2 + (2 + d)d. Finally, we note that the greatest common divisor of 2 + (2 + d)d and 2 + d is at most 2, and thus it has to be 2, as we wanted to prove.

79 4.8.4 Proof of Theorem 19

We have seen above that AP1,Γ is irreducible and has period 2. Therefore, adding the fact each edge has q3 proper out-neighbors, by the Perron-Frobenius Theorem we deduce 3 3 that q and −q are the eigenvalues of AP1,Γ with the largest absolute value. From the general Prime Geodesic Theorem (Theorem 5), we have

X 1 q6l  1 = 2q6l + o . l l p type 1 prime 2 of dimension 1 l(p)=2l

This proves the theorem.

4.8.5 Proof of Theorem 20

Γ Γ/Γ0 L 0 1 1 ρ ρ Using the fact that IndΓ = Ind{e} = ρ∈Γ[/Γ0 deg( ) we have

Y deg(ρ) Z1,1(XΓ0 , u) = Z1,1(XΓ, u) L1,1(XΓ, ρ, u) . ρ∈Γ[/Γ0 ρ6=1

From the previous proof we know that the smallest poles of Z1,1(XΓ0 , u) and Z1,1(XΓ, u), in absolute value, are 1/q3 and −1/q3 for both Zeta functions. Consequently, all the poles of the remaining L-functions in the decomposition have absolute value larger than 1/q3.

0 We have seen that the eigenvalues of AP1,Γ and AP1,Γ with the largest absolute value are q3 and −q3 for both operators. Thus, we may apply the Chebotarev Density Theorem (Theorem 8) and we obtain

|C| X 2q6l = 2l 1 + o(q6l) , | / 0| Γ Γ p type 1 prime of dimension 1 l(p)=2l Frobp=C which implies X |C| q6l q6l  1 = + o . | / 0| l l p type 1 prime Γ Γ of dimension 1 l(p)=2l Frobp=C This proves the desired result.

80 4.9 Proofs of Theorems 22 and 23

We now prove versions of the Prime Geodesic Theorem and the Chebotarev Density Theorem for type 2 geodesic cycles. Since the geodesic cycles formed by type 2 or long type 2 edges are essentially the same, we use the most convenient objects to us, which are long type 2 edges, in our arguments. As usual, we describe the long type 2 edges in the neighborhood of a vertex, and define a map that allows us to construct type 2

X A 0 geodesic cycles in Γ. Then, we prove two properties of P2,Γ and wrap up the section with the proofs of the two theorems.

4.9.1 Representatives of long type 2 edges

We describe the long type 2 edges in the neighborhood of diag(1, π, π, π2)KZ.

Long type 2 edges with initial point diag(1, π, π, π2)KZ

The long type 2 edges with initial vertex diag(1, π, π, π2)KZ are represented by the 0 2 edges gP2Z that satisfy gKZ = diag(1, π, π, π )KZ. Hence, these are parametrized by 0 2 0 P2 diag(1, π, π, π )KZ/P2Z and this quotient decomposes as

diag(1, π, π, π2)KZ =     1 1     G  aπ π  0 G  π  0 =   P Z ∪   P Z  cπ π  2  bπ cπ π  2 a,c∈OF /πOF   c∈OF /πOF   2 2 2 2 2 2 2 2 d∈OF /π OF dπ cπ −aπ π b∈OF /π OF cπ π     1 dπ 1  2    G  bπ aπ π  0 G  π  0 ∪   P Z ∪   P Z.  −π  2  π  2 a,b∈OF /πOF   d∈OF /πOF   aπ2 π2 π2

Long type 2 edges with terminal point diag(1, π, π, π2)KZ

0 2 The long type 2 edge gP2Z has terminal vertex diag(1, π, π, π )KZ if and only if diag(1, π, π, π2)KZ = g diag(1, π, π, π2)KZ. So, these edges are given by the cosets in 2 2 −1 0 diag(1, π, π, π )KZ diag(1, π, π, π ) /P2Z.

81 We also use the following matrix   π−1    1  β =   .    1  π

0 0 Note that the long type 2 edge βP2Z is P2Z in the opposite direction. As a result, the 0 0 long type 2 edges gP2Z and gβP2Z are the reverse of each other. Thus, we can multiply the representatives in the decomposition above on the right by β to obtain the long type 2 edges with terminal vertex diag(1, π, π, π2)KZ.

diag(1, π, π, π2)KZ diag(1, π, π, π2)−1 =     π−1 1     G  π a  0 G  1  0 =   P Z ∪   P Z  π c  2  π2 cπ b  2 a,c∈OF /πOF   c∈OF /πOF   2 3 2 2 2 2 d∈OF /π OF π −aπ cπ dπ b∈OF /π OF π cπ     1 π d  2    G  π aπ bπ  0 G  π  0 ∪   P2Z ∪   P2Z  −1   π  a,b∈OF /πOF   d∈OF /πOF   π2 aπ π

4.9.2 Successor map s 0 P2 2 s 0 , π, π, π KZ We now describe a map P2 from long type 2 edges with terminal vertex diag(1 ) 2 0 , π, π, π KZ s 0 gP Z to the edges with initial vertex diag(1 ) such that P2 ( 2 ) is always a proper 0 out-neighbor of gP2Z.

