DOCSLIB.ORG

THE ORDER of HIGHER BRAUER GROUPS 1. Introduction Let X Be A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
THE ORDER of HIGHER BRAUER GROUPS 1. Introduction Let X Be A THE ORDER OF HIGHER BRAUER GROUPS THOMAS JAHN Abstract. Let X be a smooth projective variety of dimension 2r over a finite field F. We prove that for every prime ℓ 6= char(F) the order of the non-divisible quotient of the ℓ-primary torsion group of the higher Brauer r group Br (X)(ℓ)nd is a square number. 1. Introduction Let X be a smooth projective surface over a finite field F, and let Br(X)= 2 Het(X, Gm) be its cohomological Brauer group. If ℓ =6 char(F) is a prime, Tate has shown that the Tate conjecture for divisors on X at the prime ℓ is equiv- alent to the ℓ-primary torsion subgroup Br(X)(ℓ) being finite. Furthermore, the Artin-Tate conjecture [13, Conjecture C] relates the order of Br(X) with the zeta function of X. More precisely, the zeta function of X may be written −s 1−s P1(X, q )P1(X, q ) ζ(X,s)= −s −s 2−s , (1 − q )P2(X, q )(1 − q ) and the Artin-Tate conjecture states that Br(X) is finite, and satisfies −s |Br(X)|·|det(Di · Dj)| 1−s ρ(X) P2(X, q ) ∼ α(X) 2 · (1 − q ) , as s → 1 ; q ·|NS(X)tor| for details, see [13]. This type of formula has been generalized to smooth projective varieties of even dimension as follows. Assuming the existence of certain complexes Γ(r) of ´etale sheaves satisfying the axioms stated by Lichten- r 2r+1 baum in [7], one can define higher Brauer groups BrΓ(X) = Het (X, Γ(r)); 1 since Γ(1) ∼ Gm[−1], we have BrΓ(X) = Br(X). Under further assumptions, including the strong Tate conjecture, Milne [9, Theorem 6.6] proved in case dim X =2r the formula (we refer to [9] for precise definitions of all quantities) |Brr (X)|·| det(D · D )| P (X, q−s) ∼ ± Γ i j (1 − qr−s)ρr , as s → r . 2r αr(X) r 2 q ·|A (X)tor| If X is a surface, i.e. r = 1, the formula given by Milne coincides with the one of the Artin-Tate conjecture [9, Remark 6.7]. In particular, these formulae relating the values of zeta functions with the order of higher Brauer groups motivate the attempt to understand the order of these groups. In the classical case, i.e. for Br(X) of a surface, Tate [13] proved that there is a canonical skew-symmetric form on the non-divisible quotient Br(X)(ℓ)nd (which, assuming the Tate conjecture for divisors at ℓ coincides with Br(X)(ℓ)). Tate conjectured that this form is also alternating which would imply that the Date: June 13, 2014. 1 2 THOMAS JAHN order of Br(X)(ℓ)nd is a square. The second statement has been proven by Urabe in [15] (using a different bilinear form). We generalize this as follows. Instead of the complexes considered by Licht- enbaum we work with the (unbounded) complex of ´etale sheaves ZX (r)et given by Bloch’s cycle complex [1] on the small ´etale site Xet. Its hypercohomology m groups define the Lichtenbaum groups HL (X, Z(r)); since ZX (1)et ∼ Gm[−1] ∼ 3 we have Br(X) = HL(X, Z(1)), and the higher Brauer we consider here are r 2r+1 the groups Br (X) = HL (X, Z(r)). As in the classical case, these groups provide obstructions to the Tate conjecture. More precisely, if X is a smooth projective variety over a finite field F of arbitrary dimension, the Tate conjec- r r ture TC (X)Qℓ for X in codimension r at ℓ holds if and only if Br (X)(ℓ) < ∞ [11], see also section 2 below. Furthermore, the complexes ZX (r)et satisfy the r formal requirements needed to prove Milne’s formula with BrΓ(X) replaced by r Br (X). Thus we are interested in the (finite) order of Br(X)(ℓ)nd. With our notion of a higher Brauer group we can generalize Urabe’s result as follows. Theorem 1.1. Let X be a smooth projective variety of dimension 2r over a r finite field F. Then |Br (X)(ℓ)nd| is a square number for every ℓ =6 char(F). The proof makes use of cohomological methods analogous to the ones used by Urabe in his proof for r = 1, i.e. a pairing induced by the cup product, Steenrod operations, and the Wu Formula for ´etale cohomology. However, in Urabe’s case a crucial role is played by the cohomology class of the canoni- cal divisor, and a crucial