The Analytic Class Number Formula for Orders in Products of Number Fields
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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]. -
An Introduction to Iwasawa Theory
An Introduction to Iwasawa Theory Notes by Jim L. Brown 2 Contents 1 Introduction 5 2 Background Material 9 2.1 AlgebraicNumberTheory . 9 2.2 CyclotomicFields........................... 11 2.3 InfiniteGaloisTheory . 16 2.4 ClassFieldTheory .......................... 18 2.4.1 GlobalClassFieldTheory(ideals) . 18 2.4.2 LocalClassFieldTheory . 21 2.4.3 GlobalClassFieldTheory(ideles) . 22 3 SomeResultsontheSizesofClassGroups 25 3.1 Characters............................... 25 3.2 L-functionsandClassNumbers . 29 3.3 p-adicL-functions .......................... 31 3.4 p-adic L-functionsandClassNumbers . 34 3.5 Herbrand’sTheorem . .. .. .. .. .. .. .. .. .. .. 36 4 Zp-extensions 41 4.1 Introduction.............................. 41 4.2 PowerSeriesRings .......................... 42 4.3 A Structure Theorem on ΛO-modules ............... 45 4.4 ProofofIwasawa’stheorem . 48 5 The Iwasawa Main Conjecture 61 5.1 Introduction.............................. 61 5.2 TheMainConjectureandClassGroups . 65 5.3 ClassicalModularForms. 68 5.4 ConversetoHerbrand’sTheorem . 76 5.5 Λ-adicModularForms . 81 5.6 ProofoftheMainConjecture(outline) . 85 3 4 CONTENTS Chapter 1 Introduction These notes are the course notes from a topics course in Iwasawa theory taught at the Ohio State University autumn term of 2006. They are an amalgamation of results found elsewhere with the main two sources being [Wash] and [Skinner]. The early chapters are taken virtually directly from [Wash] with my contribution being the choice of ordering as well as adding details to some arguments. Any mistakes in the notes are mine. There are undoubtably type-o’s (and possibly mathematical errors), please send any corrections to [email protected]. As these are course notes, several proofs are omitted and left for the reader to read on his/her own time. -
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). -
Automorphic Functions and Fermat's Last Theorem(3)
Jiang and Wiles Proofs on Fermat Last Theorem (3) Abstract D.Zagier(1984) and K.Inkeri(1990) said[7]:Jiang mathematics is true,but Jiang determinates the irrational numbers to be very difficult for prime exponent p.In 1991 Jiang studies the composite exponents n=15,21,33,…,3p and proves Fermat last theorem for prime exponenet p>3[1].In 1986 Gerhard Frey places Fermat last theorem at the elliptic curve that is Frey curve.Andrew Wiles studies Frey curve. In 1994 Wiles proves Fermat last theorem[9,10]. Conclusion:Jiang proof(1991) is direct and simple,but Wiles proof(1994) is indirect and complex.If China mathematicians had supported and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang had proved Fermat last theorem.Wiles has received many prizes and awards,he should thank China mathematicians. - Automorphic Functions And Fermat’s Last Theorem(3) (Fermat’s Proof of FLT) 1 Chun-Xuan Jiang P. O. Box 3924, Beijing 100854, P. R. China [email protected] Abstract In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means: xyznnnn+ =>(2) has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). -
Introduction to L-Functions: Dedekind Zeta Functions
Introduction to L-functions: Dedekind zeta functions Paul Voutier CIMPA-ICTP Research School, Nesin Mathematics Village June 2017 Dedekind zeta function Definition Let K be a number field. We define for Re(s) > 1 the Dedekind zeta function ζK (s) of K by the formula X −s ζK (s) = NK=Q(a) ; a where the sum is over all non-zero integral ideals, a, of OK . Euler product exists: Y −s −1 ζK (s) = 1 − NK=Q(p) ; p where the product extends over all prime ideals, p, of OK . Re(s) > 1 Proposition For any s = σ + it 2 C with σ > 1, ζK (s) converges absolutely. Proof: −n Y −s −1 Y 1 jζ (s)j = 1 − N (p) ≤ 1 − = ζ(σ)n; K K=Q pσ p p since there are at most n = [K : Q] many primes p lying above each rational prime p and NK=Q(p) ≥ p. A reminder of some algebraic number theory If [K : Q] = n, we have n embeddings of K into C. r1 embeddings into R and 2r2 embeddings into C, where n = r1 + 2r2. We will label these σ1; : : : ; σr1 ; σr1+1; σr1+1; : : : ; σr1+r2 ; σr1+r2 . If α1; : : : ; αn is a basis of OK , then 2 dK = (det (σi (αj ))) : Units in OK form a finitely-generated group of rank r = r1 + r2 − 1. Let u1;:::; ur be a set of generators. For any embedding σi , set Ni = 1 if it is real, and Ni = 2 if it is complex. Then RK = det (Ni log jσi (uj )j)1≤i;j≤r : wK is the number of roots of unity contained in 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. -
