The Riemann Hypothesis for Varieties Over Finite Fields
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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 . -
Etale and Crystalline Companions, I
ETALE´ AND CRYSTALLINE COMPANIONS, I KIRAN S. KEDLAYA Abstract. Let X be a smooth scheme over a finite field of characteristic p. Consider the coefficient objects of locally constant rank on X in ℓ-adic Weil cohomology: these are lisse Weil sheaves in ´etale cohomology when ℓ 6= p, and overconvergent F -isocrystals in rigid cohomology when ℓ = p. Using the Langlands correspondence for global function fields in both the ´etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has “companions” in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general X; building on work of Deligne, Drinfeld showed that any ´etale coefficient object has ´etale companions. We adapt Drinfeld’s method to show that any crystalline coefficient object has ´etale companions; this has been shown independently by Abe–Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of ´etale coefficient objects; this subject will be pursued in a subsequent paper. 0. Introduction 0.1. Coefficients, companions, and conjectures. Throughout this introduction (but not beyond; see §1.1), let k be a finite field of characteristic p and let X be a smooth scheme over k. When studying the cohomology of motives over X, one typically fixes a prime ℓ =6 p and considers ´etale cohomology with ℓ-adic coefficients; in this setting, the natural coefficient objects of locally constant rank are the lisse Weil Qℓ-sheaves. However, ´etale cohomology with p-adic coefficients behaves poorly in characteristic p; for ℓ = p, the correct choice for a Weil cohomology with ℓ-adic coefficients is Berthelot’s rigid cohomology, wherein the natural coefficient objects of locally constant rank are the overconvergent F -isocrystals. -
DENINGER COHOMOLOGY THEORIES Readers Who Know What the Standard Conjectures Are Should Skip to Section 0.6. 0.1. Schemes. We
DENINGER COHOMOLOGY THEORIES TAYLOR DUPUY Abstract. A brief explanation of Denninger's cohomological formalism which gives a conditional proof Riemann Hypothesis. These notes are based on a talk given in the University of New Mexico Geometry Seminar in Spring 2012. The notes are in the same spirit of Osserman and Ile's surveys of the Weil conjectures [Oss08] [Ile04]. Readers who know what the standard conjectures are should skip to section 0.6. 0.1. Schemes. We will use the following notation: CRing = Category of Commutative Rings with Unit; SchZ = Category of Schemes over Z; 2 Recall that there is a contravariant functor which assigns to every ring a space (scheme) CRing Sch A Spec A 2 Where Spec(A) = f primes ideals of A not including A where the closed sets are generated by the sets of the form V (f) = fP 2 Spec(A) : f(P) = 0g; f 2 A: By \f(P ) = 000 we means f ≡ 0 mod P . If X = Spec(A) we let jXj := closed points of X = maximal ideals of A i.e. x 2 jXj if and only if fxg = fxg. The overline here denote the closure of the set in the topology and a singleton in Spec(A) being closed is equivalent to x being a maximal ideal. 1 Another word for a closed point is a geometric point. If a point is not closed it is called generic, and the set of generic points are in one-to-one correspondence with closed subspaces where the associated closed subspace associated to a generic point x is fxg. -
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. -
THE WEIL CONJECTURES I Arxiv:1807.10810V2 [Math.AG] 26
THE WEIL CONJECTURES I By Pierre Deligne (translated by Evgeny Goncharov) [email protected],∗ [email protected] I attempted to write the full translation of this article to make the remarkable proof of Pierre Deligne available to a greater number of people. Overviews of the proofs can be found elsewhere. I especially recommend the notes of James Milne on Etale Cohomology that also contain a justifica- tion for the theory underlying this article and proofs of the results used by Deligne. The footnotes are mostly claims that some details appear in Milne, clarifications of some of the terminology or my personal struggles. I have also made a thorough overview of the proof together with more detailed explanations - https: // arxiv. org/ abs/ 1807. 