      1 1 1

  sP 0      1  0 2  1   π  0 (1)   P Z −−→     P Z  π2 cπ b  2  π2 cπ b   π  2       π2 cπ π2 cπ π2 π2     π 1 1  2 2     π π   1  0 =     P Z  b π2 cπ2 bπ2    2  ( + 1)   1  cπ3 π3 cπ3 −1

82   1    π  0 2 =   P Z, for c ∈ OF /πOF , b ∈ OF /π OF  b π cπ π  2  ( + 1)  cπ2 π2

      1 1 1

 2  sP 0  2   2  π aπ bπ  0 2 π aπ bπ  bπ π  0 (2)   P Z −−→     P Z  −  2  −   π π  2  1  1   π2 aπ π2 aπ π2 −bπ3 π2     π π 1 1 bπ  2 3 2 2 2 3   2 2  (a + 1)π bπ (a − b π )π bπ   −bπ −1 − b π 1  0 =     P Z  −π2 bπ3 −π2  − −bπ  2    1  bπ4 (a + 1)π3 −abπ4 aπ3 −1   1   (a + 1)π π  0 =   P Z, for a, b ∈ OF /πOF  π  2   bπ3 −(a + 1)π2 π2

  π−1

  sP 0  π a  2 (3)   −−→  π    π3 −aπ2 (−1 + bπ)π     π−1 1      π a  −π π  0     P Z  π  bπ2 π  2     π3 −aπ2 (−1 + bπ)π bπ3 π2 π2   bπ2 π π  3 3 2 2   bπ abπ (1 + a)π aπ  =   ·  −π2 π2    (1 + a)π3 (−a − bπ + b2π2)π3 (−1 + bπ)π3 (−1 + bπ)π3   1 1 − bπ 1    1 − bπ 1  0 ·   P Z  − bπ b2π2 −bπ  2  1 + 1  −1

83   1  2   bπ (1 + a)π π 0 =   P Z, for a, b ∈ OF /πOF  −π  2   (1 + a)π2 π2

      π d π d 1

  sP 0      π  0 2  π   π  0 (4)   P Z −−→     P Z  π  2  π   π  2       π π π2 π2     1 + dπ dπ 1 − dπ + d2π2 −dπ      π   1  0 =     P Z  π    2    1  π2 π2 −d2π2 1 + dπ   1    π  0 =   P Z, for d ∈ OF /πOF  π  2   (1 − dπ)π2 π2

      π−1 π−1 1

  sP 0      π  0 2  π   π  0 (5)   P Z −−→     P Z  π  2  π   π  2       π3 dπ · π π3 dπ2 dπ · π2 π2     dπ2 π 1 − d2π2 −dπ  2     π   1  0 =     P Z  π2    2    1  (1 + d2π2)π3 dπ4 d3π3 1 + d2π2   dπ 1    π  0 =   P Z, for d ∈ OF /πOF  π  2   π2

84   π−1

  sP 0  π a  0 2 (6)   P Z −−→  π c  2   π3 −aπ2 cπ2 dπ     π−1 1      π a   π  0     P Z  π c   π  2     π3 −aπ2 cπ2 dπ π2 π2     π π 1 1  2 2 2     aπ π aπ   1  0 =     P Z  cπ2 π2 cπ2    2    1  (d + 1)π3 −aπ3 cπ3 dπ3 −1   1    aπ π  0 2 =   P Z, for a, c ∈ OF /πOF , d ∈ OF /π OF  cπ π  2   (d + 1)π2 cπ2 −aπ2 π2

except when a ≡ c ≡ d ≡ 0 (mod πOF ), or c ≡ 0 and d ≡ −1 (mod πOF )

s 0 X S K In order to extend P2 to Γ, we choose a set Γ( ) of representatives of all special 2 vertices of XΓ. Then, we set δ = diag(1, π, π, π ), and define

−1 0 −1 0 0 −1 0 s 0 aδ gP Z aδ s 0 gP Z , ∀a ∈ S K , gP Z ∈ δKZδ /P Z. P2 ( 2 ) = P2 ( 2 ) Γ( ) 2 2

−1 0 This map does indeed define a bijection, as aδ gP2Z covers all the long type 2 edges −1 0 aKZ aδ s 0 gP Z with terminal vertex , and P2 ( 2 ) covers all the long type 2 edges with 0 −1 0 initial vertex aKZ, when gP2Z runs through the cosets in δKZδ /P2Z.

4.9.3 Properties of A 0 P2,Γ

A 0 Now, we show two properties of P2,Γ.

Proposition 20. The adjacency operator A 0 has two nonzero blocks and each of them P2,Γ is irreducible.

Proof. Note that all vertices in a path of long type 2 edges have the same type. This

A 0 separates P2,Γ into two blocks, corresponding to the vertices of type 0 and 2, respectively.

85 Now, we prove that each block is irreducible. Since XΓ is connected, it is sufficient to prove that there is a type 2 geodesic path connecting a long type 2 edge to any of its 0 0 out-neighbors. To check that, consider g0P2Z and g1P2Z two edges so that the terminal 0 0 0 vertex of g0P2Z coincides with the initial vertex of g1P2Z, but g1P2Z is not a proper 0 0 0 out-neighbor of g0P2Z. We wish to find an edge g0P2Z satisfying

0 0 0 (i) g1P2Z is a proper out-neighbor of g0P2Z ;

0 0 0 s 0 g P Z g P Z (ii) P2 ( 0 2 ) is a proper out-neighbor of 0 2 .

4 3 2 0 3 2 Note that out of q + q + q + q possibilities for g0P2Z there are q + q + q exceptions for condition (i) and the same number for condition (ii), by Propositions 11 and 12. Thus, the number of edges that satisfy both of these conditions is at least

q4 + q3 + q2 + q − 2(q3 + q2 + q) = q(q3 − q2 − q − 1) = q(q − 1)(q2 − 1) > 0 .

0 0 0 For such a choice g0P2Z we may consider a type 2 geodesic path starting from g0P2Z, k 0 0 1,3 and then following the sequence (s 0 (g0P2Z))k≥1. Moreover, since X is finite and sP 0 P2 Γ 2 is a bijection, the sequence must be periodic. As a result, there is a geodesic path like the following

0 0 0 2 0 0 0 0 0 g0P Z → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z → g1P Z. 2 2 0 2 P2 0 2 0 2 2

This concludes the proof of the proposition.

