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On Functional Equations of Euler Systems
ON FUNCTIONAL EQUATIONS OF EULER SYSTEMS DAVID BURNS AND TAKAMICHI SANO Abstract. We establish precise relations between Euler systems that are respectively associated to a p-adic representation T and to its Kummer dual T ∗(1). Upon appropriate specialization of this general result, we are able to deduce the existence of an Euler system of rank [K : Q] over a totally real field K that both interpolates the values of the Dedekind zeta function of K at all positive even integers and also determines all higher Fitting ideals of the Selmer groups of Gm over abelian extensions of K. This construction in turn motivates the formulation of a precise conjectural generalization of the Coleman-Ihara formula and we provide supporting evidence for this conjecture. Contents 1. Introduction 1 1.1. Background and results 1 1.2. General notation and convention 4 2. Coleman maps and local Tamagawa numbers 6 2.1. The local Tamagawa number conjecture 6 2.2. Coleman maps 8 2.3. The proof of Theorem 2.3 10 3. Generalized Stark elements and Tamagawa numbers 13 3.1. Period-regulator isomorphisms 13 3.2. Generalized Stark elements 16 3.3. Deligne-Ribet p-adic L-functions 17 3.4. Evidence for the Tamagawa number conjecture 18 arXiv:2003.02153v1 [math.NT] 4 Mar 2020 4. The functional equation of vertical determinantal systems 18 4.1. Local vertical determinantal sytems 19 4.2. The functional equation 20 5. Construction of a higher rank Euler system 24 5.1. The Euler system 25 5.2. The proof of Theorem 5.2 26 6. -
Euler Systems and Refined Conjectures of Birch Swinnerton
Contemporary Mathematics Volume 00, 0000 Euler Systems and Refined Conjectures of Birch Swinnerton-Dyer Type HENRI DARMON Abstract. The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their L- functions poses many deep and difficult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain refinements that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic fields involving Heegner points, and explain how the fundamental work of V.A. Kolyvagin sheds light on these new conjectures. 1 Preliminaries. x The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their L-functions poses many deep and subtle questions. The theme of this paper is the Birch Swinnerton-Dyer conjecture, which concerns the case where the arithmetic object in question is an elliptic curve defined over a global field. Let E be an elliptic curve defined over the rational numbers. The conjecture of Shimura-Taniyama-Weil asserts that E is modular, i.e., is equipped with a rational map ' : X (N) E; 0 −! where X0(N) is the modular curve of level N, defined over Q, which parameter- izes elliptic curves with a distinguished cyclic N-isogeny. We assume that E has this property. (For a specific E this can be checked by a finite computation.) 1991 Mathematics Subject Classification. Primary 11G40; Secondary 11G05. The ideas for this paper are part of the author's Harvard PhD thesis; he gratefully ac- knowledges the support of Harvard University, and in particular of his advisor, B.H. -
Anticyclotomic Euler Systems for Unitary Groups
ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS ANDREW GRAHAM AND SYED WAQAR ALI SHAH Abstract. Let n ≥ 1 be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature (1; 2n − 1). Contents 1. Introduction 1 2. Preliminaries 7 3. Cohomology of unitary Shimura varieties 13 4. Horizontal norm relations 15 5. Interlude: cohomology functors 25 6. Vertical norm relations at split primes 31 7. The anticyclotomic Euler system 41 Appendix A. Continuous ´etalecohomology 49 Appendix B. Shimura{Deligne varieties 55 Appendix C. Ancona's construction for Shimura{Deligne varieties 60 References 64 1. Introduction In [Kol89], Kolyvagin constructs an anticyclotomic Euler system for (modular) elliptic curves over Q which satisfy the so-called \Heegner hypothesis". One can view the classes in this construction as images under the modular parameterisation of certain divisors on modular curves arising from the embedding of reductive groups (1.1) ResE=Q Gm ,! GL2 where E is an imaginary quadratic number field. These divisors (Heegner points) are defined over ring class fields of E and satisfy trace compatibility relations as one varies the field of definition. In particular, Kolyvagin shows that if the bottom Heegner point is non-torsion, then the group of E-rational points has rank equal to one. Combining this so-called \Kolyvagin step" with the Gross{Zagier formula that relates the height of this point to the derivative of the L-function at the central value, one obtains instances of the Birch{ Swinnerton-Dyer conjecture in the analytic rank one case. The above construction has been generalised to higher weight modular forms and to situations where a more general hypothesis is placed on the modular form (see [Sch86], [Nek92], [BDP13] and [Zha01]). -
