ZETAS 2018 : Zeta Functions, Polyzeta Functions, Arithmetical Series : Applications to Motives and Number Theory
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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 . -
In Search of the Riemann Zeros
In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes Michel L. Lapidus 2000 Mathematics Subject Classification. P rim a ry 1 1 A 4 1 , 1 1 G 20, 1 1 M 06 , 1 1 M 26 , 1 1 M 4 1 , 28 A 8 0, 3 7 N 20, 4 6 L 5 5 , 5 8 B 3 4 , 8 1 T 3 0. F o r a d d itio n a l in fo rm a tio n a n d u p d a tes o n th is b o o k , v isit www.ams.org/bookpages/mbk-51 Library of Congress Cataloging-in-Publication Data Lapidus, Michel L. (Michel Laurent), 1956 In search o f the R iem ann zero s : string s, fractal m em b ranes and no nco m m utativ e spacetim es / Michel L. Lapidus. p. cm . Includes b ib lio g raphical references. IS B N 97 8 -0 -8 2 18 -4 2 2 2 -5 (alk . paper) 1. R iem ann surfaces. 2 . F unctio ns, Z eta. 3 . S tring m o dels. 4 . N um b er theo ry. 5. F ractals. 6. S pace and tim e. 7 . G eo m etry. I. T itle. Q A 3 3 3 .L3 7 2 0 0 7 515.93 dc2 2 2 0 0 7 0 60 8 4 5 Cop ying and rep rinting. Indiv idual readers o f this pub licatio n, and no npro fi t lib raries acting fo r them , are perm itted to m ak e fair use o f the m aterial, such as to co py a chapter fo r use in teaching o r research. -
The Weil Conjectures for Curves
The Weil Conjectures for Curves Caleb Ji Summer 2021 1 Introduction We will explain Weil’s proof of his famous conjectures for curves. For the Riemann hypothe- sis, we will follow Grothendieck’s argument [1]. The main tools used in these proofs are basic results in algebraic geometry: Riemann-Roch, intersection theory on a surface, and the Hodge index theorem. For references: in Section 2 we used [3], while the rest can be found in Hartshorne [2] V.1 and Appendix C (some of it in the form of exercises). 1.1 Statements of the Weil conjectures We recall the statements. Let X be a smooth projective variety of dimension n over Fq. We define its zeta function by 1 ! X tr Z(X; t) := exp N ; r r r=1 where Nr is the number of closed points of X where considered over Fqr . Theorem 1.1 (Weil conjectures). Use the above notation. 1. (Rationality) Z(X; t) is a rational function of t. 2. (Functional equation) Let E be the Euler characteristic of X considered over C. Then 1 Z = ±qnE=2tEZ(t): qnt 3. (Riemann hypothesis) We can write P (t) ··· P (t) Z(t) = 1 2n−1 P0(t) ··· P2n(t) n where P0(t) = 1 − t; P2n(t) = 1 − q t and all the Pi(t) are integer polynomials that can be written as Y Pi(t) = (1 − αijt): j i=2 Finally, jαijj = q . 4. (Betti numbers) The degree of the polynomials Pi are the Betti numbers of X considered over C. 1 Caleb Ji The Weil Conjectures for Curves Summer 2021 d P1 r Note that dt log Z(X; t) = r=0 Nr+1t . -
Counting Points and Acquiring Flesh
Innovations in Incidence Geometry Volume 00 (XXXX), Pages 000–000 ISSN 1781-6475 Counting points and acquiring flesh Koen Thas Abstract This set of notes is based on a lecture I gave at “50 years of Finite Ge- ometry — A conference on the occasion of Jef Thas’s 70th birthday,” in November 2014. It consists essentially of three parts: in a first part, I in- troduce some ideas which are based in the combinatorial theory underlying F1, the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over F1 which was designed by Deitmar. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area. Keywords: Field with one element, Deitmar scheme, loose graph, zeta function, Weyl geometry MSC 2000: 11G25, 11D40, 14A15, 14G15 Contents 1 Introduction 2 2 Combinatorial theory 5 3 Deninger-Manin theory 8 4 Deitmar schemes 10 5 Acquiring flesh (1) 14 arXiv:1508.03997v1 [math.AG] 17 Aug 2015 6 Kurokawa theory 15 7 Graphs and zeta functions 18 2 Thas 8 Acquiring flesh (2) — The Weyl functor depicted 25 1 Introduction For a class of incidence geometries which are defined (for instance coordina- tized) over fields, it often makes sense to consider the “limit” of these geome- tries when the number of field elements tends to 1. As such, one ends up with a guise of a “field with one element, F1” through taking limits of geometries. A general reference for F1 is the recent monograph [21]. -
![Arxiv:1604.01256V4 [Math.NT] 14 Aug 2017 As P Tends to Infinity, We Are Happy to Ignore Such Primes, Which Are Necessarily finite in Number](https://docslib.b-cdn.net/cover/1914/arxiv-1604-01256v4-math-nt-14-aug-2017-as-p-tends-to-in-nity-we-are-happy-to-ignore-such-primes-which-are-necessarily-nite-in-number-521914.webp)