impetus in our generalization is the construction of cohomology classes with similar properties in all codimension r. Notations and Conventions. In what follows, X always denotes a smooth projective variety of dimension 2r over a finite field F of characteristic p, and ℓ a prime number different from p. If F is an algebraic closure of F, we write X = X ×F F for the base change. We let G = Gal(F/F) be the Galois group and denote by σ the Frobenius. If M is a G-Module, we define M G to be the kernel and MG to be the cokernel of (1 − σ): M → M. For an abelian group A we use the following conventions: An denotes the n- n torsion subgroup, i.e. An = ker(A → A), Ator = ∪n≥1An, and Afree = A/Ator; n if q is a prime number, A(q) = n>0 Aq is the q-primary torsion subgroup. We write Adiv for the subgroup ofS divisible elements, and set And = A/Adiv. Acknowledgement. The author is grateful to his advisor Professor Andreas Rosenschon for suggesting the problem and his helpful advice. 2. Preliminaries We first give the precise definition of the higher Brauer groups used in this paper, and explain the relation of these groups to the Tate conjecture. Let Y be a equi-dimensional smooth quasi-projective variety over a field k and let zn(Y, •) the cycle complex of abelian groups defined by Bloch in [1]. The assertion zn(−, •): U 7→ zn(U, •) defines a complex of sheaves on n the flat site of Y , thus a complex z (−, •)et of sheaves on the ´etale site of Y THE ORDER OF HIGHER BRAUER GROUPS 3 (which is unbounded on the left). Given an abelian group A, we set AY (n)et = n (z (−, •)et ⊗ A)[−2n]. The Lichtenbaum cohomology groups are defined as m m HL (Y, A(n)) = Het (Y, AY (n)et); for the definition of the hypercohomology of an unbounded complex, see [12, 3 2 p. 121]. Since ZY (1)et ∼ Gm[−1] we have HL(Y, Z(1)) = Het(Y, Gm) = r 2r+1 r Br(Y ). Set Br (Y ) = HL (Y, Z(r)). Note that the Br (Y ) are torsion groups 2r (as quotients of HL (Y, Q/Z(r)); this uses that with rational coefficients the Lichtenbaum groups coincide with the usual motivic cohomology groups, thus 2r+1 HL (Y, Q(r)) = 0). Moreover, for every prime ℓ we have the exact sequence m ℓ m 0 → ZY (n)et → ZY (n)et → (Z/ℓ Z)Y (n)et → 0 . m If ℓ =6 char(k), it follows from this and the quasi-isomorphism (Z/ℓ Z)Y (n)et ∼ ⊗n µℓm [5, Theorem 1.5] that there are short exact universal coefficient sequences ⊗ i m i n i+1 m (1) 0 → HL(Y, Z(n)) ⊗ Z/ℓ Z → Het(Y,µℓm ) → HL (Y, Z(n))ℓ → 0 . In particular, in bidegree (i, n)=(2r, r) we obtain the short exact sequence ⊗ 2r m 2r r r m 0 → HL (Y, Z(r)) ⊗ Z/ℓ Z → Het (Y,µℓm ) → Br (Y )ℓ → 0 which generalizes the short exact sequence for the classical Brauer group com- ing from the Kummer sequences. r We use this to describe the non-divisible quotient Br (Y )(ℓ)nd in terms of ´etale cohomology groups. Lemma 2.1. Let k be a field and let Y/k be a smooth quasi-projective variety. If ℓ =6 char(k) is a prime, there are isomorphisms ∼ ∼ r = 2r = 2r+1 Br (Y )(ℓ)nd ← Het (Y, Qℓ/Zℓ(r))nd → Het (Y, Zℓ(r))tor . Proof. From (1) we obtain by taking the evident direct limit the exact sequence 2r 2r r 0 → HL (Y, Z(r)) ⊗ Qℓ/Zℓ → Het (Y, Qℓ/Zℓ(n)) → Br (Y )(ℓ) → 0 which immediately implies the first isomorphism. Consider the exact sequence 2r ϕ 2r δ 2r+1 ψ 2r+1 Het (Y, Qℓ(r)) → Het (Y, Qℓ/Zℓ(r)) → Het (Y, Zℓ(r)) → Het (Y, Qℓ(r)) associated with 0 → Zℓ(r) → Qℓ(r) → Qℓ/Zℓ(r) → 0. Since im ϕ is the max- 2r imal divisible subgroup of Het (Y, Qℓ/Zℓ(r)) and ker ψ is the torsion subgroup 2r+1 of Het (Y, Zℓ(r)), the boundary map δ induces the second isomorphism. Finally, we sketch how the higher Brauer groups provide obstructions to the Tate conjecture [14]. Let k be a field which is finitely generated over its prime field, let Y/k be a smooth, projective, geometrically integral variety, and set Y = Y ×k k, where k is a separable closure of k. Denote by G = Gal(k/k) the m Galois group. The Tate conjecture TC (Y )Qℓ for Y in codimension m at the prime ℓ =6 char(k) states that the ℓ-adic cycle map from Chow groups to ´etale cohomology m m 2m G cQℓ : CH (Y ) ⊗ Qℓ → Het (Y, Qℓ(m)) 4 THOMAS JAHN is surjective.
