An Introduction to Tate's Thesis
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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 . -
Algebraic Hecke Characters and Compatible Systems of Abelian Mod P Galois Representations Over Global fields
Algebraic Hecke characters and compatible systems of abelian mod p Galois representations over global fields Gebhard B¨ockle January 28, 2010 Abstract In [Kh1, Kh2, Kh3], C. Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke Characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare's result for arbitrary global fields. In a sequel we shall apply this result to compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms. 1 Introduction In [Kh1, Kh2, Kh3], C. Khare studied compatible systems of abelian mod p Galois rep- resentations of a number field. He showed that the semisimplification of any such arises from a direct sum of algebraic Hecke characters, as was suggested by the framework of motives. On the one hand, this result shows that with minimal assumptions such as only knowing the mod p reductions, one can reconstruct a motive from an abelian compati- ble system. On the other hand, which is also remarkable, the association of the Hecke characters to the compatible systems is based on fairly elementary tools from algebraic number theory. The first such association was based on deep transcendence results by Henniart [He] and Waldschmidt [Wa] following work of Serre [Se]. -
Hecke Characters, Classically and Idelically
HECKE CHARACTERS CLASSICALLY AND IDELICALLY` Hecke's original definition of a Gr¨ossencharakter, which we will call a Hecke character from now on, is set in the classical algebraic number theory environment. The definition is as it must be to establish the analytic continuation and functional equation for a general number field L-function X Y L(χ, s) = χ(a)Na−s = (1 − χ(p)Np−s)−1 a p analogous to Dirichlet L-functions. But the classical generalization of a Dirich- let character to a Hecke character is complicated because it must take units and nonprincipal ideals into account, and it is difficult to motivate other than the fact that it is what works. By contrast, the definition of a Hecke character in the id`elic context is simple and natural. This writeup explains the compatibility of the two definitions. Most of the ideas here were made clear to me by a talk that David Rohrlich gave at PCMI in 2009. Others were explained to me by Paul Garrett. The following notation is in effect throughout: • k denotes a number field and O is its ring of integers. • J denotes the id`elegroup of k. • v denotes a place of k, nonarchimedean or archimedean. 1. A Multiplicative Group Revisited This initial section is a warmup whose terminology and result will fit into what follows. By analogy to Dirichlet characters, we might think of groups of the form (O=f)×; f an ideal of O as the natural domains of characters associated to the number field k. This idea is na¨ıve, because O needn't have unique factorization, but as a starting point we define a group that is naturally isomorphic to the group in the previous display. -
Nonvanishing of Hecke L-Functions and Bloch-Kato P
NONVANISHING OF HECKE L{FUNCTIONS AND BLOCH-KATO p-SELMER GROUPS ARIANNA IANNUZZI, BYOUNG DU KIM, RIAD MASRI, ALEXANDER MATHERS, MARIA ROSS, AND WEI-LUN TSAI Abstract. The canonical Hecke characters in the sense of Rohrlich form a set of algebraic Hecke characters with important arithmetic properties. In this paper, we prove that for an asymptotic density of 100% of imaginary quadratic fields K within certain general families, the number of canonical Hecke characters of K whose L{function has a nonvanishing central value is jdisc(K)jδ for some absolute constant δ > 0. We then prove an analogous density result for the number of canonical Hecke characters of K whose associated Bloch-Kato p-Selmer group is finite. Among other things, our proofs rely on recent work of Ellenberg, Pierce, and Wood on bounds for torsion in class groups, and a careful study of the main conjecture of Iwasawa theory for imaginary quadratic fields. 1. Introduction and statement of results p Let K = Q( −D) be an imaginary quadratic field of discriminant −D with D > 3 and D ≡ 3 mod 4. Let OK be the ring of integers and letp "(n) = (−D=n) = (n=D) be the Kronecker symbol. × We view " as a quadratic character of (OK = −DOK ) via the isomorphism p ∼ Z=DZ = OK = −DOK : p A canonical Hecke character of K is a Hecke character k of conductor −DOK satisfying p 2k−1 + k(αOK ) = "(α)α for (αOK ; −DOK ) = 1; k 2 Z (1.1) (see [R2]). The number of such characters equals the class number h(−D) of K. -
Notes on the Riemann Hypothesis Ricardo Pérez-Marco
Notes on the Riemann Hypothesis Ricardo Pérez-Marco To cite this version: Ricardo Pérez-Marco. Notes on the Riemann Hypothesis. 2018. hal-01713875 HAL Id: hal-01713875 https://hal.archives-ouvertes.fr/hal-01713875 Preprint submitted on 21 Feb 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. NOTES ON THE RIEMANN HYPOTHESIS RICARDO PEREZ-MARCO´ Abstract. Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture. Contents 1. Euler transalgebraic world. 2 2. Riemann's article. 8 2.1. Meromorphic extension. 8 2.2. Value at negative integers. 10 2.3. First proof of the functional equation. 11 2.4. Second proof of the functional equation. 12 2.5. The Riemann Hypothesis. 13 2.6. The Law of Prime Numbers. 17 3. On Riemann zeta-function after Riemann. -