Arxiv:1604.01256V4 [Math.NT] 14 Aug 2017 As P Tends to Infinity, We Are Happy to Ignore Such Primes, Which Are Necessarily finite in Number
SATO-TATE DISTRIBUTIONS ANDREW V.SUTHERLAND ABSTRACT. In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures≤ presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry. 1. AN INTRODUCTION TO SATO-TATE DISTRIBUTIONS Before discussing the Sato-Tate conjecture and Sato-Tate distributions for abelian varieties, we first consider the more familiar setting of Artin motives, which can be viewed as varieties of dimension zero. 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d; for example, we may take 3 2 f (x) = x x + 1. Since f has integer coefficients, we can reduce them modulo any prime p to obtain − a polynomial fp with coefficients in the finite field Z=pZ Fp. For each prime p define ' Nf (p) := # x Fp : fp(x) = 0 , f 2 g which we note is an integer between 0 and d. We would like to understand how Nf (p) varies with p. 3 The table below shows the values of Nf (p) when f (x) = x x + 1 for primes p < 60: − p : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Nf (p) 00111011200101013 There does not appear to be any obvious pattern (and we should know not to expect one, the Galois group lurking behind the scenes is nonabelian). -
Decomposition Spaces, Incidence Algebras and M\" Obius Inversion I
DECOMPOSITION SPACES, INCIDENCE ALGEBRAS AND MOBIUS¨ INVERSION I: BASIC THEORY IMMA GALVEZ-CARRILLO,´ JOACHIM KOCK, AND ANDREW TONKS Abstract. This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and M¨obius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in ∆. Just as the Segal condition expresses composition, the new exactness condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1-simplices, and with coefficients in ∞-groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen S•-construction of an abelian (or stable infinity) category is shown to be a decomposition space. In the second paper in this series we impose further conditions on decomposition spaces, to obtain a general M¨obius inversion principle, and to ensure that the various constructions and results admit a homotopy cardinality. In the third paper we show that the Lawvere–Menni Hopf algebra of M¨obius intervals is the homotopy cardinality of a certain universal decomposition space. -
Combinatorics
09.10.74 Combinatorics Lectures of Mark Dukes Typing and layout by Líney Halla Kristinsdóttir University of Iceland Spring Term 2008 Contents 1 Enumeration1 1.1 Counting Arguments...............................2 1.2 Elementary Counting Coecients........................3 1.3 Compositions & Partitions of an Integer....................7 1.4 The Reection Principle.............................9 1.5 Stirling Numbers................................. 10 1.5.1 Stirling Numbers of the 2nd Kind: S(n; k) ............... 10 1.5.2 Signless Stirling Numbers of the 1st Kind............... 12 1.6 Multinomial Coecients............................. 14 1.7 Generating Functions I.............................. 15 1.8 Generating Functions II............................. 22 2 Permutations & Permutation Statistics 27 2.1 Descents & the Eulerian Polynomial...................... 27 2.2 Cycle structure & left-to-right maxima..................... 32 2.3 Excedances & weak excedances......................... 33 2.4 Inversions & the Major index.......................... 35 2.5 Multisets, Permutations & q-series....................... 39 2.6 Subspaces of a Vector Space.......................... 43 2.7 Permutations as Increasing Binary Trees.................... 45 2.8 Standard Young Tableaux............................ 46 2.9 Robinson-Schensted-Knuth (RSK) correspondence............... 48 i 3 Partially Ordered Sets (Posets) 53 3.1 Posets...................................... 53 3.2 Lattices..................................... 60 3.3 Modular Lattices................................ -
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. -
LTCC Enumerative Combinatorics 5 Posets and M¨Obius Inversion