Load more

Recommended publications
  • Tate's Conjecture, Algebraic Cycles and Rational K-Theory In
    K-Theory 13: 109–122, 1998. 109 © 1998 Kluwer Academic Publishers. Printed in the Netherlands. Tate’s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p THOMAS GEISSER? Institute for Experimental Mathematics, Ellernstr. 29, 45326 Essen, Germany e-mail: [email protected] (Received: May 1997) Abstract. The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate’s conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate’s conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin’s conjecture). Parshin’s conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result. Mathematics Subject Classifications (1991). 19E08, 14C35, 19E15. Key words: higher algebraic K-theory, Milnor K-theory, finite fields, Tate’s conjecture, Beilinson’s conjecture, Parshin’s conjecture, Chow groups. 1. Introduction There are three results which allow to deduce properties of the category of motives for numerical equivalence: Deligne’s proof of the Weil conjectures, Jannsen’s semi- simplicity theorem, and the existence of the Frobenius automorphism πM for motives M. If one further assumes Tate’s conjecture, one can give a very precise description of this category (Milne [10]). For example, simple motives are determined by their Frobenius endomorphism, and one can recover the endomorphism algebra with the Frobenius.
    [Show full text]
  • Monodromy and the Tate Conjecture-1
    Monodromy and the Tate conjecture-1 Monodromy and ttthe Tattte conjjjecttture::: Piiicard numbers and Mordellllll-Weiiilll ranks iiin famiiillliiies A. Johan de Jong and Nicholas M. Katz Intttroductttiiion We use results of Deligne on …-adic monodromy and equidistribution, combined with elementary facts about the eigenvalues of elements in the orthogonal group, to give upper bounds for the average "middle Picard number" in various equicharacteristic families of even dimensional hypersurfaces, cf. 6.11, 6.12, 6.14, 7.6, 8.12. We also give upper bounds for the average Mordell- Weil rank of the Jacobian of the generic fibre in various equicharacteristic families of surfaces fibred over @1, cf. 9.7, 9.8. If the relevant Tate Conjecture holds, each upper bound we find for an average is in fact equal to that average The paper is organized as follows: 1.0 Review of the Tate Conjecture 2.0 The Tate Conjecture over a finite field 3.0 Middle-dimensional cohomology 4.0 Hypersurface sections of a fixed ambient variety 5.0 Smooth hypersurfaces in projective space 6.0 Families of smooth hypersurfaces in projective space 7.0 Families of smooth hypersurfaces in products of projective spaces 8.0 Hypersurfaces in @1≠@n as families over @1 9.0 Mordell-Weil rank in families of Jacobians References 1...0 Reviiiew of ttthe Tattte Conjjjecttture 1.1 Let us begin by recalling the general Tate Conjectures about algebraic cycles on varieties over finitely generated ground fields, cf. Tate's articles [Tate-Alg] and [Tate-Conj]. We start with a field k, a separable closure äk of k, and Gal(äk/k) its absolute galois group.
    [Show full text]
  • Arxiv:1810.06480V3 [Math.AG] 10 May 2021 1.2.1
    A NOTE ON THE BEHAVIOUR OF THE TATE CONJECTURE UNDER FINITELY GENERATED FIELD EXTENSIONS EMILIANO AMBROSI ABSTRACT. We show that the `-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the `-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher codimension are given. 1. INTRODUCTION Let k be a field of characteristic p ≥ 0 with algebraic closure k and write π1(k) for the absolute Galois group of k.A k-variety is a reduced scheme, separated and of finite type over k. For a k-variety Z i write Zk := Z ×k k and CH (Zk) for the group of algebraic cycles of codimension i modulo rational equivalence. Let ` 6= p be a prime. 1.1. Conjectures. Recall the following versions of the Grothendieck-Serre-Tate conjectures ([Tat65], [And04, Section 7.3]): Conjecture 1.1.1. If k is finitely generated and Z is a smooth proper k-variety, then: • T (Z; i; `): The `-adic cycle class map [ 0 c : CHi(Z ) ⊗ ! H2i(Z ; (i))π1(k ) Zk k Q` k Q` [k0:k]<+1 is surjective; 2i • S(Z; i; `): The action of π1(k) on H (Zk; Q`(i)) is semisimple; 2i π1(k) 2i • WS(Z; i; `): The inclusion H (Zk; Q`(i)) ⊆ H (Zk; Q`(i)) admits a π1(k)-equivariant splitting. For a field K, one says that T (K; i; `) holds if for every finite field extension K ⊆ L and every smooth proper L-variety Z, T (Z; i; `) holds.