Proposition 21. Each block of the adjacency operator A 0 has period . P2,Γ 1

0 Proof. Recall that the out-neighbors of the long type 2 edge gP2Z are obtained by multiplying the cosets below on the left by g

2 0 −1 0 diag(1, π, π, π )KZ/P2Z = diag(π , 1, 1, π)KZ/P2Z.

The following matrices belong to some of the cosets above (cf. section 4.9.1)

      π−1 −1 π−1 1 π−1        1   1   1  β =   , ε− =   , ε+ =   .        1   1   1  π π π

86 Moreover, note that right multiplication by these matrices gives out-neighbors that are not proper out-neighbors. We observe the following relations       π−1 π−1 1       2  1   1   1  0 β =     =   ∈ P Z ;       2  1   1   1  π π 1

ε−ε+ε− =         −1 π−1 1 π−1 −1 π−1 −1          1   1   1   1  0       =   ∈ P Z.         2  1   1   1   1  π π π 1

Using the adjacent edges given by the relations above, we will prove that the period

A 0 of each block of P2,Γ divides 2 and 3, thus can only be 1. We begin by considering 0 0 0 0 0 0 a path starting at gP2Z and we write g0P2Z for gP2Z, g1P2Z for gβP2Z, and g2P2Z 2 0 0 for gβ P2Z = g0P2Z. Then, using the notation from the previous proof we have the following geodesic cycle:

0 g0P2Z 0 0 2 0 0 0 0 → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z 2 0 2 P2 0 2 0 2 0 → g1P2Z 0 0 2 0 0 0 0 → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z 2 1 2 P2 1 2 1 2 0 0 → g2P2Z = g0P2Z.

Express the total length of this geodesic cycle as 2 + d. Then, if we repeat the two intermediate geodesic loops 2 + d times, the resulting cycle will still be geodesic, and its length will be 2 + (2 + d)d. Finally, one can easily verify that the greatest common divisor 2 + (2 + d)d and 2 + d divides 2. 0 0 0 0 0 On the other hand, we can also use g0P2Z for gP2Z, g3P2Z for gε−P2Z, g4P2Z for 0 0 0 0 gε−ε+P2Z, and g5P2Z for gε−ε+ε−P2Z = g0P2Z. Again, using the notation from the previous proof, we obtain the following geodesic cycle:

0 g0P2Z 0 0 2 0 0 0 0 → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z 2 0 2 P2 0 2 0 2

87 0 → g3P2Z 0 0 2 0 0 0 0 → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z 2 3 2 P2 3 2 3 2 0 → g4P2Z 0 0 2 0 0 0 0 → sP 0 (g P Z) → s 0 (g P Z) → · · · → g P Z 2 4 2 P2 4 2 4 2 0 0 → g5P2Z = g0P2Z.

Let us say that the total length of this geodesic cycle is 3 + d. If we repeat the three intermediate geodesic loops 3 + d times, the resulting cycle will still be geodesic, and its length will be 3 + (3 + d)d. Consequently, one can easily verify that the greatest common divisor of 3 + (3 + d)d and 3 + d divides 3. This ends the proof of the proposition.

4.9.4 Proof of Theorem 22

A 0 We have seen above that P2,Γ has two connected components and, in each one of them, the periods are equal to 1. Furthermore, since each edge has q4 proper out-neighbors, 4 q A 0 by the Perron-Frobenius Theorem is the eigenvalue of P2,Γ with the largest absolute value and it has multiplicity 2. Then, by the general Prime Geodesic Theorem (Theorem 5), we obtain X 1 q4l  1 = 2q4l + o . l l p type 2 prime of dimension 1 l(p)=l This ends the proof of the theorem.

4.9.5 Proof of Theorem 23

From the proof of the previous theorem, we know that the largest eigenvalues in absolute 4 A 0 A 0 0 q value of both P2,Γ and P2,Γ consist of just , with multiplicity 2. Hence, we may apply the Chebotarev Density Theorem (Theorem 8) and we obtain

|C| X 2q4l = 2l 1 + o(q4l) , | / 0| Γ Γ p type 2 prime of dimension 1 l(p)=l Frobp=C which implies X |C| q4l q4l  1 = + o . | / 0| l l p type 2 prime Γ Γ of dimension 1 l(p)=l Frobp=C

88 This concludes the proof of the theorem.

4.10 Proofs of Theorems 26 and 27

In this section, we prove the Prime Geodesic Theorem and Chebotarev Density Theorem for 2-dimensional primes of spin type. We start by describing the spin type chambers in the neighborhood of a type 2 edge. Then, we define a map that we use to construct spin type geodesic galleries over XΓ. Finally, we prove some properties of AI1,Γ and use those to prove the desired theorems.

4.10.1 Representatives of spin type chambers

We now list the spin type chambers in the neighborhood of a type 2 edge.

Spin type chambers with initial edge ts1P2Z

The proper out-neighbors of spin type chambers form geodesic galleries where any two consecutive chambers intersect along a type 2 edge. This gives a notion of initial and terminal edge for each spin type chamber. Recall that s1P2Z is one of the sides of I1Z, and may be interpreted as the initial edge of I1Z, whereas ts1P2Z = τP2Z is the terminal edge. Then, the initial edge of the spin type chamber gI1Z is gs1P2Z and the spin type chambers with initial edge ts1P2Z are described in the following decomposition

−1 ts1P2s1 I1Z = ts1P2s1Z     1 a 1     G  1  G  1  =   I1Z ∪   I1Z. aπ −π  −aπ π  a,d∈OF /πOF   a∈OF /πOF   dπ aπ π π

Spin type chamber with terminal edge ts1P2Z

As seen above, the spin type chamber gI1Z has terminal edge gts1P2Z. Hence, the spin type chambers with terminal edge ts1P2Z are given by the cosets in

−1 ts1P2(ts1) I1Z/I1Z.