LTCC Enumerative Combinatorics Notes 5 Alex Fink Fall 2015 5 Posets and Mobius¨ inversion Mobius¨ inversion can be viewed as a generalisation of the inclusion-exclusion principle with an apparatus to keep track of how the conditions intersect, as an apparatus to reduce the number of terms. The apparatus, that of partial orders, turns out to be of great combinatorial utility in its own right. 5.1 The inclusion-exclusion principle Often we are in the situation where we have a number of conditions on a set of combinatorial objects, and we have information about the number of objects which satisfy various combinations of these conditions (inclusion), while we want to count the objects satisfying none of the conditions (exclusion), or perhaps satis- fying some but not others. What is known as the sieve method is of general use in this situation: overcount the objects satisfying the conditions, and then make cor- rections and subtract off elements that have been multiply counted, and so forth. The sieve of Eratosthenes gave its name to the class (although, alone, it’s not es- pecially helpful for the enumeration of primes): the primes are the integers which satify none of the conditions of having the forms 2n;3n;5n;7n;::: for n ≥ 2. Let A1;:::;An be subsets of a finite set X. For any non-empty subset J of the index set [n], we put \ AJ = A j; j2J and take A/0 = X by convention. Theorem 5.1 (Inclusion-Exclusion Principle) The number of elements of X ly- ing in none of the sets Ai is equal to jJj ∑ (−1) jAJj: J⊆[n] Proof The expression in the theorem is a linear combination of the cardinalities of the sets AJ, and so we can calculate it by working out, for each x 2 X, the contribution of x to the sum. -
Arithmetic Zeta-Function
Arithmetic Zeta-Function Gaurish Korpal1 [email protected] Summer Internship Project Report 14th year Int. MSc. Student, National Institute of Science Education and Research, Jatni (Bhubaneswar, Odisha) Certificate Certified that the summer internship project report \Arithmetic Zeta-Function" is the bona fide work of \Gaurish Korpal", 4th year Int. MSc. student at National Institute of Science Ed- ucation and Research, Jatni (Bhubaneswar, Odisha), carried out under my supervision during June 4, 2018 to July 4, 2018. Place: Mumbai Date: July 4, 2018 Prof. C. S. Rajan Supervisor Professor, Tata Institute of Fundamental Research, Colaba, Mumbai 400005 Abstract We will give an outline of the motivation behind the Weil conjectures, and discuss their application for counting points on projective smooth curves over finite fields. Acknowledgements Foremost, I would like to express my sincere gratitude to my advisor Prof. C. S. Rajan for his motivation. I am also thankful to Sridhar Venkatesh1, Rahul Kanekar 2 and Monalisa Dutta3 for the enlightening discussions. Last but not the least, I would like to thank { Donald Knuth for TEX { Michael Spivak for AMS-TEX { Sebastian Rahtz for TEX Live { Leslie Lamport for LATEX { American Mathematical Society for AMS-LATEX { H`anTh^e´ Th`anhfor pdfTEX { Heiko Oberdiek for hyperref package { Steven B. Segletes for stackengine package { David Carlisle for graphicx package { Javier Bezos for enumitem package { Hideo Umeki for geometry package { Peter R. Wilson & Will Robertson for epigraph package { Jeremy Gibbons, Taco Hoekwater and Alan Jeffrey for stmaryrd package { Lars Madsen for mathtools package { Philipp Khl & Daniel Kirsch for Detexify (a tool for searching LATEX symbols) { TeX.StackExchange community for helping me out with LATEX related problems 1M.Sc. -
Elementary Proof of a Theorem of Hawkes, Isaacs And¨Ozaydin
Elementary Proof of a Theorem of Hawkes, Isaacs and Ozaydin¨ Matth´evan der Lee Abstract We present an elementary proof of the theorem of Hawkes, Isaacs and Ozaydin,¨ which states that Σ µG(H, K) ≡ 0 mod d, where µG denotes the M¨obius function for the subgroup lattice of a finite group G, H ranges over the conjugates of a given subgroup F of G with [G : F ] divisible by d, and K over the supergroups of the H for which [K : H] divides d. We apply the theorem to obtain a result on the number of solutions of |hH,gi| | n, for said H and a natural number n. The present version of the article includes an additional result on a quantity studied by K.S. Brown. Keywords: M¨obius function, arithmetic functions, subgroup lattice 2010 MSC: 05E15, 11A25, 20D30 1 Introduction The purpose of this note is to present a simple proof for the theorem of Hawkes, Isaacs and Ozaydin,¨ an important tool in the area of counting problems in finite groups. We refer to [3], where the result has appeared as Theorem 5.1, for background information regarding the subject. The exposition uses minimal group theory. The ingredients for the proof are Burnside’s Lemma, the incidence algebra of a finite partially ordered set (of which we use only the basic properties), and a feature of arithmetic functions: Corollary 3.4. Next, we derive a result on the number of solutions g of |hH,gi| | n, for a finite group G, a subgroup H ≤ G, and n | |G| (Theorem 6.1), and some results on a quantity studied by K.S.