DOCSLIB.ORG

Towards the Triviality of $X 0^+ (P^R) (\Mathbb{Q})

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Towards the Triviality of $X 0^+ (P^R) (\Mathbb{Q}) Compositio Math. 141 (2005) 561–572 DOI: 10.1112/S0010437X04001022 + r Towards the triviality of X0 (p )(Q)forr>1 Pierre J. R. Parent Abstract We give a criterion to check if, given a prime power pr with r>1, the only rational points + r of the modular curve X0 (p ) are trivial (i.e. cusps or points furnished by complex multi- plication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit(p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves. 1. Introduction r r Let p beapowerofaprimenumberp,withr>1, and X0(p ) be the classical modular curve + r + r over Q.LetX0 (p ) be its quotient by the Atkin–Lehner involution. We say that a point of X0 (p )(C) is trivial if it is a cusp, or if the underlying elliptic curves have complex multiplication. In this + r paper, we state a criterion to check whether X0 (p )(Q) is trivial (Proposition 3.2). Then we prove that this criterion is verified if p satisfies some congruences. Explicitly, set A := {primes that are simultaneously a square mod 3, mod 4, mod 7, and a square mod at least five of the following: 8, 11, 19, 43, 67, 163}. Our main theorem is the following. Theorem 1.1. If pr is a prime power such that r>1, p 11, p =13 ,andp does not belong to the + r above set A,thenX0 (p )(Q) is trivial. + r+2 + r (Note that the existence of Q-morphisms X0 (p ) → X0 (p )(see[Mom86, p. 443]) shows that this result actually boils down to the case r =2andr =3.) Theorem 1.1 applies in particular to the modular curve Xsplit(p), which is isomorphic over Q + 2 to X0 (p ). This special result was our first motivation for this work, because of its relation with the following problem of Serre. Let E be an elliptic curve over a number field K without complex mul- tiplication over K. The Galois action induces a representation Gal(K/K) → GL(E[p]) GL2(Fp). In his famous article [Ser72], Serre proved that there exists an integer CE such that this repre- sentation is surjective if p>CE. In the same paper Serre asked if the integer CE can be chosen to depend only on K, not on E ([Ser72, p. 299]; see also [Maz77, Introduction]). This question boils down to determining whether the K-rational points of several modular curves of level p are trivial (in the above sense) for large enough p. These curves are X0(p), Xsplit(p), Xnon-split(p), and maybe ‘exceptional’ ones. The latter case (of exceptional curves) was ruled out by Serre (see [Maz77, Introduction]), and it is a celebrated theorem of Mazur [Maz78]thatX0(p)(Q) is made of cusps for p>163 (and is trivial for p>37). In the cases of Xsplit(p)andXnon-split(p), a new difficulty arises from the fact that elliptic curves over Q with complex multiplication always provide rational points on one of those two modular curves. From the above, our criterion for Xsplit(p)(Q)tobetrivial (Proposition 3.2) can be verified for p 11, p ∈{13, 37},wherep does not belong to the set A. Received 6 November 2003, accepted in final form 1 March 2004, published online 21 April 2005. 2000 Mathematics Subject Classification 11G18, 11G05, 14G10. Keywords: elliptic curves, modular curves, rational points, Gross–Heegner points. This journal is c Foundation Compositio Mathematica 2005. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 30 Sep 2021 at 23:05:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X04001022 P. J. R. Parent The density of the prime numbers p in Theorem 1.1 is (1−7.2−9) 0.986 .... At the moment, we are unable to prevent a positive density of primes from escaping our methods, which use quadratic imaginary orders of trivial class number (as one guesses from the shape of A). Still in § 6 we indicate a procedure that could asymptotically improve this density. For general p we also prove a + r quantitative result giving explicit upper bounds for the number of non-trivial points in X0 (p )(Q) (Theorem 6.2). Our approach is based on the well-known method introduced by Mazur. We also make use of + r previous works by Momose on X0 (p )(Q), Kolyvagin–Logachev’s (or now Kato’s) theorem on the Birch and Swinnerton-Dyer conjecture, and a recent application by Merel of the graph method for X0(p) of Mestre and Oesterl´e. Our criterion is in fact almost the same as that of Merel [Mer01, Proposition 4], which arose in a different context. A new tool we use is a formula of Gross, generalized by Zhang, on special values of L-functions, which allows us to describe the cotangent space of Je (conjecturally the largest quotient of J0(p) having rank 0 over Q) in terms of Heegner points (Proposition 4.2). More precisely, the text is organized as follows. In § 2 we reduce our problem to a question on the fiber of X0(p)atp.In§ 3 we give our criterion (Propositions 3.1 and 3.2). In § 4 we concentrate on the cotangent space of Je in order to reformulate our criterion with the help of the Gross formula. In § 5 we restrict to primes satisfying the congruences of Theorem 1.1 and apply the graph method to build special elements that make the criterion work. Finally, in § 6 we prove our + r quantitative improvement (Theorem 6.2), we describe an algorithm to verify triviality of X0 (p )(Q) + r for any specified prime p, and we discuss the example of X0 (37 ). In the course of this paper we also give new elementary proofs of a result by Ahlgren and Ono on Weierstrass points of X0(p)(Q) (Theorem 3.3), and of a theorem by Vatsal for the equidistribution of Heegner points (Theorem 4.3). 2. Reducing to the bad fiber of X0(p) For any positive integer N, recall that X0(N) is the modular curve over Q corresponding to the congruence subgroup a b Γ0(N):= ∈ SL2(Z),c≡ 0modN . c d This curve deprived of its cusps is the coarse moduli space over Q of the isomorphism classes of elliptic curves equipped with an N-isogeny. If M|N,wewriteπN,M : X0(N) → X0(M)forthede- generacy morphism, which is defined functorially as (E,CN ) → (E,CM ), where CM := E[M] ∩ CN . In this paper, the model of X0(N)overZ that we consider is the modular one, which is obtained 1 1 by taking the normalization of PZ in X0(N)Q via the morphism πN,1 : X0(N)Q → X0(1)Q PQ. Models over arbitrary schemes of this modular curve will be deduced by base change. We denote by sm X0(p)Z the smooth part of X0(p)Z, obtained by removing the singular points in the fiber at p.IfM is a divisor of N such that M and N/M are relatively prime, we write wM for the corresponding + Atkin–Lehner involution, and X0 (N):=X0(N)/wN .Asusual,wewriteJ0(N) for the Jacobian − over Q of X0(N), and J0 (N):=J0(N)/(1+wN )J0(N). The models we use for abelian varieties over rings of integers are N´eron models. One defines similarly the curve Xsp.C.(N) associated to the split Cartan subgroup ∗ 0 Γsp C (N):= γ ∈ SL2(Z),γ≡ mod N , . 0 ∗ and the curve Xsplit(N) corresponding to the normalizer of the above group: ∗ 0 0 ∗ Γsplit(N):= γ ∈ SL2(Z),γ≡ or mod N . 0 ∗ ∗ 0 562 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 30 Sep 2021 at 23:05:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X04001022 + r Towards the triviality of X0 (p )(Q) for r>1 The curve Xsp.C.(N) (respectively, Xsplit(N)) parametrizes elliptic curves endowed with an ordered (respectively, unordered) pair of independent N-isogenies. There is an involution w on Xsp.C.(N), defined functorially by (E,(A, B)) → (E,(B,A)), such that Xsplit(N)=Xsp.C.(N)/w.Themap 2 z → Nz on the upper half-plane induces a Q-isomorphism X0(N ) Xsp.C.(N)(an‘exotic’mapin 2 the language of [KM85]), and Xsplit(N) X0(N )/wN 2 . r + r We now fix a prime power p , r>1. Let P be a non-cuspidal point in X0 (p )(Q)andz ∈ r X0(p )(K) a lifting of P ,forK a quadratic number field (or for K = Q, but this case is possible only for p 163 by Mazur’s theorem). The point z corresponds to a couple (E,Cpr )overK,by[DR73, − Proposition VI.3.2]. Set π := πpr,p, x := wpπ(z)andx0 := πwpr (z) ∈ X0(p)(K). Let y ∈ J0 (p)(K) be the image of the divisor class of (x) − (x0)inJ0(p)(K). Lemma 2.1. The point y is Q-rational. Proof. This is a straightforward calculation: if σ is the non-trivial element of Gal(K/Q), then cl(y − σ(y)) = cl((wpπ(z)) − (πwpr (z)) − (wpπwpr (z)) + (π(z))) =(1+wp)cl((π(z)) − (πwpr (z))).