Arxiv:1506.05408V1 [Math.AT]
NOVIKOV’S CONJECTURE JONATHAN ROSENBERG Abstract. We describe Novikov’s “higher signature conjecture,” which dates back to the late 1960’s, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in high-dimensional manifold topology, but more importantly, vari- ants and analogues permeate many other areas of mathematics, from geometry to operator algebras to representation theory. 1. Origins of the Original Conjecture The Novikov Conjecture is perhaps the most important unsolved problem in the topology of high-dimensional manifolds. It was first stated by Sergei Novikov, in various forms, in his lectures at the International Congresses of Mathematicians in Moscow in 1966 and in Nice in 1970, and in a few other papers [84, 87, 86, 85]. For an annotated version of the original formulation, in both Russian and English, we refer the reader to [37]. Here we will try instead to put the problem in context and explain why it might be of interest to the average mathematician. For a nice book- length exposition of this subject, we recommend [65]. Many treatments of various aspects of the problem can also be found in the many papers in the collections [38, 39]. For the typical mathematician, the most important topological spaces are smooth manifolds, which were introduced by Riemann in the 1850’s. However, it took about 100 years for the tools for classifying manifolds (except in dimension 1, which is trivial, and dimension 2, which is relatively easy) to be developed. The problem is that manifolds have no local invariants (except for the dimension); all manifolds of the same dimension look the same locally. -
ON the TRIANGULATION of MANIFOLDS and the HAUPTVERMUTUNG 1. the First Author's Solution of the Stable Homeomorphism Con- Jecture
ON THE TRIANGULATION OF MANIFOLDS AND THE HAUPTVERMUTUNG BY R. C. KIRBY1 AND L. C. SIEBENMANN2 Communicated by William Browder, December 26, 1968 1. The first author's solution of the stable homeomorphism con jecture [5] leads naturally to a new method for deciding whether or not every topological manifold of high dimension supports a piecewise linear manifold structure (triangulation problem) that is essentially unique (Hauptvermutung) cf. Sullivan [14]. At this time a single obstacle remains3—namely to decide whether the homotopy group 7T3(TOP/PL) is 0 or Z2. The positive results we obtain in spite of this obstacle are, in brief, these four: any (metrizable) topological mani fold M of dimension ^ 6 is triangulable, i.e. homeomorphic to a piece- wise linear ( = PL) manifold, provided H*(M; Z2)=0; a homeo morphism h: MI—ÏMÎ of PL manifolds of dimension ^6 is isotopic 3 to a PL homeomorphism provided H (M; Z2) =0; any compact topo logical manifold has the homotopy type of a finite complex (with no proviso) ; any (topological) homeomorphism of compact PL manifolds is a simple homotopy equivalence (again with no proviso). R. Lashof and M. Rothenberg have proved some of the results of this paper, [9] and [l0]. Our work is independent of [l0]; on the other hand, Lashofs paper [9] was helpful to us in that it showed the relevance of Lees' immersion theorem [ll] to our work and rein forced our suspicions that the Classification theorem below was correct. We have divided our main result into a Classification theorem and a Structure theorem. -
Die Hauptvermutung Der Kombinatorischen Topologie
Abel Prize Laureate 2011 John Willard Milnor Die Hauptvermutung der kombinatorischen Topolo- gie (Steiniz, Tietze; 1908) Die Hauptvermutung (The main Conjecture) of combinatorial topology (now: algebraic topology) was published in 1908 by the German mathematician Ernst Steiniz and the Austrian mathematician Heinrich Tietze. The conjecture states that given two trian- gulations of the same space, there is always possible to find a common refinement. The conjecture was proved in dimension 2 by Tibor Radó in the 1920s and in dimension 3 by Edwin E. Moise in the 1950s. The conjecture was disproved in dimension greater or equal to 6 by John Milnor in 1961. Triangulation by points. Next you chose a bunch of connecting “Norges Geografiske Oppmåling” (NGO) was lines, edges, between the reference points in order established in 1773 by the military officer Hein- to obtain a tiangular web. The choices of refer- rich Wilhelm von Huth with the purpose of meas- ence points and edges are done in order to ob- uring Norway in order to draw precise and use- tain triangles where the curvature of the interior ful maps. Six years later they started the rather landscape is neglectable. Thus, if the landscape is elaborate triangulation task. When triangulating hilly, the reference points have to be chosen rath- a piece of land you have to pick reference points er dense, while flat farmland doesn´t need many and compute their coordinates relative to near- points. In this way it is possible to give a rather accurate description of the whole landscape. The recipe of triangulation can be used for arbitrary surfaces. -
