L-Functions of the Selberg Class S
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Arithmetic Equivalence and Isospectrality
ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK . -
Selberg's Orthogonality Conjecture and Symmetric
SELBERG'S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS PENG-JIE WONG Abstract. Let π be a cuspidal representation of GL2pAQq defined by a non-CM holomorphic newform of weight k ¥ 2. Let K{Q be a totally real Galois extension with Galois group G, and let χ be an irreducible character of G. In this article, we show that Selberg's orthogonality conjecture implies that the symmetric power L-functions Lps; Symm πq are primitive functions in the Selberg class. Moreover, assuming the automorphy for Symm π, we show that Selberg's orthogonality conjecture implies that each Lps; Symm π ˆ χq is automorphic. The key new idea is to apply the work of Barnet-Lamb, Geraghty, Harris, and Taylor on the potential automorphy of Symm π. 1. Introduction Thirty years ago, Selberg [21] introduced a class S of L-functions, called the Selberg class nowadays. The class S consists of functions F psq of a complex variable s enjoying the following properties: (Dirichlet series and Euler product). For Repsq ¡ 1, 8 a pnq 8 F psq “ F “ exp b ppkq{pks ns F n“1 p k“1 ¸ ¹ ´ ¸ ¯ k kθ 1 with aF p1q “ 1 and bF pp q “ Opp q for some θ ă 2 , where the product is over rational primes p. Moreover, both Dirichlet series and Euler product of F converge absolutely on Repsq ¡ 1. (Analytic continuation and functional equation). There is a non-negative integer mF such that ps ´ 1qmF F psq extends to an entire function of finite order. Moreover, there are numbers rF , QF ¡ 0, αF pjq ¡ 0, RepγF pjqq ¥ 0, so that rF s ΛF psq “ QF ΓpαF pjqs ` γF pjqqF psq j“1 ¹ satisfies ΛF psq “ wF ΛF p1 ´ sq for some complex number wF of absolute value 1. -
Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods
Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2016 Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods Kühn, Patrick Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-126347 Dissertation Published Version Originally published at: Kühn, Patrick. Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods. 2016, University of Zurich, Faculty of Science. UNIVERSITÄT ZÜRICH Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von Patrick KÜHN von Mendrisio (TI) Promotionskomitee: Prof. Dr. Paul-Olivier DEHAYE (Vorsitz und Leitung) Prof. Dr. Ashkan NIKEGHBALI Prof. Dr. Andrew KRESCH Zürich, 2016 Declaration of Authorship I, Patrick KÜHN, declare that this thesis titled, ’Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods’ and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree • at this University. Where any part of this thesis has previously been submitted for a degree or any • other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- • tributed. Where I have quoted from the work of others, the source is always given. With • the exception of such quotations, this thesis is entirely my own work. -
Introduction to L-Functions: the Artin Cliffhanger…
Introduction to L-functions: The Artin Cliffhanger. Artin L-functions Let K=k be a Galois extension of number fields, V a finite-dimensional C-vector space and (ρ, V ) be a representation of Gal(K=k). (unramified) If p ⊂ k is unramified in K and p ⊂ P ⊂ K, put −1 −s Lp(s; ρ) = det IV − Nk=Q(p) ρ (σP) : Depends only on conjugacy class of σP (i.e., only on p), not on P. (general) If G acts on V and H subgroup of G, then V H = fv 2 V : h(v) = v; 8h 2 Hg : IP With ρjV IP : Gal(K=k) ! GL V . −1 −s Lp(s; ρ) = det I − Nk=Q(p) ρjV IP (σP) : Definition For Re(s) > 1, the Artin L-function belonging to ρ is defined by Y L(s; ρ) = Lp(s; ρ): p⊂k Artin’s Conjecture Conjecture (Artin’s Conjecture) If ρ is a non-trivial irreducible representation, then L(s; ρ) has an analytic continuation to the whole complex plane. We can prove meromorphic. Proof. (1) Use Brauer’s Theorem: X χ = ni Ind (χi ) ; i with χi one-dimensional characters of subgroups and ni 2 Z. (2) Use Properties (4) and (5). (3) L (s; χi ) is meromorphic (Hecke L-function). Introduction to L-functions: Hasse-Weil L-functions Paul Voutier CIMPA-ICTP Research School, Nesin Mathematics Village June 2017 A “formal” zeta function Let Nm, m = 1; 2;::: be a sequence of complex numbers. 1 m ! X Nmu Z(u) = exp m m=1 With some sequences, if we have an Euler product, this does look more like zeta functions we have seen. -