And Siegel's Theorem We Used Positivi
Math 259: Introduction to Analytic Number Theory A nearly zero-free region for L(s; χ), and Siegel's theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s; χ). Better zero-free regions can be obtained with some more effort by working with the L(s; χ) individually. The situation is most satisfactory for 2 complex χ, that is, for characters with χ = χ0. (Recall that real χ were also the characters that gave us the most difficulty6 in the proof of L(1; χ) = 0; it is again in the neighborhood of s = 1 that it is hard to find a good zero-free6 region for the L-function of a real character.) To obtain the zero-free region for ζ(s), we started with the expansion of the logarithmic derivative ζ 1 0 (s) = Λ(n)n s (σ > 1) ζ X − − n=1 and applied the inequality 1 0 (z + 2 +z ¯)2 = Re(z2 + 4z + 3) = 3 + 4 cos θ + cos 2θ (z = eiθ) ≤ 2 it iθ s to the phases z = n− = e of the terms n− . To apply the same inequality to L 1 0 (s; χ) = χ(n)Λ(n)n s; L X − − n=1 it it we must use z = χ(n)n− instead of n− , obtaining it 2 2it 0 Re 3 + 4χ(n)n− + χ (n)n− : ≤ σ Multiplying by Λ(n)n− and summing over n yields L0 L0 L0 0 3 (σ; χ ) + 4 Re (σ + it; χ) + Re (σ + 2it; χ2) : (1) ≤ − L 0 − L − L 2 Now we see why the case χ = χ0 will give us trouble near s = 1: for such χ the last term in (1) is within O(1) of Re( (ζ0/ζ)(σ + 2it)), so the pole of (ζ0/ζ)(s) at s = 1 will undo us for small t . -
Arxiv:1802.06193V3 [Math.NT] 2 May 2018 Etr ...I Hsppr Efi H Udmna Discrimi Fundamental the fix Fr We Quadratic Paper, Binary I This a Inverse by in the Squares 1.4.]
THE LEAST PRIME IDEAL IN A GIVEN IDEAL CLASS NASER T. SARDARI Abstract. Let K be a number field with the discriminant DK and the class number hK , which has bounded degree over Q. By assuming GRH, we prove that every ideal class of K contains a prime ideal with norm less 2 2 than hK log(DK ) and also all but o(hK ) of them have a prime ideal with 2+ǫ norm less than hK log(DK ) . For imaginary quadratic fields K = Q(√D), by assuming Conjecture 1.2 (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of log(D) from our bounds and show that these bounds are optimal. Contents 1. Introduction 1 1.1. Motivation 1 1.2. Repulsion of the prime ideals near the cusp 5 1.3. Method and Statement of results 6 Acknowledgements 7 2. Hecke L-functions 7 2.1. Functional equation 7 2.2. Weil’s explicit formula 8 3. Proof of Theorem 1.8 8 4. Proof of Theorem 1.1 11 References 12 1. Introduction 1.1. Motivation. One application of our main result addresses the optimal upper arXiv:1802.06193v3 [math.NT] 2 May 2018 bound on the least prime number represented by a binary quadratic form up to a power of the logarithm of its discriminant. Giving sharp upper bound of this form on the least prime or the least integer representable by a sum of two squares is crucial in the analysis of the complexity of some algorithms in quantum compiling. -
Class Number Formula for Dihedral Extensions 2
CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS LUCA CAPUTO AND FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO Abstract. We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory. 1. Introduction Let k be a number field and let M/k be a finite Galois extension. For a subgroup H Gal(M/k), let χH denote the rational character of the permutation representation defined by the Gal(M/k)-set⊆ Gal(M/k)/H. If M/k is non-cyclic, then there exists a relation n χ =0 H · H XH where nH Z are not all 0 and H runs through the subgroups of Gal(M/k). Using the formalism of Artin L-functions∈ and the formula for the residue at 1 of the Dedekind zeta-function, Brauer showed in [Bra51] that the above relation translates into the following formula nH H nH wM (1.1) (hM H ) = RM H YH YH where, for a number field E, hE, RE, wE denote the class number, the regulator and the order of the group of roots of unity of E, respectively. -
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.