Load more

Recommended publications
  • Ordinary Differential Equations
    Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license is that You are free to: Share copy and redistribute the material in any medium or format • Adapt remix, transform, and build upon the material for any purpose, even commercially. • The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes • were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Notices: You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. To view a full copy of this license, visit https://creativecommons.org/licenses/by/4.0/legalcode.
    [Show full text]
  • 256B Algebraic Geometry
    256B Algebraic Geometry David Nadler Notes by Qiaochu Yuan Spring 2013 1 Vector bundles on the projective line This semester we will be focusing on coherent sheaves on smooth projective complex varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre duality 4. Reconstruction theorems (Bondal-Orlov, Tannaka, Gabriel) 5. Hochschild homology, Chern classes, Grothendieck-Riemann-Roch For now we'll introduce enough background to talk about vector bundles on P1. We'll regard varieties as subsets of PN for some N. Projective will mean that we look at closed subsets (with respect to the Zariski topology). The reason is that if p : X ! pt is the unique map from such a subset X to a point, then we can (derived) push forward a bounded complex of coherent sheaves M on X to a bounded complex of coherent sheaves on a point Rp∗(M). Smooth will mean the following. If x 2 X is a point, then locally x is cut out by 2 a maximal ideal mx of functions vanishing on x. Smooth means that dim mx=mx = dim X. (In general it may be bigger.) Intuitively it means that locally at x the variety X looks like a manifold, and one way to make this precise is that the completion of the local ring at x is isomorphic to a power series ring C[[x1; :::xn]]; this is the ring where Taylor series expansions live.
    [Show full text]
  • Partial Differential Equations I: Basics and Separable Solutions
    18 Partial Differential Equations I: Basics and Separable Solutions We now turn our attention to differential equations in which the “unknown function to be deter- mined” — which we will usually denote by u — depends on two or more variables. Hence the derivatives are partial derivatives with respect to the various variables. (By the way, it may be a good idea to quickly review the A Brief Review of Elementary Ordinary Differential Equations, Appendex A of these notes. We will be using some of the material discussed there.) 18.1 Intro and Examples Simple Examples If we have a horizontally stretched string vibrating up and down, let u(x, t) the vertical position at time t of the bit of string at horizontal position x , = and make some almost reasonable assumptions regarding the string, the universe and the laws of physics, then we can show that u(x, t) satisfies ∂2u ∂2u c2 0 ∂t2 − ∂x2 = where c is some positive constant dependent on the physical properties of the stretched string. This equation is called the one-dimensional wave equation (with no external forces). If, instead, we have a uniform one-dimensional heat conducting rod along the X–axis and let u(x, t) the temperature at time t of the bit of rod at horizontal position x , = then, after applying suitable assumptions about heat flow, etc., we get the one-dimensional heat equation ∂u ∂2u κ f (x, t) . ∂t − ∂x2 = Here, κ is a positive constant dependent on the material properties of the rod, and f (x, t) describes the thermal contributions due to heat sources and/or sinks in the rod.