On $ P $-Adic String Amplitudes in the Limit $ P $ Approaches To
On p-adic string amplitudes in the limit p approaches to one M. Bocardo-Gaspara1, H. Garc´ıa-Compe´anb2, W. A. Z´u˜niga-Galindoa3 Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional aDepartamento de Matem´aticas, Unidad Quer´etaro Libramiento Norponiente #2000, Fracc. Real de Juriquilla. Santiago de Quer´etaro, Qro. 76230, M´exico bDepartamento de F´ısica, P.O. Box 14-740, CP. 07000, M´exico D.F., M´exico. Abstract In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, conse- quently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa’s local zeta function gives rise to an object called topological zeta func- tion. By using Denef-Loeser’s theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (repro- ducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. -
On the Selberg Class of L-Functions
ON THE SELBERG CLASS OF L-FUNCTIONS ANUP B. DIXIT Abstract. The Selberg class of L-functions, S, introduced by A. Selberg in 1989, has been extensively studied in the past few decades. In this article, we give an overview of the structure of this class followed by a survey on Selberg's conjectures and the value distribution theory of elements in S. We also discuss a larger class of L-functions containing S, namely the Lindel¨ofclass, introduced by V. K. Murty. The Lindel¨ofclass forms a ring and its value distribution theory surprisingly resembles that of the Selberg class. 1. Introduction The most basic example of an L-function is the Riemann zeta-function, which was intro- duced by B. Riemann in 1859 as a function of one complex variable. It is defined on R s 1 as ∞ 1 ( ) > ζ s : s n=1 n It can be meromorphically continued to( the) ∶= wholeQ complex plane C with a pole at s 1 with residue 1. The unique factorization of natural numbers into primes leads to another representation of ζ s on R s 1, namely the Euler product = −1 1 ( ) ( ) > ζ s 1 : s p prime p The study of zeta-function is vital to( ) understanding= M − the distribution of prime numbers. For instance, the prime number theorem is a consequence of ζ s having a simple pole at s 1 and being non-zero on the vertical line R s 1. ( ) = In pursuing the analogous study of distribution( ) = of primes in an arithmetic progression, we consider the Dirichlet L-function, ∞ χ n L s; χ ; s n=1 n ( ) for R s 1, where χ is a Dirichlet character( ) ∶= moduloQ q, defined as a group homomorphism ∗ ∗ χ Z qZ C extended to χ Z C by periodicity and setting χ n 0 if n; q 1. -
International Conference on Number Theory and Discrete Mathematics (ONLINE) (ICNTDM) 11 - 14, DECEMBER-2020
International Conference on Number Theory and Discrete Mathematics (ONLINE) (ICNTDM) 11 - 14, DECEMBER-2020 Organised by RAMANUJAN MATHEMATICAL SOCIETY Hosted by DEPARTMENT OF MATHEMATICS RAJAGIRI SCHOOL OF ENGINEERING & TECHNOLOGY (AUTONOMOUS) Kochi, India International Conference on Number Theory and Discrete Mathematics PREFACE The International Conference on Number Theory and Discrete Mathematics (ICNTDM), organised by the Ramanujan Mathematical Society (RMS), and hosted by the Rajagiri School of Engineering and Technology (RSET), Cochin is a tribute to the legendary Indian mathematician Srinivasa Ramanujan who prematurely passed away a hundred years ago on 26th April 1920 leaving a lasting legacy. Conceived as any usual conference as early as June 2019, the ICNTDM was compelled to switch to online mode due to the pandemic, like most events round the globe this year. The International Academic Programme Committee, consisting of distinguished mathematicians from India and abroad, lent us a helping hand every possible way, in all aspects of the conference. As a result, we were able to evolve a strong line up of reputed speakers from India, Austria, Canada, China, France, Hungary, Japan, Russia, Slovenia, UK and USA, giving 23 plenary talks and 