89 Now, we use the matrix   1    1  ω =   .  π    π

Observe that ωts1P2Z = s1P2Z and ωs1P2Z = ts1P2Z, so the spin type chamber ωI1Z is I1Z in the opposite direction. As a consequence, the spin type chamber gωI1Z is gI1Z in the opposite direction. Hence, we may use the representatives from the previous decomposition, and multiply them on the right by ω (or ωπ−1), to obtain a description of the spin type chambers with terminal edge ts1P1Z. In particular, note that s1ωts1 ∈ P2Z, −1 −1 so P2(ts1) Z = P2s1ωπ Z and we have

−1 ts1P2(ts1) I1Z = ts1P2s1ωZ     π−1 1 a     G  1  G  1  =   I1Z ∪   I1Z.  −1 a   1 −a a,d∈OF /πOF   a∈OF /πOF   π aπ d 1

4.10.2 Successor map sI1

In this section we verify the existence of a bijective map from ts1P2s1ωZ/I1Z to ts1P2s1Z/I1Z that maps each spin type chamber to one of its proper out-neighbors.

We define sI1 as follows:       1 a 1 a 1   s      1  I1  1   1  (1)   I1Z −−→     I1Z  −a  −a aπ −π   1   1    1 1 aπ π       1 a 1 −a 1    2     1   1 −2a −a  1  =     I1Z =   I1Z aπ −π −a2π −aπ   aπ −π     1    aπ π 1 aπ π

for a ∈ OF /πOF

90       π−1 π−1 1   s      1  I1  1   1  (2)   I1Z −−→     I1Z  − a   − a   −π   1   1    π aπ d π aπ d π π     1 1 1 1      1   −1  =     I1Z  aπ π aπ      1  (d + 1)π aπ dπ −1   1    1  =   I1Z, for a, d ∈ OF /πOF , d 6≡ −1 (mod πOF )  aπ −π    (d + 1)π aπ π

      π−1 π−1 1   s      1  I1  1   1  (3)   I1Z −−→     I1Z  − a   − a  aπ −π   1   1    π aπ −1 π aπ −1 aπ π       a 1 1 1 a 1    2     1   1 −a   1  =     =   I1Z, −aπ π a2π aπ    −aπ π     1    π −π 1 π

for a ∈ OF /πOF

2,spin We can also extend sI1 to all Γ\X . To do so, we let SΓ(P2) be a set of represen- tatives of the orbits in Γ\X1,2. Then, we define

sI1 (as1ω · gI1Z) = as1ω · sI1 (gI1Z) , ∀a ∈ SΓ(P2), gI1Z ∈ ts1P2s1ωZ/I1Z.

Note that the spin type chamber as1ω · gI1Z ends on the edge aP2Z and as1ω · sI1 (gI1Z) starts on the edge aP2Z, when we vary gI1Z. Thus, this map is well defined for all spin type chambers of XΓ and is bijective, like we wished to prove.

4.10.3 Properties of AI1,Γ

We now show two properties of AI1,Γ.

Proposition 22. The adjacency operator AI1,Γ is irreducible.

91 Proof. First of all we note that XΓ is connected. So, in order to verify the irreducibility of AI1,Γ, it is enough to prove two things: (1) there is a geodesic gallery connecting a spin type chamber to any of its out-neighbors, and (2) there is a geodesic gallery connecting a spin type chamber to its different orientations.

For the first statement, consider two spin type chambers g0I1Z and g1I1Z, such that the terminal edge of g0I1Z coincides with the initial edge of g1I1Z, but g1I1Z is not a 0 proper out-neighbor of g0I1Z. We wish to find a spin type chamber g0I1Z satisfying

0 (i) g1I1Z is a proper out-neighbor of g0I1Z ;

0 (ii) sI1 (g0I1Z) is a proper out-neighbor of g0I1Z .

2 0 Note that out of q + q possibilities for g0I1Z there are q exceptions for condition (i) and the same number of exceptions for condition (ii), by Propositions 14 and 15. Thus, the number of spin type chambers that satisfy both of these conditions is at least

q2 + q − 2q = q2 − q > 0 .

0 Given g0I1Z, we may construct a spin type geodesic gallery starting from g0I1 and then sk g0 I Z X2,spin s going along the sequence ( I1 ( 0 1 ))k≥1. Furthermore, since Γ is finite and I1 is a bijection, the sequence must be periodic. Therefore, there is a spin type geodesic gallery like the following

0 2 0 0 g0I1Z → sI1 (g0I1Z) → sI1 (g0I1Z) → · · · → g0I1Z → g1I1Z, as we wanted to prove.

Now, for the second statement we prove that a spin type chamber gI1Z can be connected to gs1I1Z or gτI1Z using geodesic galleries. Recall that the actions of s1 and

τ on I1Z are reflexions along its diagonals. We have the following relations         1 1 1 1 1 1 π          1   1   1   −π       =   ∈ τI1Z ; −π π  −π π  −π π  −π2          π π π π2         1 1 1 1 1 1 1          1   1   1   1          = −π π  −π π  −π π   π          π π π π

92   π2  2  −π  =   ∈ s1I1Z.  −π2   π2

Each term in the multiplications above corresponds to an out-neighbor of I1Z (cf. section 4.10.1). Therefore, if we apply the first statement to each multiplication, we prove the second claim, as we wanted to show.

Proposition 23. The adjacency operator AI1,Γ has period 2.