    [Show full text]
  • Zero and Triviality Andreas Haida1 and Tue Trinh2 1 ELSC, HUJI, Jerusalem, IL 2 Leibniz-ZAS, Berlin, DE Corresponding Author: Tue Trinh ([email protected])
    a journal of Haida, Andreas and Tue Trinh. 2020. Zero and general linguistics Glossa triviality. Glossa: a journal of general linguistics 5(1): 116. 1–14. DOI: https://doi.org/10.5334/gjgl.955 RESEARCH Zero and triviality Andreas Haida1 and Tue Trinh2 1 ELSC, HUJI, Jerusalem, IL 2 Leibniz-ZAS, Berlin, DE Corresponding author: Tue Trinh ([email protected]) This paper takes issue with Bylinina & Nouwen’s (2018) hypothesis that the numeral zero has the basic weak meaning of ‘zero or more.’ We argue, on the basis of empirical observation and theoretical consideration, that this hypothesis implies that exhaustification can circum- vent L-triviality, and that exhaustification cannot circumvent L-triviality. We also provide some experimental results to support our argument. Keywords: numerals; modification; exhaustification; L-Triviality 1 Theses and antitheses 1.1 Exhaustification The aim of this squib is to discuss some issues regarding the hypothesis about zero put for- ward by Bylinina & Nouwen (2018). These authors assume that numerals have the weak, ‘at least’ meaning as basic and the strengthened, ‘exactly’ meaning as derived by way of exhaustification, and propose thatzero be considered no different from other numerals in this respect.1 This is illustrated by (1a) and (1b). (1) a. Zero students smoked | ⟦students⟧ ⟦smoked⟧ | ≥ 0 ‘the number of students⟺ who smoked is zero or greater’ ⋂ b. EXH [Zero students smoked] | ⟦students⟧ ⟦smoked⟧ | ≥ 0 | ⟦students⟧ ⟦smoked⟧ | ≥| 1 ‘the number of students who smoked⟺ is zero and not greater’ ⋂ ∧ ⋂ Before we go on, let us be explicit about the notion of exhaustification which we will adopt for the purpose of this discussion.
    [Show full text]
  • Arxiv:1509.06425V4 [Math.AG] 9 Dec 2017 Nipratrl Ntegoercraiaino Conformal of Gen Realization a Geometric the (And in Fact Role This Important Trivial
    TRIVIALITY PROPERTIES OF PRINCIPAL BUNDLES ON SINGULAR CURVES PRAKASH BELKALE AND NAJMUDDIN FAKHRUDDIN Abstract. We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of π1 of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve. 1. Introduction It is a consequence of a theorem of Harder [Har67, Satz 3.3] that generically trivial principal G-bundles on a smooth affine curve C over an arbitrary field k are trivial if G is a semisimple and simply connected algebraic group. When k is algebraically closed and G reductive, generic triviality, conjectured by Serre, was proved by Steinberg[Ste65] and Borel–Springer [BS68]. It follows that principal bundles for simply connected semisimple groups over smooth affine curves over algebraically closed fields are trivial. This fact (and a generalization to families of bundles [DS95]) plays an important role in the geometric realization of conformal blocks for smooth curves as global sections of line bundles on moduli-stacks of principal bundles on the curves (see the review [Sor96] and the references therein). An earlier result of Serre [Ser58, Th´eor`eme 1] (also see [Ati57, Theorem 2]) implies that this triviality property is true if G “ SLprq, and C is a possibly singular affine curve over an arbitrary field k.
    [Show full text]
  • Virtual Knot Invariants Arising from Parities
    Virtual Knot Invariants Arising From Parities Denis Petrovich Ilyutko Department of Mechanics and Mathematics, Moscow State University, Russia [email protected] Vassily Olegovich Manturov Faculty of Science, People’s Friendship University of Russia, Russia [email protected] Igor Mikhailovich Nikonov Department of Mechanics and Mathematics, Moscow State University, Russia [email protected] Abstract In [12, 15] it was shown that in some knot theories the crucial role is played by parity, i.e. a function on crossings valued in {0, 1} and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than Z2 and investigate them for different knot theories. For some knot theories we arXiv:1102.5081v1 [math.GT] 24 Feb 2011 show that there is the universal parity, i.e. such a parity that any other parity factors through it. We realise that in the case of flat knots all parities originate from homology groups of underlying surfaces and, at the same time, allow one to “localise” the global homological information about the ambient space at crossings. We prove that there is only one non-trivial parity for free knots, the Gaussian parity. At the end of the paper we analyse the behaviour of some invariants constructed for some modifications of parities. 1 Contents 1 Introduction 2 2 Basic definitions 3 2.1 Framed 4-graphs and chord diagrams .