11 invited talks, in addition to the 39 speakers in the contributed session. This booklet consists of abstracts of all these talks and two invited articles; one describing the various academic activities that RMS is engaged in, tracing history from its momentous beginning in 1985, and a second article on Srinivasa Ramanujan. It is very appropriate to record here that, in conjunction with the pronouncement of the year 2012 { the 125th birth anniversary year of Ramanujan { as the `National Mathematics Year', RMS heralded an excellent initiative of translating the cele- brated book `The Man Who Knew Infinity' into many Indian languages. -
The Riemann Hypothesis for Varieties Over Finite Fields
The Riemann Hypothesis for Varieties over Finite Fields Sander Mack-Crane 16 July 2015 Abstract We discuss the Weil conjectures, especially the Riemann hypothesis, for varieties over finite fields. Particular detail is devoted to the proof of the Riemann hypothesis for cubic threefolds in projective 4-space, as given by Bombieri and Swinnerton-Dyer. The main idea is to relate the number of points on a cubic threefold to the trace of Frobenius on an associated abelian variety, and we develop the necessary machinery of abelian varieties. 1 Introduction The Weil conjectures are a set of conjectures (now proven) describing the number of points of varieties over finite fields, as encoded in the zeta function of the variety. They give an analog in finite fields of both the Riemann hypothesis of analytic number theory and the cohomology theory of complex varieties. We will introduce both of these perspectives in turn, partly following [2]. But first, we review an element of algebraic geometry that is essential for our discussion. 1.1 Fields of Definition In the classical setting, let n Ak = f(a1,..., an) j ai 2 kg be n-dimensional affine space over a field k, f fig ⊂ k[x1,..., xn] some set of polynomials, and n n V = V( fi) = f(a1,..., an) 2 Ak j fi(a1,..., an) = 0 for all ig ⊂ Ak the variety defined by the polynomials f fig. For any extension K of k, we have k[x0,..., xn] ⊂ K[x0,..., xn], and in particular our polynomials fi defining X are in K[x0,..., xn]. -
An Introduction to the Theory of Local Zeta Functions from Scratch
Revista Integración, temas de matemáticas Escuela de Matemáticas Universidad Industrial de Santander H ◦ Vol. 37, N 1, 2019, pág. 45–76 DOI: http://dx.doi.org/10.18273/revint.v37n1-2019004 An Introduction to the Theory of Local Zeta Functions from Scratch Edwin León-Cardenal a, W. A. Zúñiga-Galindo b∗ a Centro de Investigación en Matemáticas, Unidad Zacatecas, Quantum, Ciudad del Conocimiento, Zacatecas, México. b Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Unidad Querétaro, Departamento de Matemáticas, Libramiento Norponiente #2000, Fracc. Real de Juriquilla. Santiago de Querétaro, Qro. 76230, México. Abstract. This survey article aims to provide an introduction to the theory of local zeta functions in the p-adic framework for beginners. We also give an extensive guide to the current literature on local zeta functions and its connections with other fields in mathematics and physics. Keywords: Local zeta functions, p-adic analysis, local fields, stationary phase formula. MSC2010: 11S40, 11S80, 11M41, 14G10. Una Introducción a la Teoría de las Funciones Zeta Locales para principiantes Resumen. En este artículo panorámico brindamos una introducción a la teoría de las funciones zeta locales p-ádicas para principiantes. También se presenta una revisión extensiva a la literatura especializada sobre funciones zeta locales y sus conexiones con otros campos de las matemáticas y la física. Palabras clave: Funciones zeta locales, análisis p-ádico, cuerpos locales, fór- mula de la fase estacionaria. 1. Introduction In these notes we provide an introduction to the theory of local zeta functions from scratch. We assume essentially a basic knowledge of algebra, metric spaces and basic analysis, mainly measure theory. -
Arxiv:1711.06671V1 [Math.NT] 17 Nov 2017 Function Enetnieysuid[1 Nrcn Ie.Tesuyo Au Dist Value of Study the of Times