Present Trends in Pure Mathematics
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 27, 235-255 (1978) Present Trends in Pure Mathematics JEAN DIEUDONNB Villa Nancago, Corniche Fleurie, 06200 Nice, France Over 1500 papers in pure mathematics are now reviewed every month in Mathematical Reviews; is it possible to discern any trends at all in such an explosive development? This is the question which I will try to answer in the following pages; it is clear that such an answer can only be a highly subjective one. If most problems and theorems of mathematics can be compared to individual atoms, trends may only appear when these atoms are connected in somemanner. A comparison with the way astrophysicists describe the evolution of stars may be helpful here : They tell us that in the beginning lumps of diffuse matter concentrate under the influence of gravitation until their density is high enough for the nuclear fuel to ignite; the star is then born, and enters the “Main se- quence, ” where it remains for a variable period of time in a fairly steady state, until it has exhausted its regular nuclear fuel, and has to leave the “Main sequence” for various fates. In mathematics, we may compare the birth of a star to the coalescence of several problems into some general method, or to their organization under an axiomatic theory; the method (or theory) then belongs to what we may call the “Main stream” of mathematics, which can be characterized by the many connections between its components and the many influences they exert on one another. -
TRIANGULATIONS of MANIFOLDS in Topology, a Basic Building Block for Spaces Is the N-Simplex. a 0-Simplex Is a Point, a 1-Simplex
TRIANGULATIONS OF MANIFOLDS CIPRIAN MANOLESCU In topology, a basic building block for spaces is the n-simplex. A 0-simplex is a point, a 1-simplex is a closed interval, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. In general, an n-simplex is the convex hull of n + 1 vertices in n-dimensional space. One constructs more complicated spaces by gluing together several simplices along their faces, and a space constructed in this fashion is called a simplicial complex. For example, the surface of a cube can be built out of twelve triangles|two for each face, as in the following picture: Apart from simplicial complexes, manifolds form another fundamental class of spaces studied in topology. An n-dimensional topological manifold is a space that looks locally like the n-dimensional Euclidean space; i.e., such that it can be covered by open sets (charts) n homeomorphic to R . Furthermore, for the purposes of this note, we will only consider manifolds that are second countable and Hausdorff, as topological spaces. One can consider topological manifolds with additional structure: (i)A smooth manifold is a topological manifold equipped with a (maximal) open cover by charts such that the transition maps between charts are smooth (C1); (ii)A Ck manifold is similar to the above, but requiring that the transition maps are only Ck, for 0 ≤ k < 1. In particular, C0 manifolds are the same as topological manifolds. For k ≥ 1, it can be shown that every Ck manifold has a unique compatible C1 structure. Thus, for k ≥ 1 the study of Ck manifolds reduces to that of smooth manifolds. -
AN INTRODUCTION to P-ADIC L-FUNCTIONS
AN INTRODUCTION TO p-ADIC L-FUNCTIONS by Joaquín Rodrigues Jacinto & Chris Williams Contents General overview......................................................................... 2 1. Introduction.............................................................................. 2 1.1. Motivation. 2 1.2. The Riemann zeta function. 5 1.3. p-adic L-functions. 6 1.4. Structure of the course.............................................................. 11 Part I: The Kubota–Leopoldt p-adic L-function.................................... 12 2. Measures and Iwasawa algebras.......................................................... 12 2.1. The Iwasawa algebra................................................................ 12 2.2. p-adic analysis and Mahler transforms............................................. 14 2.3. An example: Dirac measures....................................................... 15 2.4. A measure-theoretic toolbox........................................................ 15 3. The Kubota-Leopoldt p-adic L-function. 17 3.1. The measure µa ..................................................................... 17 3.2. Restriction to Zp⇥ ................................................................... 18 3.3. Pseudo-measures.................................................................... 19 4. Interpolation at Dirichlet characters..................................................... 21 4.1. Characters of p-power conductor. 21 4.2. Non-trivial tame conductors........................................................ 23 4.3. Analytic -