Proof. Recall that the proper out-neighbors of a spin type chamber gI1Z can be written in the form ghI1Z, where h ∈ I1tI1, and so ordπ(det h) ≡ 2 (mod 4). Moreover, since ordπ(det Γ) ⊂ 4Z, this implies that any spin type closed geodesic gallery has length divisible by 2. Furthemore, we shall verify that the greatest common divisor of the lengths of closed galleries divides 2, thus it has to be equal to 2. We use multiplication by ω (given in section 4.9.1), and the same reasoning that was applied in the previous proposition. Write g0I1Z for gI1Z and g1I1Z for gωI1Z, then there is a closed geodesic running through the following spin type chambers in the specified order

g0I1Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I1 ( 0 1 ) I1 ( 0 1 ) 0 1 → g1I1Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I1 ( 1 1 ) I1 ( 1 1 ) 1 1 → g0I1Z.

Write the total length of this geodesic as 2 + d, where d is the sum of the lengths of the two intermediate geodesic loops in the geodesic gallery shown above. Then, if we repeat each loop d + 2 times, we still obtain a spin type closed geodesic gallery that has total length 2 + (2 + d)d. Consequently, the greatest common divisor of the lengths of such closed geodesic galleries divides 2, and the period must be 2.

4.10.4 Proof of Theorem 26

We showed above that AI1,Γ is irreducible and has period 2. Additionally, since each spin type chamber has q2 proper out-neighbors, by the Perron-Frobenius Theorem q2 and 2 −q are the eigenvalues of AI1,Γ with the largest absolute value. By the general Prime

93 Geodesic Theorem (Theorem 5), we conclude that

X 1 q4l  1 = 2q4l + o . l l p spin type prime 2 of dimension 2 l(p)=2l

This ends the proof.

4.10.5 Proof of Theorem 27

0 We saw in the previous sections that both AI1,Γ and AI1,Γ are irreducible and have period 2 0 2. For that reason, the largest eigenvalues in absolute value of AI1,Γ and AI1,Γ are q and −q2. Applying the Chebotarev Density Theorem (Theorem 8), we conclude that

|C| X 2q4l = 2l 1 + o(q4l) . | / 0| Γ Γ p spin type prime of dimension 2 l(p)=2l Frobp=C

Equivalently,

X |C| q4l q4l  1 = + o . | / 0| l l p spin type prime Γ Γ of dimension 2 l(p)=2l Frobp=C

This proves the theorem.

4.11 Proofs of Theorems 30 and 31

Here, we prove versions of the Prime Geodesic Theorem and the Chebotarev Density Theorem for 2-dimensional primes of standard type. We begin by describing the standard type chambers in the neighborhood of a type 1 edge, and then we construct standard type geodesic galleries over the whole XΓ. After that, we will be able to prove some properties of AI2,Γ and to characterize its eigenvalues, that will be used to prove the desired results.

4.11.1 Standard type chamber representatives

We describe the standard type chambers in the neighborhood of a type 1 edge.

94 Standard type chambers with terminal edge P1Z

The proper out-neighbors of standard type chambers give rise to geodesic galleries, where any two consecutive standard type chambers intersect along a type 1 edge. This induces a notion of initial and terminal edge of a standard type chamber. Recall that the type

1 edge P1Z is one of the sides of I2Z, which may be interpreted as its terminal edge, whereas s0s2P1Z is its initial edge. Then, the terminal edge of the standard type chamber gI2Z is gP1Z and its initial edge is gs0s2P1Z. Consequently, the standard type chambers with terminal edge P1Z are described by the following partition     a 1 d 1     G  1  G  1 b  P1Z =   I2Z ∪   I2Z. bπ −a 1  1  a,b,d∈OF /πOF   b,d∈OF /πOF   1 dπ 1

Standard type chambes with initial edge P1Z

In order to give a description of the chambers ending on the edge P1Z, we note that the matrices     π−1 −π−1      1   1  ν =   , and s0s2 =      −   1   1  π π only differ by multiplication by a diagonal matrix in K. Furthermore, they satisfy

νP1Z = s0s2P1Z and ν(s0s2)P1Z = P1Z so, in a sense, νI2Z is I2Z in the opposite direction. Thus, the initial edge of the standard type chamber gI2Z is the same as the terminal edge of gνI2Z, and vice-versa. For that reason we can multiply the representatives of the cosets in the previous decomposition by ν to obtain a parametrization of the standard type chambers starting on the edge P1Z, which is described below.     d 1 aπ−1 π−1  −1    G  π  G  b 1  P1νI2Z =   I2Z ∪   I2Z π −a b   1  a,b,d∈OF /πOF   b,d∈OF /πOF   1 π d

4.11.2 Successor map sI2

In this section we define a bijective map from P1Z/I2Z to P1νI2Z/I2Z that sends each standard type chamber to one of its proper out-neighbors. We define the map sI2 as

95 follows.       1 1 b 1   s    −1  1 b  I2  1 b   π  (1)   I2Z −−→     I2Z     π d   1   1    dπ 1 dπ 1 1     b 1 1 + bdπ bd2  −1    bπ π (1 + bdπ)  1 − bdπ −dπ  =     I2Z  π d   b2dπ bdπ     1 +  1 + bdπ dπ −bπ2 1 − bdπ   b 1  −1  π  =   I2Z, for b, d ∈ OF /πOF π d    1

      −1 1 d −1 1 d d 1   s    −1  1  I2  1   π  (2)   I2Z −−→     I2Z bπ  bπ  π b   1 1  1 1   1 1 1     dπ −d −1 π−1(1 + bdπ) 1 + bdπ −bd −b b2d      d 1   −π 1 − bdπ −bπ −b  =     I2Z  π bdπ bπ b   dπ bd2π bdπ bd   1 +   1 +  π b −dπ2 dπ π 1 − bdπ   π−1    d 1  =   , for b, d ∈ OF /πOF    1  π b

      a 1 d a 1 d 1 π−1   s    −1  1  I2  1   π  (3)   I2Z −−→     I2Z bπ −a  bπ −a  π −   1  1  1  1 1 1

96     dπ −d a (a + 1)π−1 1  −1     1 π  π −1  =     I2Z −aπ a bπ b   −   + 1   1  π −1 π 1   d 1 (a + 1)π−1  −1   π  =   I2Z, π − a b   ( + 1)  1

for a, b, d ∈ OF /πOF and a 6≡ −1 (mod πOF )

We may extend this map to all standard type chambers in XΓ. In order to do so, let 1,1 SΓ(P1) be a set of representatives of the orbits in Γ\X . Then, we define

sI2 (a · gI2Z) = a · sI2 (gI2Z) , ∀a ∈ SΓ(P1), gI2Z ∈ P1Z/I2Z.

