Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 129 Zeta and L-functions in Number Theory and Combinatorics Wen-Ching Winnie Li with support from the 10.1090/cbms/129 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 129 Zeta and L-functions in Number Theory and Combinatorics Wen-Ching Winnie Li Published for the Conference Board of the Mathematical Sciences by the with support from the NSF-CBMS Regional Conference in the Mathematical Sciences on Combinatorial Zeta and L-functions held at the Sundance Resort, Utah, May 12–16, 2014 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of the Mathematical Sciences and NSF grant #DMS-1341413. Any opinions, findings, and conclusions or recommenda- tions expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. 2010 Mathematics Subject Classification. Primary 11M26, 05E45, 11M36, 11F41, 11Z05, 11F66. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-129 Library of Congress Cataloging-in-Publication Data Names: Li, W. C. Winnie (Wen-Ching Winnie), author. | Conference Board of the Mathematical Sciences. | National Science Foundation (U.S.) Title: Zeta and L-functions in number theory and combinatorics / Wen-Ching Winnie Li. Description: Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2019] | Series: CBMS regional conference series in mathematics ; number 129 | “Support from the National Science Foundation.” | Includes bibliographical references and index. Identifiers: LCCN 2018048915 | ISBN 9781470449001 (alk. paper) Subjects: LCSH: Functions, Zeta. | L-functions. | Number theory. | Combinatorial number theory. Classification: LCC QA403 .J67 2018 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2018048915 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 242322212019 Contents Preface v Lecture 1. Number theoretic zeta and L-functions 1 1.1. The Riemann zeta function 1 1.2. The Dirichlet L-functions 2 1.3. Finite extensions of Q 3 1.4. The Artin L-functions 7 1.5. Zeta functions for varieties over finite fields 9 Lecture 2. The Selberg zeta function 15 2.1. The action of SL2(R)15 2.2. Classification of elements in SL2(R)15 2.3. Compact Riemann surfaces arising as quotients of H 17 2.4. Geodesic cycles in XΓ 18 2.5. The Selberg zeta function attached to XΓ 21 2.6. Properties of the Laplacian operator 24 Lecture 3. L-functions in geometry 27 3.1. General setup 27 3.2. Anosov flow 29 3.3. L-functions and distributions of prime geodesics 31 Lecture 4. The Ihara zeta function 35 4.1. The Ihara zeta function 35 4.2. Ihara’s theorem 37 4.3. Hashimoto’s theorem 38 4.4. A cohomological proof of the zeta identity for a graph 39 Lecture 5. Spectral graph theory 43 5.1. Eigenvalues of a regular graph 43 5.2. The behavior of nontrivial eigenvalues 43 5.3. Ramanujan graphs and the Riemann hypothesis 45 5.4. Friedman’s theorem 46 Lecture 6. Explicit constructions of Ramanujan graphs 47 6.1. Expanding constant and spectral gap 47 6.2. Some nice features of Ramanujan graphs 48 6.3. Cayley graphs 49 6.4. PGL2(Qp)/P GL2(Zp)asa(p + 1)-regular tree 50 6.5. Ramanujan graphs as finite quotients of PGL2(Qp)51 iii iv CONTENTS 6.6. Hashimoto’s class number formula 53 6.7. Biregular bipartite Ramanujan graphs 56 Lecture 7. Artin L-functions and prime distributions for graphs 61 7.1. Artin L-functions for graphs 61 7.2. Unramified covers of X 62 7.3. Distribution of primes for graphs 64 7.4. A characterization for finite unramified Galois covers 68 Lecture 8. Zeta and L-functions of complexes 69 8.1. The building attached to PGLn(F )69 8.2. Spectral theory of regular complexes from Bn(F )70 8.3. Ramanujan complexes as finite quotients of Bn(F )70 8.4. Zeta and L-functions of finite quotients of B3(F )74 8.5. Zeta functions of finite quotients of the building Δ(F )ofSp4(F )80 8.6. Distribution of primes in finite quotients of B3(F )andΔ(F )83 Bibliography 87 Index 93 Preface The zeta and L-functions play a central role in number theory. They provide important information of arithmetic nature. For example, the analytic behavior of the Riemann zeta