L-FUNCTIONS and CYCLOTOMIC UNITS 1. Introduction
L-FUNCTIONS AND CYCLOTOMIC UNITS TOM WESTON, UNIVERSITY OF MICHIGAN 1. Introduction Let K be a number field with r1 real embeddings and r2 pairs of complex con- jugate embeddings. If ζK (s) is the Dedekind zeta function of K, then ζK (s) has a zero of order r1 + r2 1 at s = 0, and the value of the first non-zero derivative is given by the Dirichlet− class number formula: (r1+r2 1) hK RK ζK − (0) = : − wK Here RK is the regulator of K, wK is the number of roots of unity in K and hK is the class number of K. This formula is a striking connection between arithmetic and analysis, and there have been many attempts to generalize it to other L-functions: one expects that the value of a \motivic" L-function at an integer point should involve a transcendental factor, a boring rational factor, and an interesting rational factor. In the case of the Dedekind zeta function, these roles are played by RK , wK , and hK , respectively. In the general case one expects that the interesting rational factor is the order of a certain Selmer group. 2. Selmer groups and Kolyvagin systems Let us recall the definition of the Selmer group of a p-adic Galois representation. Let T be a free Zp-module with a continuous action of the absolute Galois group ¯ GK = Gal(K=K). We define the Cartier dual of T by T ∗ = HomZp (T; µp1 ) with the adjoint Galois action. Let be a Selmer structure on T . Recall that consists of choices of local Selmer structuresF F 1 1 H (Kv;T ) H (Kv;T ) F ⊆ for each place v of K; these are assumed to coincide with the unramified choice at almost all places v. -
Kato's Euler System and Rational Points on Elliptic Curves I: a P-Adic
KATO'S EULER SYSTEM AND RATIONAL POINTS ON ELLIPTIC CURVES I: A p-ADIC BEILINSON FORMULA MASSIMO BERTOLINI AND HENRI DARMON Abstract. This article is the ¯rst in a series devoted to Kato's Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ¸ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic ¶etalecohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato's explicit reciprocity law. Contents 1. Introduction 1 2. Complex L-series 2 2.1. Eisenstein series 3 2.2. Rankin's method 4 2.3. A factorisation of critical values 6 2.4. Beilinson's formula 7 2.5. Algebraicity 8 3. p-adic L-functions 9 3.1. Rankin's p-adic L-functions 9 3.2. Factorisation of p-adic L-functions 10 4. p-adic regulators 11 4.1. The regulator on K2 12 4.2. The general case 14 5. The p-adic Beilinson formula 15 5.1. The main results 15 5.2. Relation with the work of Brunault and Gealy 15 References 17 1. Introduction This article is the ¯rst in a series devoted to Kato's Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. -
Lecture Notes in Triangulation Conjecture
Lecture Notes In Triangulation Conjecture Is Win always misleading and nary when unify some tilery very guilefully and anywise? Hagan reproduces her fulness grossly, she formatted it transcriptionally. Spectrological and directive Vilhelm tackles some philosophers so especially! Your experience on this approach compares to lecture notes in the same work with the dual graded graphs, the appendices to get the book If request that you an abelian varieties. It may be equivalent to delete all selected works you are, triangulation conjecture is also be solved exactly in. Proceedings of lectures we give an understanding and conjecture. We are interested in the set date all subdivisions or triangulations of sympathy given polytope P and with. A triangulation of a space maybe a homeomorphism from a simplicial complex shield the space. Manifolds in a conjecture of notes on topological manifolds and triangulations are lecture notes for any. Lecture 1 Triangulated categories of motives 3 1 Triangulated categories 4 2. The Hauptvermutung for sparse singular homeomorphisms. In conduct, especially pray that different with intersection theory. We show that lecture notes in one can to read full content visible, triangulation conjecture that are shown that support from each chapter. This is close joint capacity with Jiming Ma. Equivalences to the triangulation conjecture 1 Introduction. Progress in Mathematics, Vol. Please feel allegiance to email me we talk to source about details. The triangulated category, gradient vector bundles, thus providing some familiarity with vertical boundary between combinatorial theory and affine coxeter group and does not assume much as well. We thus present our solutions to both questions, Vol.