    [Show full text]
  • Triviality Results and the Relationship Between Logical and Natural
    Triviality Results and the Relationship between Logical and Natural Languages Justin Khoo Massachusetts Institute of Technology [email protected] Matthew Mandelkern All Souls College, Oxford [email protected] Inquiry into the meaning of logical terms in natural language (‘and’, ‘or’, ‘not’, ‘if’) has generally proceeded along two dimensions. On the one hand, semantic theories aim to predict native speaker intuitions about the natural language sentences involving those logical terms. On the other hand, logical theories explore the formal properties of the translations of those terms into formal languages. Sometimes these two lines of inquiry appear to be in tension: for instance, our best logical investigation into conditional connectives may show that there is no conditional operator that has all the properties native speaker intuitions suggest ‘if’ has. Indicative conditionals have famously been the source of one such tension, ever since the triviality proofs of both Lewis (1976) and Gibbard (1981) established conclusions which are prima facie in tension with ordinary judgements about natural language indicative conditionals. In a recent series of papers, Branden Fitelson (2013, 2015, 2016) has strengthened both triviality results, revealing a common culprit: a logical schema known as IMPORT-EXPORT. Fitelson’s results focus the tension between the logical results and ordinary judgements, since IMPORT-EXPORT seems to be supported by intuitions about natural language. In this paper, we argue that the intuitions which have been taken to support IMPORT-EXPORT are really evidence for a closely related but subtly different principle. We show that the two principles are independent by showing how, given a standard assumption about the conditional operator in the formal language in which IMPORT- EXPORT is stated, many existing theories of indicative conditionals validate one but not the other.
    [Show full text]
  • PARITY of the PARTITION FUNCTION 1. Introduction And
    PARITY OF THE PARTITION FUNCTION KEN ONO Abstract. Although much is known about the partition function, little is known about its parity. For the polynomials D(x) := (Dx2 + 1)=24, where D ≡ 23 (mod 24), we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras. 1. Introduction and statement of results A partition of a non-negative integer n is a non-increasing sequence of positive integers that sum to n. Ramanujan investigated [43, 44] p(n), the number of partitions of n, and he discovered congruences in certain very special arithmetic progressions such as: p(5n + 4) ≡ 0 (mod 5); (1.1) p(7n + 5) ≡ 0 (mod 7); p(11n + 6) ≡ 0 (mod 11): Ramanujan's congruences have inspired many works (for example, see [2, 3, 4, 5, 6, 7, 8, 9, 15, 19, 20, 24, 26, 28, 31, 32, 39, 51] to name a few). Atkin [6] and Watson [51] notably proved Ramanujan's conjectures concerning congruences mod powers of 5, 7 and 11, while Andrews and Garvan unearthed [5] Dyson's [19] elusive crank, thereby providing a procedure for dividing the partitions into 5, 7, and 11 groups of equal size in (1.1).
    [Show full text]
  • BEYOND CALCULUS Math21a, O
    5/7/2004 CALCULUS BEYOND CALCULUS Math21a, O. Knill DISCRETE SPACE CALCULUS. Many ideas in calculus make sense in a discrete setup, where space is a graph, curves are curves in the graph and surfaces are collections of "plaquettes", polygons formed by edges of the graph. One can look at functions on this graph. Scalar functions are functions defined on the vertices of the INTRODUCTION. Topics beyond multi-variable calculus are usually labeled with special names like "linear graphs. Vector fields are functions defined on the edges, other vector fields are defined as functions defined on algebra", "ordinary differential equations", "numerical analysis", "partial differential equations", "functional plaquettes. The gradient is a function defined on an edge as the difference between the values of f at the end analysis" or "complex analysis". Where one would draw the line between calculus and non-calculus topics is points. not clear but if calculus is about learning the basics of limits, differentiation, integration and summation, then Consider a network modeled by a planar graph which forms triangles. A scalar function assigns a value f multi-variable calculus is the "black belt" of calculus. Are there other ways to play this sport? n to each node n. An area function assigns values fT to each triangle T . A vector field assign values Fnm to each edge connecting node n with node m. The gradient of a scalar function is the vector field Fnm = fn fm. HOW WOULD ALIENS COMPUTE? On an other planet, calculus might be taught in a completely different − The curl of a vector field F is attaches to each triangle (k; m; n) the value curl(F )kmn = Fkm + Fmn + Fnk.