Note that the standard type chamber a · gI2Z ends on the type 1 edge aP1Z and a · sI2 (gI2Z) starts on aP1Z. Thus, by varying gI2Z we conclude that this defines sI2 for all standard type chambers in XΓ, and the resulting map is a bijection.

4.11.3 Properties of AI2,Γ

Next, we prove two useful properties of AI2,Γ.

Proposition 24. The adjacency operator AI2,Γ has two nonzero blocks and both are irreducible.

Proof. Note that the proper out-neighbors of a standard type chamber gI2Z are given 0 by ghI2Z, where h ∈ I2t I2, and so ordπ(det h) ≡ 0 (mod 4). This means that we can separate the standard type geodesic galleries into two sets depending on whether they contain standard type chambers of the form gI2Z with ordπ(det g) ≡ 0 or 2 (mod 4).

This proves that AI2,Γ has two nonzero blocks. Now, we prove that each of the blocks is irreducible. To do this we will prove that (1) a standard type chamber can be connected to any of its out-neighbors using a geodesic gallery, and (2) a standard type chamber can be connected to its different orientations, using a geodesic gallery. For the first statement, consider two standard type chambers g0I2Z and g1I2Z, such that the terminal edge of g0I2Z coincides with the initial edge of g1I2Z, but g1I2Z is not a proper out-neighbor of g0I2Z. Then, we wish to find a standard 0 type chamber g0I2Z with the same terminal edge as g0I2Z that satisfies

97 0 (i) g1I2Z is a proper out-neighbor of g0I2Z ;

0 (ii) sI2 (g0I2Z) is a proper out-neighbor of g0I2Z .

3 2 0 2 Notice that out of the q + q possibilities for g0I2Z there are q exceptions for (i) and q2 exceptions for (ii), by Propositions 16 and 17. Hence, there is at least the following number of choices q3 + q2 − 2q2 = q3 − q2 > 0 .

For the second claim, it is enough to see that any chamber gI2Z can be connected to gs0I2Z and gs2I2Z by spin type geodesic galleries. We use the representatives of the out-neighbors of I2Z (cf. section 4.11.1). We have

        π−1 π−1 π−1 π−1          −1 1   1 1   −1 1   −1        =            1   1   1   1  π π π π

∈ s0I2Z ;

        π−1 π−1 π−1 −1          1   1   1   1        =   ∈ s2I2Z.          1   1   1   1  π 1 π −1 π 1 1 Where multiplication by each of the matrices above represents a movement to an out- neighbor. Therefore, using claim (1) for each multiplication proves claim (2), as we wanted to show.

Proposition 25. Each block of AI2,Γ has period 1.

Proof. We will use the following matrices in our calculations     π−1 π−1      1 1   −1 1  ε+ =   , ε− =   .      1   1  π 1 π −1

In order to determine the period of each component of AI2,Γ, we will check that the period divides 2 and 3, thus can only be 1. Let gI2Z be a standard type chamber and write g0I2Z for gI2Z and g1I2Z for gνI2Z, where ν was given in section 4.11.1. Note that g1I2Z is an out-neighbor of g0I2Z, which is not a proper out-neighbor, and vice-versa.

98 Then, using the observations from the previous proof, we see that there is a standard type closed geodesic gallery running through the following chambers in the shown order

g0I2Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I2 ( 0 2 ) I2 ( 0 2 ) 0 2 → g1I2Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I2 ( 1 2 ) I2 ( 1 2 ) 1 2 → g0I2Z.

Now, we let 2 + d be the length of the above closed geodesic gallery. If we repeat the two intermediate geodesic loops 2 + d times, we still obtain a closed geodesic gallery that has length 2 + (2 + d)d. Thus, the period divides 2. On the other hand note that

ε+ε−ε+ ∈ I2Z.

Notice that right multiplication of a standard type chamber gI2Z by ε− or ε+ corresponds to taking an out-neighbor that is not a proper out-neighbor (cf. section 4.11.1). In this context, we write g0I2Z for gI2Z, g2I2Z for gε+I2Z, and g3I2Z for gε+ε−I2Z. Then, using observations from the previous proof, we obtain a standard type geodesic gallery like the following

g0I2Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I2 ( 0 2 ) I2 ( 0 2 ) 0 2 → g2I2Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I2 ( 2 2 ) I2 ( 2 2 ) 2 2 → g3I2Z → s g0 I Z → s2 g0 I Z → ... → g0 I Z I2 ( 3 2 ) I2 ( 3 2 ) 3 2 → g0I2Z.

Let 3 + d be the total length of the closed geodesic gallery above. If we repeat the three geodesic loops 3 + d times we still obtain a closed geodesic, this time with length

3 + d(d + 3). We conclude that the period of AI2,Γ divides 3 as well, which together with the previous claim, means that it has to be 1.

4.11.4 Proof of Theorem 30

We have seen above that AI2,Γ has two irreducible components and each of them has period 1. Furthermore, we know that each standard type chamber has q3 proper out-

99 3 neighbors, so by the Perron-Frobenius Theorem, q is the eigenvalue of AI2,Γ with the largest absolute value, and it has multiplicity 2. Then, by the general Prime Geodesic Theorem (Theorem 5) we have

X 1 q3l  1 = 2q3l + o . l l p standard type prime of dimension 2 l(p)=l

This proves the theorem.