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. -
Tate's Thesis
学士学位论文 Tate 的论文 作者姓名: 刘浩浩 学科专业: 数学与应用数学 导师姓名: Benoît Stroh 教授 欧阳毅 教授 完成时间: 二〇一九年五月一日 University of Science and Technology of China A dissertation for bachelor’s degree Tate’s Thesis Author: Haohao Liu Speciality: Mathematics and Applied Mathematics Supervisors: Prof. Benoît Stroh, Prof. Yi Ouyang Finished time: May 1, 2019 中国科学技术大学本科毕业论文 Acknowledgements 撰写论文期间,我得到了两位老师的指导,他们是:中国科学技术大学的欧 阳毅教授和巴黎六大的 Benoît Stroh 教授。我感谢欧阳老师所给的选题建议,让 我有机会仔细学习 Tate’s thesis 这一优美而重要的工作。欧阳老师也为我提供了 关于参考资料的宝贵意见,论文修改也花费了他诸多时间。Stroh 老师具体指导 我的学习,他看待问题观点新颖,和他的讨论每每让我眼前一亮。他指出我理解 的不足,并且制定了论文的具体内容。在此对两位老师深表谢意。 Please send me an email if you note any mistakes: [email protected]. 中国科学技术大学本科毕业论文 Contents 中文内容摘要 ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 3 英文内容摘要 ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 4 Chapter 1 简介/Introduction ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 6 Chapter 2 经典理论/Classical Theory ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 7 2.1 Dedekind zeta function ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 7 2.1.1 General Minkowski space ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 7 2.1.2 Functional equation ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 10 2.2 Zeta function of global function field ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 12 2.2.1 Rational expression ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 15 2.2.2 Functional equation ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ 16 2.3 Dirichlet Lfunction ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ ꞏ -
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. -
Algebraic Hecke Characters Motivation
Algebraic Hecke Characters Motivation This motivation was inspired by the excellent article [Serre-Tate, x 7]. Our goal is to prove the main theorem of complex multiplication. The Galois theoretic formulation is as follows: Theorem (Main Theorem of Complex Multiplication). Let (A; α) be a CM abelian variety over a number field K ⊂ Q; and let (L; Φ) be its CM-type. (i) The reflex field E ⊂ Q is contained in K: × (ii) There exists a unique algebraic Hecke character : AK ! L such that for each `; the continuous homomorphism ab × φ` : Gal(K =K) ! L` −1 φ`(a) = (a)alg(a` ) ab is equal to the `-adic representation of Gal(K =K) on the `-adic Tate module V`(A): (iii) The algebraic part of is alg = NΦ ◦ NmK=E; where NΦ : ResE=QGm ! ResL=QGm denotes the reflex norm. (v) If v is a finite place of K where A has good reduction, say A/κv; then for any uniformizer πv of OKv ; we have (πv) = F rA;qv ; where #κv = qv: Let's meditate on this statement to understand the need for algebraic Hecke characters. We start with (v): Because A=K has CM by L; we have a diagram 0 redv 0 i : L,! End (A) −−! End (AL) where the second map is the reduction at v map. The second map is injective. Indeed, for any m prime to qv;A[m] is a finite etale group scheme over OKv ; for such group schemes, the reduction map is always isomorphism. Now F rA;qv commutes with every kv endomorphism of A: In particular, it centralizes the image of i in End0(A): But because A is CM, so in particular, [iL : Q] = 2 dim(A); we must have 1 0 F rA;qv 2 im(i): Let πv 2 L denote the unique preimage of F rA;qv : All Galois representations φ` on 0 V`(A) are completely encoded in the collection fπvg; by the Chebotarev density theorem. -
Birch and Swinnerton-Dyer Conjecture in the Complex Multiplication Case
BIRCH AND SWINNERTON-DYER CONJECTURE IN THE COMPLEX MULTIPLICATION CASE AND THE CONGRUENT NUMBER PROBLEM KAZUMA MORITA Abstract. For an elliptic curve E over K, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group E(K) is equal to the order of the zero of L(E/K ,s) at s = 1. In this paper, we shall give a proof for elliptic curves with complex multiplications. The key method of the proof is to reduce the Galois action of infinite order on the Tate module of an elliptic curve to that of finite order by using the p-adic Hodge theory. As a corollary, we can determine whether a given natural number is a congruent number (con- gruent number problem). This problem is one of the oldest unsolved problems in mathematics. 1. Introduction A natural number N is called a congruent number if it is the area of a rational right-angled triangle. To determine whether N is a congruent number is a difficult problem which can be traced back at least a millennium and is the oldest major unsolved problem in number theory. For example, it is easy to verify that 6 is a congruent number since the sides lengths (3, 4, 5) give the right-angled triangle whose area is 6. Furthermore, 5 and 7 are shown to be congruent numbers by Fibonacci and Euler. On the other hand, Fermat showed that the square numbers are never congruent numbers and, based on this experience, led to the famous Fermat’s conjecture. In modern language, a natural number N is a congruent number if and only if there exists a rational point (x, y) with y =6 0 on the elliptic arXiv:1803.11074v3 [math.NT] 30 May 2019 curve y2 = x3 − N 2x. -
L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field. -
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.