10812 . Enjoy! Abstract In this article I prove the Weil conjecture about the eigenvalues of Frobenius endomor- phisms. The precise formulation is given in (1.6). I tried to make the demonstration as geometric and elementary as possible and included reminders: only the results of paragraphs 3, 6, 7 and 8 are original. In the article following this one I will give various refinements of the intermediate results and the applications, including the hard Lefschetz theorem (on the iterated cup products by the cohomology class of a hyperplane section). The text faithfully follows from the six lectures given at Cambridge in July 1973. I thank N.Katz for allowing me to use his notes. Contents 1 The theory of Grothendieck: a cohomological interpretation of L-functions 2 2 The theory of Grothendieck: Poincare duality 7 arXiv:1807.10810v2 [math.AG] 26 Jan 2019 3 The main lemma (La majoration fondamentale) 10 4 Lefschetz theory: local theory 13 5 Lefschetz theory: global theory 15 6 The rationality theorem 20 7 Completion of the proof of (1.7) 23 8 First applications 26 ∗Cambridge University, Center for Mathematical Sciences, Wilberforce Road, Cambridge, UK yNational Research University Higher School of Economics (NRU HSE), Usacheva 6, Moscow, Russia. -
The Role of the Ramanujan Conjecture in Analytic Number Theory
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 267–320 S 0273-0979(2013)01404-6 Article electronically published on January 14, 2013 THE ROLE OF THE RAMANUJAN CONJECTURE IN ANALYTIC NUMBER THEORY VALENTIN BLOMER AND FARRELL BRUMLEY Dedicated to the 125th birthday of Srinivasa Ramanujan Abstract. We discuss progress towards the Ramanujan conjecture for the group GLn and its relation to various other topics in analytic number theory. Contents 1. Introduction 267 2. Background on Maaß forms 270 3. The Ramanujan conjecture for Maaß forms 276 4. The Ramanujan conjecture for GLn 283 5. Numerical improvements towards the Ramanujan conjecture and applications 290 6. L-functions 294 7. Techniques over Q 298 8. Techniques over number fields 302 9. Perspectives 305 J.-P. Serre’s 1981 letter to J.-M. Deshouillers 307 Acknowledgments 313 About the authors 313 References 313 1. Introduction In a remarkable article [111], published in 1916, Ramanujan considered the func- tion ∞ ∞ Δ(z)=(2π)12e2πiz (1 − e2πinz)24 =(2π)12 τ(n)e2πinz, n=1 n=1 where z ∈ H = {z ∈ C |z>0} is in the upper half-plane. The right hand side is understood as a definition for the arithmetic function τ(n) that nowadays bears Received by the editors June 8, 2012. 2010 Mathematics Subject Classification. Primary 11F70. Key words and phrases. Ramanujan conjecture, L-functions, number fields, non-vanishing, functoriality. The first author was supported by the Volkswagen Foundation and a Starting Grant of the European Research Council. The second author is partially supported by the ANR grant ArShiFo ANR-BLANC-114-2010 and by the Advanced Research Grant 228304 from the European Research Council. -
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. -
Hasse-Weil Zeta-Function in a Special Case
Universit´eParis-Sud 11 Universiteit Leiden Master Erasmus Mundus ALGANT HASSE-WEIL ZETA-FUNCTION IN A SPECIAL CASE Weidong ZHUANG Advisor: Professor M. Harris 1 HASSE-WEIL ZETA-FUNCTION IN A SPECIAL CASE WEIDONG ZHUANG I sincerely thank Prof. M. Harris for guiding me to this fantastic topic. I wish to thank all those people who have helped me when I am studying for the thesis. I also appreciate ERASMUS MUNDUS ALGANT, especially Universit´eParis-Sud 11 and Universiteit Leiden for all conveniences afforded. 2 Contents 1. Introduction 3 2. Preliminaries 5 2.1. Elliptic curves 5 2.2. Moduli space with level structure in good reduction 5 2.3. Moduli space with level structure in bad reduction 6 3. Basics of representations 7 3.1. Representations of GL2 7 3.2. Weil group 10 3.3. The Bernstein center 11 4. Harmonic analysis 12 4.1. Basics of integration 12 4.2. Character of representations 13 4.3. Selberg trace formula 14 5. Advanced tools 15 5.1. Crystalline cohomology 15 5.2. Nearby cycles 17 5.3. Base change 18 5.4. Counting points over finite fields 21 6. The semi-simple trace and semi-simple local factor 22 6.1. Basics of the semi-simple trace and semi-simple local factor 23 6.2. Nearby cycles 24 6.3. The semi-simple trace of Frobenius as a twisted orbital integral 27 6.4. Lefschetz number 29 7. The Hasse-Weil zeta-function 31 7.1. Contributions from the boundary 31 7.2. The Arthur-Selberg Trace formula 32 7.3. -
Motives and Motivic L-Functions