VALUE DISTRIBUTION OF L-FUNCTIONS ANUP B. DIXIT Abstract. In 2002, V. Kumar Murty [8] introduced a class of L-functions, namely the Lindel¨of class, which has a ring structure attached to it. In this paper, we establish some results on the value distribution of L-functions in this class. As a corollary, we also prove a uniqueness theorem in the Selberg class. 1. Introduction In 1992, Selberg [10] formulated a class of L-functions, which can be regarded as a model for L-functions originating from arithmetic objects. The value distribution of such L-functions has been extensively studied [11] in recent times. The study of value distribution is concerned with the zeroes of L-functions and more generally, with the set of pre-images L−1(c) := {s ∈ C : L(s)= c} where c is any complex number, which Selberg called the c-values of L. For any two meromorphic functions f and g, we say that they share a value c ignoring multiplicity(IM) if f −1(c) is the same as g−1(c) as sets. We further say that f and g share a value c counting multiplicity(CM) if the zeroes of f(x) − c and g(x) − c are the same with multiplicity. The famous Nevanlinna theory [9] establishes that any two meromorphic functions of finite order sharing five values IM must be the same. Moreover, if they share four values CM, then one must be a M¨obius transform of the other. The numbers four and five are the best possible for meromorphic functions. -
Analytic Problems for Elliptic Curves
ANALYTIC PROBLEMS FOR ELLIPTIC CURVES E. KOWALSKI Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. Contents 1. Introduction 1 2. Some local invariants for elliptic curves 2 3. Totally split primes 7 4. Elliptic twins 20 5. Curves with complex multiplication 25 6. Local study of totally split primes 41 7. Numerical examples 57 8. Conclusion 63 References 64 1. Introduction This paper introduces and discusses some problems of analytic number theory which are related to the arithmetic of elliptic curves over number fields. One can see them as analogues of some very classical problems about the distribution of prime numbers, especially primes in arithmetic progressions to large moduli. The motivation comes both from these analogies and from the conjecture of Birch and Swinnerton-Dyer. To explain this, consider an elliptic curve E defined over Q, given by a (minimal) Weierstrass equation ([Si-1, VII-1]) 2 3 2 (1.1) y + a1xy + a3y = x + a2x + a4x + a6 with ai ∈ Z. For all primes p we can consider the reduced curve Ep modulo p, which for almost all p will be an elliptic curve over the finite field Fp = Z/pZ. We wish to study the behavior of sums of the type X (1.2) ιE(p) p6X as X → +∞, where ιE(p) is some invariant attached to the reduced curve Ep and to its finite group of Fp-rational points in particular. -
Uncoveringanew L-Function Andrew R
UncoveringaNew L-function Andrew R. Booker n March of this year, my student, Ce Bian, plane, with the exception of a simple pole at s = 1. announced the computation of some “degree Moreover, it satisfies a “functional equation”: If 1 3 transcendental L-functions” at a workshop −s=2 γ(s) = ÐR(s) := π Ð (s=2) and Λ(s) = γ(s)ζ(s) at the American Institute of Mathematics (AIM). This article aims to explain some of the then I (1) (s) = (1 − s). motivation behind the workshop and why Bian’s Λ Λ computations are striking. I begin with a brief By manipulating the ζ-function using the tools of background on L-functions and their applications; complex analysis (some of which were discovered for a more thorough introduction, see the survey in the process), one can deduce the famous Prime article by Iwaniec and Sarnak [2]. Number Theorem, that there are asymptotically x ≤ ! 1 about log x primes p x as x . L-functions and the Selberg Class Other L-functions reveal more subtle properties. There are many objects that go by the name of For instance, inserting a multiplicative character L-function, and it is difficult to pin down exactly χ : (Z=qZ)× ! C in the definition of the ζ-function, what one is. In one of his last published papers we get the so-called Dirichlet L-functions, [5], the late Fields medalist A. Selberg tried an Y 1 L(s; χ) = : axiomatic approach, basically by writing down the 1 − χ(p)p−s common properties of the known examples.