function ζ(s) on the closed right half-plane (s) ≥ 1 leads to the estimate of the number of prime numbers up to x by the logarithmic integral Li(x), and the error of this estimate is controlled by the location of the zeros of ζ(s)inthe critical strip 0 < (s) < 1. The celebrated Riemann Hypothesis asserts that these zeros should all lie on the line of symmetry (s)=1/2. Under this assumption one can bound the error by O(x1/2 log x). This is the Prime Number Theorem. Fur- thermore, given a finite Galois extension of Q with Galois group G, we can partition the prime numbers according to their associated Frobenius conjugacy classes in G. For a conjugacy class C of G,denotebySC the collection of prime numbers whose associated Frobenius conjugacy classes equal C. The distribution of prime num- bers in SC is described by the Chebotarev Density Theorem, which says that SC has natural density |C|/|G|. In particular, the prime numbers are equidistributed among the sets SC when the Galois group G is abelian. Specialized to cyclotomic extensions of Q, this is Dirichlet’s Theorem on primes in arithmetic progressions. The Chebotarev Density Theorem follows from the holomorphy and nonvanishing of the Artin L-functions attached to nontrivial irreducible representations of G on the half-plane (s) ≥ 1. On the geometric side, to a d-dimensional projective smooth irreducible variety V defined over a finite field Fq, Artin and Smith attached the zeta function Z(V,u), which counts points on V with coordinates in finite extensions of Fq.Asshownby Grothendieck, Z(V,u) is an alternating product of polynomials Pi(u), 0 ≤ i ≤ 2d, arising geometrically; and Deligne proved the Riemann Hypothesis as conjectured −i/2 by Weil; that is, the zeros of a nonconstant Pi(u) have absolute value q . Like the Riemann zeta function, Z(V,u) has an Euler product over closed points, which play the role of primes of V .WhenV is a curve, the analytic behavior of Z(V,u) and similarly defined Artin L-functions give rise to analogues of the Prime Number Theorem and the Chebotarev Density Theorem. These remarkable achievements in number theory are reviewed in Lecture 1. The purpose of this monograph is to provide a systematic and comprehensive ac- count of the developments of these topics in geometry and combinatorics for grad- uate students and researchers. We shall highlight interactions between number theory and other fields and compare similarities and dissimilarities under different settings. This is done in chronological order. Lecture 2 introduces the first instance, considered by Selberg in the 1950s, on compact Riemann surfaces arising as quo- tients of the upper half-plane H by discrete torsion-free cocompact subgroups Γ of SL2(R). In this setting the primes of Γ\H are primitive closed geodesic cycles not v vi PREFACE counting the starting points, and the fundamental group Γ plays the role of abso- lute Galois group. Selberg introduced zeta functions attached to finite-dimensional representations of Γ as suitable products over such primes. These functions have nice analytic properties like ζ(s), and they satisfy the Riemann Hypothesis for all except possibly finitely many real zeros in the critical strip. The Prime Geodesic Theorem and the Chebotarev Density Theorem for Γ\H are established by Huber and Sarnak. Lecture 3 is a digression to finite-dimensional compact Riemannian manifolds and the connection to dynamical systems. We shall see that the distribution of primes is related to the analytic behavior of the associated L-functions of Artin type, which are variations of the Ruelle zeta function in dynamical systems. While this is a much explored topic in dynamical systems, our exposition will stay close to the theme of this monograph. The remainder of this monograph is devoted to the combinatorial setting. In the 1960s, by interpreting the upper half-plane H as the homogeneous space PGL2(R)/P O2(R), Ihara extended Selberg’s results from the real field R to a nonar- chimedean local field F with q elements in its residue field.