Motives and Motivic L-functions Christopher Elliott December 6, 2010 1 Introduction This report aims to be an exposition of the theory of L-functions from the motivic point of view. The classical theory of pure motives provides a category consisting of `universal cohomology theories' for smooth projective varieties defined over { for instance { number fields. Attached to every motive we can define a function which is holomorphic on a subdomain of C which at least conjecturally satisfies similar properties to the Riemann zeta function: for instance meromorphic continuation and functional equation. The properties of this function are expected to encode deep arithmetic properties of the underlying variety. In particular, we will discuss a conjecture of Deligne and its more general forms due to Beilinson and others, that relates certain \special" values of the L-function { values at integer points { to a subtle arithmetic invariant called the regulator, vastly generalising the analytic class number formula. Finally I aim to describe potentially fruitful ways in which these exciting conjectures might be rephrased or reinterpreted, at least to make things clearer to me. There are several sources from which I have drawn ideas and material. In particular, the book [1] is a very good exposition of the theory of pure motives. For the material on L-functions, and in particular the Deligne and Beilinson conjectures, I referred to the surveys [9], [11] and [4] heavily. In particular, my approach to the Deligne conjecture follows the survey [11] of Schneider closely. Finally, when I motivate the definition of the motivic L-function I use the ideas and motivation in the introduction to the note [5] of Kim heavily. -
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. -
Deligne's Proof of the Weil-Conjecture
Deligne's Proof of the Weil-conjecture Prof. Dr. Uwe Jannsen Winter Term 2015/16 Inhaltsverzeichnis 0 Introduction 3 1 Rationality of the zeta function 9 2 Constructible sheaves 14 3 Constructible Z`-sheaves 21 4 Cohomology with compact support 24 5 The Frobenius-endomorphism 27 6 Deligne's theorem: Formulation and first reductions 32 7 Weights and determinant weights 36 8 Cohomology of curves and L-series 42 9 Purity of real Q`-sheaves 44 10 The formalism of nearby cycles and vanishing cycles 46 11 Cohomology of affine and projective spaces, and the purity theorem 55 12 Local Lefschetz theory 59 13 Proof of Deligne's theorem 74 14 Existence and global properties of Lefschetz pencils 86 0 Introduction The Riemann zeta-function is defined by the sum and product X 1 Y 1 ζ(s) = = (s 2 C) ns 1 − p−s n≥1 p which converge for Re(s) > 1. The expression as a product, where p runs over the rational prime numbers, is generally attributed to Euler, and is therefore known as Euler product formula with terms the Euler factors. Formally the last equation is easily achieved by the unique decomposition of natural numbers as a product of prime numbers and by the geometric series expansion 1 1 X = p−ms : 1 − p−s m=0 The { to this day unproved { Riemann hypothesis states that all non-trivial zeros of ζ(s) 1 should lie on the line Re(s) = 2 . This is more generally conjectured for the Dedekind zeta functions X 1 Y 1 ζ (s) = = : K Nas 1 − Np−s a⊂OK p Here K is a number field, i.e., a finite extension of Q, a runs through the ideals =6 0 of the ring OK of the integers of K, p runs through the prime ideals =6 0, and Na = jOK =aj, where jMj denotes the cardinality numbers of a finite set M. -
The Weil Conjectures for Elliptic Curves
The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures The Weil Conjectures for Elliptic Curves Zeta function Statement Counting points over finite fields Main lemma Gaurish Korpal An example The University of Arizona, Tucson December 10, 2020 Outline The Weil Conjectures for Elliptic Curves Gaurish Korpal 1 Elliptic curves Elliptic curves Definition Definition Group structure Group structure Weil conjectures Zeta function Statement Counting points 2 Weil conjectures over finite fields Zeta function Main lemma An example Statement 3 Counting points over finite fields Main lemma An example The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures Zeta function Statement Counting points over finite fields Elliptic curves Main lemma An example Definition The Weil Conjectures for Elliptic Curves Since we won't be discussing any proofs in this seminar, it will be Gaurish Korpal sufficient to restrict ourselves to the following definition of elliptic Elliptic curves curves: Definition Group structure Elliptic curve Weil conjectures Zeta function An elliptic curve over a field K is a nonsingular projective plane curve Statement over K given by an affine equation of the form Counting points over finite fields 2 3 2 Main lemma y + a1xy + a3y = x + a2x + a4x + a6 An example with an extra point O = [0 : 1 : 0] at infinity. Therefore, in homegenous coordinates the equation will be: 2 2 3 2 2 3 Y Z + a1XYZ + a3YZ = X + a2X Z + a4XZ + a6Z where x = X =Z