4.11.5 Proof of Theorem 31

0 From the calculations in the previous sections, we have seen that both AI2,Γ and AI2,Γ have two irreducible components with period 1. Consequently, this implies that the larger 3 0 eigenvalues of AI2,Γ and AI2,Γ in absolute value are only q , with multiplicity 2. Using the Chebotarev Density Theorem (Theorem 8), we conclude that

|C| X 2q3l = l 1 + o(q3l) . |Γ0/Γ| p standard type prime of dimension 2 l(p)=l Frobp=C

This in turn implies

X |C| 2q3l q3l  1 = + o , |Γ/Γ0| l l p standard type prime of dimension 2 l(p)=l Frobp=C as we wanted to prove.

100 Appendix | Perron-Frobenius Theorem for nonnegative matrices

In this appendix, we provide some results for real n × n matrices with nonnegative entries. We begin by introducing some concepts.

Definition. Let A be a real n × n matrix with nonnegative entries. We say that A is reducible if it can be turned into a block upper diagional matrix by conjugating by a permutation matrix, that is, there exists a permutation matrix P ∈ Mn(R) such that the following holds: ! T BC P AP = ,B ∈ Mr(R),D ∈ Mn−r(R) . D

Otherwise, A is called irreducible.

k k Definition. For a real nonnegative n × n matrix A = (aij), write A = (aij). We say that A is strongly connected if for any indices 1 ≤ i, j ≤ n, there is an integer k > 0 such k that aij > 0. Proposition 26. A real nonnegative n × n matrix A is irreducible if and only if it is strongly connected.

Now, we state the main result.

Theorem 32 (cf. [HJ13]). Let A be an n × n irreducible nonnegative matrix, and write

k k A = (aij).

k For 1 ≤ i ≤ n, let hi be the greatest common divisor of {k ∈ N : aii > 0}. Then, the hi are independent of i, called the period h. Furthermore, the eigenvalues of A with

101 maximum absolute value λ are given by

2πit λe h , 1 ≤ t ≤ h , each with multiplicity one.

102 Bibliography

[Alo86] N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83–96, 1986. Theory of computing (Singer Island, Fla., 1984).

[Apo76] Tom M. Apostol. Introduction to analytic number theory. Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics.

[Bas92] Hyman Bass. The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math., 3(6):717–797, 1992.

[Bro89] Kenneth S. Brown. Buildings. Springer-Verlag, New York, 1989.

[CF77] J. W. S. Cassels and A. Fröhlich. Hans Arnold Heilbronn. Bull. London Math. Soc., 9(2):219–232. (1 plate), 1977.

[Cio06] Sebastian M. Cioabă. Eigenvalues of graphs and a simple proof of a theorem of Greenberg. Linear Algebra Appl., 416(2-3):776–782, 2006.

[CSZ03] Donald I. Cartwright, Patrick Solé, and Andrzej Żuk. Ramanujan geometries of type Ãn. Discrete Math., 269(1-3):35–43, 2003.

[Dei98] Anton Deitmar. Geometric zeta-functions on p-adic groups. Math. Japon., 47(1):1–17, 1998.

[Del74] Pierre Deligne. La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., (43):273–307, 1974.

[DH06] Anton Deitmar and J. William Hoffman. The Ihara-Selberg zeta function for PGL3 and Hecke operators. Internat. J. Math., 17(2):143–155, 2006.

[Dwo60] Bernard Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J. Math., 82:631–648, 1960.

[ES85] I. Efrat and P. Sarnak. The determinant of the Eisenstein matrix and Hilbert class fields. Trans. Amer. Math. Soc., 290(2):815–824, 1985.

[FLW13] Yang Fang, Wen-Ching Winnie Li, and Chian-Jen Wang. The zeta functions of complexes from Sp(4). Int. Math. Res. Not. IMRN, (4):886–923, 2013.

[Gar97] Paul Garrett. Buildings and classical groups. Chapman & Hall, London, 1997.

103 [Gre95] Yoseph Greenberg. On the Spectrum of Graphs and Their Universal Covering. dissertation, Hebrew University, 1995.

[Gro65] Alexander Grothendieck. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Fasc. I: Exposés 1–8; Fasc. II: Exposés 9–13. Séminaire de Géométrie Algébrique 1962. Troisième édition, corrigée. Rédigé par un groupe d’auditeurs. Institut des Hautes Études Scientifiques, Paris, 1965.

[Has89] Ki-ichiro Hashimoto. Zeta functions of finite graphs and representations of p-adic groups. In Automorphic forms and geometry of arithmetic varieties, volume 15 of Adv. Stud. Pure Math., pages 211–280. Academic Press, Boston, MA, 1989.

[Has90] Ki-ichiro Hashimoto. On zeta and L-functions of finite graphs. Internat. J. Math., 1(4):381–396, 1990.

[Has92] Ki-ichiro Hashimoto. Artin type L-functions and the density theorem for prime cycles on finite graphs. Internat. J. Math., 3(6):809–826, 1992.

[Hej76] Dennis A. Hejhal. The Selberg trace formula for PSL(2,R). Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.

[HH89] Ki-ichiro Hashimoto and Akira Hori. Selberg-Ihara’s zeta function for p-adic discrete groups. In Automorphic forms and geometry of arithmetic varieties, volume 15 of Adv. Stud. Pure Math., pages 171–210. Academic Press, Boston, MA, 1989.

[HJ13] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013.

[HL] Hau-Wen Huang and Wen-Ching W. Li. Unramified graph cover of finite degree. preprint.

[Hub61] Heinz Huber. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II. Math. Ann., 142:385–398, 1960/1961.