    [Show full text]
  • Scaling Qualitative Probability Mark Burgin University of California, Los Angeles 405 Hilgard Ave
    Scaling Qualitative Probability Mark Burgin University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA 90095 Abstract There are different approaches to qualitative probability, which includes subjective probability. We developed a representation of qualitative probability based on relational systems, which allows modeling uncertainty by probability structures and is more coherent than existing approaches. This setting makes it possible proving that any comparative probability is induced by some probability structure (Theorem 2.1), that classical probability is a probability structure (Theorem 2.2) and that inflated, i.e., larger than 1, probability is also a probability structure (Theorem 2.3). In addition, we study representation of probability structures by classical probability. Keywords: probability, qualitative probability, quantitative probability, axiom, random event, permissible event, set algebra. 1. Introduction Qualitative probability and its particular case - subjective probability – plays important role in psychology, decision making and cognitive sciences. However, the mainstream of probability theory and statistics treats probability as a numeric function or measure with specific properties (cf., for example, (Devore, 2015; Hogg, et al, 2014; Walpole, et al, 2011)). Kolmogorov rigorously represented these properties by a system of axioms treating probability as an additive measure on a system of random events (Kolmogorov, 1933). Later researchers suggested other systems of axioms for the conventional probability theory (cf., for example, (Whittle, 2000)), as well as for extended probability theory, which included negative probabilities (Burgin, 2009; 2016), and for larger than 1 (inflated) probabilities (Burgin and Meissner, 2012). At the same time, to reflect uncertainty and probabilistic aspects of decision-making and other cognitive processes, various researchers developed and applied the theory of qualitative probability as an approach to mathematical modeling and investigation of uncertainty.
    [Show full text]
  • R-Triviality of Groups of Type F4 Arising from the First Tits Construction
    R-TRIVIALITY OF GROUPS OF TYPE F4 ARISING FROM THE FIRST TITS CONSTRUCTION SEIDON ALSAODY, VLADIMIR CHERNOUSOV, AND ARTURO PIANZOLA Abstract. Any group of type F4 is obtained as the automorphism group of an Albert algebra. We prove that such a group is R-trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras. 1. Introduction The notion of path connectedness for topological spaces is natural and well- known. In algebraic geometry, this notion corresponds to that of R-triviality. Recall that the notion of R-equivalence, introduced by Manin in [9], is an important birational invariant of an algebraic variety defined over an arbitrary field and is defined as follows: if X is a variety over a field K with X(K) 6= ;, two K-points x; y 2 X(K) are called elementarily R-equivalent if there is a 1 path from x to y, i.e. if there exists a rational map f : A 99K X defined at 0; 1 and mapping 0 to x and 1 to y. This generates an equivalence relation R on X(K) and one can consider the set X(K)=R. If X has in addition a group structure, i.e. X = G is an algebraic group over a field K, then the set G(K)=R has a natural structure of an (abstract) group and this gives rise to a functor G=R : F ields=K −! Groups; where F ields=K is the category of field extensions of K, and G=R is given by L=K ! G(L)=R.
    [Show full text]
  • Local Triviality for Continuous Field C∗-Algebras Joan
    J. Bosa and M. Dadarlat. () “Local triviality for continuous field C∗-algebras,” International Mathematics Research Notices, Vol. , Article ID , 10 pages. doi:10.1093/imrn/ Local triviality for continuous field C∗-algebras Joan Bosa1 and Marius Dadarlat2 1 Departament de Matematiques,` Universitat Autonoma` de Barcelona, 08193 Bellaterra, Barcelona, Spain and 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA Correspondence to be sent to: [email protected] Let X be a finite dimensional compact metrizable space. Let A be a separable continuous field C∗-algebra over X with all fibers isomorphic to the same stable Kirchberg algebra D. We show that if D has finitely generated K-theory and it satisfies the Universal Coefficient Theorem in KK-theory, then there exists a dense open subset U of X such that the ideal A(U) is locally trivial. The assumptions that the space X is finite dimensional and that the K-theory of the fiber is finitely generated are both necessary. 1 Introduction Continuous field C∗-algebras play the role of bundles of C∗-algebras (in the sense of topology) as explained in [1]. Continuous field C∗-algebras appear naturally since any separable C∗-algebra A with Hausdorff primitive spectrum X has a canonical continuous field structure over X with fibers the primitive quotients of A [5]. The bundle structure that underlies a continuous field C∗-algebra is typically not locally trivial even if all fibers are mutually isomorphic, see [4, Example 8.4]. A point x 2 X is called singular for A if A(U) is nontrivial (i.e.
    [Show full text]