[Iha66] Yasutaka Ihara. On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan, 18:219–235, 1966.

[KL14] Ming-Hsuan Kang and Wen-Ching Winnie Li. Zeta functions of complexes arising from PGL(3). Adv. Math., 256:46–103, 2014.

[KL15] Ming-Hsuan Kang and Wen-Ching Winnie Li. Artin L-functions on finite quotients of PGL3. Int. Math. Res. Not. IMRN, (19):9251–9276, 2015.

[KLW] Ming-Hsuan Kang, Wen-Ching Winnie Li, and Chian-Jen Wang. Zeta and L-functions of finite quotients of apartments and buildings. preprint.

104 [KLW10] Ming-Hsuan Kang, Wen-Ching Winnie Li, and Chian-Jen Wang. The zeta functions of complexes from PGL(3): a representation-theoretic approach. Israel J. Math., 177:335–348, 2010.

[Kue15] Thilo Kuessner. Multicomplexes, bounded cohomology and additivity of simplicial volume. Bull. Korean Math. Soc., 52(6):1855–1899, 2015.

[Lan94] Serge Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathe- matics. Springer-Verlag, New York, second edition, 1994.

[Li96] W. C. Winnie Li. Number theory with applications, volume 7 of Series on University Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

[Li04] W.-C. W. Li. Ramanujan hypergraphs. Geom. Funct. Anal., 14(2):380–399, 2004.

[LLR18] Alexander Lubotzky, Zur Luria, and Ron Rosenthal. On groups and simplicial complexes. European J. Combin., 70:408–444, 2018.

[LN98] Alexander Lubotzky and Tatiana Nagnibeda. Not every uniform tree covers Ramanujan graphs. J. Combin. Theory Ser. B, 74(2):202–212, 1998.

[LPS88] A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.

[LSV05a] Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type Aed. European J. Combin., 26(6):965–993, 2005.

[LSV05b] Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Ramanujan complexes of type Aed. Israel J. Math., 149:267–299, 2005. Probability in mathematics.

[Mar69] G. A. Margulis. Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funkcional. Anal. i Priložen., 3(4):89–90, 1969.

[Mar88] G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1):51–60, 1988.

[MV07] Hugh L. Montgomery and Robert C. Vaughan. Multiplicative number theory. I. Classical theory, volume 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007.

[Neu99] Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

105 [Sar80] Peter Clive Sarnak. PRIME GEODESIC THEOREMS. ProQuest LLC, Ann Arbor, MI, 1980. Thesis (Ph.D.)–Stanford University.

[Sar87] Peter Sarnak. Determinants of Laplacians. Comm. Math. Phys., 110(1):113–120, 1987.

[Sel56] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.), 20:47–87, 1956.

[Ser80] Jean-Pierre Serre. Trees. Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell.

[Set08] Alison Setyadi. Distance in the affine buildings of SLn and Spn. Integers, 8:A48, 17, 2008.

[Som15] Marius Somodi. On Sunada equivalent graph coverings. J. Comb. Number Theory, 7(2):79–94, 2015.

[ST96] H. M. Stark and A. A. Terras. Zeta functions of finite graphs and coverings. Adv. Math., 121(1):124–165, 1996.

[ST00] H. M. Stark and A. A. Terras. Zeta functions of finite graphs and coverings. II. Adv. Math., 154(1):132–195, 2000.

[Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985.

[Sun86] Toshikazu Sunada. L-functions in geometry and some applications. In Cur- vature and topology of Riemannian manifolds (Katata, 1985), volume 1201 of Lecture Notes in Math., pages 266–284. Springer, Berlin, 1986.

[Sun87] Toshikazu Sunada. Fundamental group and Laplacians. S¯ugaku, 39(3):193–203, 1987.

[Ter11] Audrey Terras. Zeta functions of graphs, volume 128 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2011. A stroll through the garden.

[TS07] A. A. Terras and H. M. Stark. Zeta functions of finite graphs and coverings. III. Adv. Math., 208(1):467–489, 2007.

[Vor87] A. Voros. Spectral functions, special functions and the Selberg zeta function. Comm. Math. Phys., 110(3):439–465, 1987.

[Wei49] André Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497–508, 1949.

106 Vita João Correia Matias

Education The Pennsylvania State University, State College, PA, USA

• Ph.D. in Mathematics, August 2012 – December 2018 Advisor: Wen-Ching Winnie Li

Instituto Superior Técnico, Technical University of Lisbon (UTL), Lisbon, Portugal

• Master in Applied Mathematics, September 2010 – July 2012 Advisor: Carlos Florentino

• BSc in Applied Mathematics and Computation, September 2007 – July 2010

Publications • P. Lopes, J. Matias, Minimum number of colors: the Turk’s head knots case study, Discrete Math. Theor. Comput. Sci. 17 (2015), no. 2, (arXiv:1002.4722).

• P. Lopes, J. Matias, Minimum Number of Fox Colors for Small Primes, J. Knot Theory Ramifications, 21 (2012) no. 3, (arXiv:1001.1334).

Experience • Graduate Teaching Assistant, The Pennsylvania State University 2012 – 2018 Taught courses at an undergraduate level ranging from college algebra to multivariable calculus, while conducting my own research. Also managed the associated online platform by uploading extra materials and grades.

• Participation in competitive programming websites including Google Code Jam, Codeforces, Hackerrank, and Leetcode.

• Multiple Research Fellowships (Undergraduate Student) 2007 – 2011 Studied various extra-curricular subjects including: knot theory, number theory, plane geometry and algebraic geometry. These projects led to two publications in peer-reviewed journals.

• Math Olympiads 2002 – 2009 Represented Portugal in international competitions, and won medals multiple times.

Programming Languages

• C++, C, Python, Mathematica, LATEX, Git, SQL, MatLab, Sage, Java, HTML.