    [Show full text]
  • A REMARK on UNIFORM BOUNDEDNESS for BRAUER GROUPS 1. Introduction 1.1. Let K Be a Field of Characteristic P ≥ 0 and Let K Be A
    A REMARK ON UNIFORM BOUNDEDNESS FOR BRAUER GROUPS ANNA CADORET AND FRANC¸OIS CHARLES Abstract. The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of `-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors { e.g. abelian varieties and K3 surfaces. 1. Introduction 1.1. Let k be a field of characteristic p ≥ 0 and let k be a separable closure of k. Let X be a smooth proper variety over k. For any prime number ` 6= p, consider the cycle class map 2 c1 : Pic(Xk) ⊗ Q` ! H (Xk; Q`(1)): 2 The absolute Galois group Gal(k=k) of k acts naturally on the ´etalecohomology group H (Xk; Q`(1)), and the image of the cycle class map c1 is contained in the group [ 2 U H (Xk; Q`(1)) ; U where U runs through the open subgroups of Gal(k=k). We say that X satisfies the `-adic Tate conjecture for divisors if the image of c1 is equal to the group above. As is well-known, the Tate conjecture for divisors is related to finiteness results for the Brauer group. Indeed, given X and ` as above, X satisfies the `-adic Tate conjecture for divisors if and only if for every subgroup U of finite index in Gal(k=k), the group U 1 Br(Xk) [` ] is finite. 1.2. In this paper, we investigate whether one might be able to find uniform bounds on the Brauer groups above when the variety X varies in a family.
    [Show full text]
  • The Mumford-Tate Conjecture for Drinfeld-Modules
    MUMFORD-TATE CONJECTURE 1 THE MUMFORD-TATE CONJECTURE FOR DRINFELD-MODULES by Richard PINK* Abstract Consider the Galois representation on the Tate module of a Drinfeld module over a finitely generated field in generic characteristic. The main object of this paper is to determine the image of Galois in this representation, up to commensurability. We also determine the Dirichlet density of the set of places of prescribed reduction type, such as places of ordinary reduction. §0. Introduction Let F be a finitely generated field of transcendence degree 1 over a finite field of characteristic p. Fix a place ∞ of F , and let A be the ring of elements of F which are regular outside ∞. Consider a finitely generated extension K of F and a Drinfeld module ϕ : A → EndK (Ga) of rank n ≥ 1 (cf. Drinfeld [10]). In other words K is a finitely generated field of transcendence degree ≥ 1 over Fp, and ϕ has “generic characteristic”. sep Let K ⊂ K¯ denote a separable, respectively algebraic closure of K. Let Fλ denote the completion of F at a place λ. If λ 6= ∞ we have a continuous representation sep ρλ : Gal(K /K) −→ GLn(Fλ) Communicated by Y. Ihara, September 20, 1996. 1991 Mathematics Subject Classifications: 11G09, 11R58, 11R45 * Fakult¨at f¨ur Mathematik und Informatik, Universit¨at Mannheim, D-68131 Mannheim, Germany 2 RICHARD PINK which describes the Galois action on the λ-adic Tate module of ϕ. The main goal of this article is to give a qualitative characterization of the image of ρλ. Here the term “qualitative” refers to properties that are shared by all open subgroups, i.e.
    [Show full text]
  • Sato-Tate Distributions
    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 we 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 in the context of abelian vari- eties, let us first consider the more familiar setting of Artin motives (varieties of dimension zero). 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d. For each prime p, let 2 fp (Z=pZ)[x] Fp[x] denote the reduction of f modulo p, and define 2 ' 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, because the Galois group of f is nonabelian).
    [Show full text]
  • Higher Brauer Groups
    Higher Brauer Groups Thomas Sebastian Alexander Benedikt Jahn Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München vorgelegt von Thomas Sebastian Alexander Benedikt Jahn am 2. Juni 2015 Higher Brauer Groups Thomas Sebastian Alexander Benedikt Jahn Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München vorgelegt von Thomas Sebastian Alexander Benedikt Jahn am 2. Juni 2015 Erstgutachter: Prof. Dr. Andreas Rosenschon, München Zweitgutachter: Prof. Dr. Alexander Schmidt, Heidelberg Tag der mündlichen Prüfung: 21. Juli 2015 Contents Zusammenfassung/Abstract vii 1. Introduction 1 2. Higher Brauer Groups 5 2.1. Brauer groups . 6 2.2. Zeta functions and Tate’s conjecture . 10 2.3. Lichtenbaum’s complex . 23 2.4. Lichtenbaum and motivic cohomology . 27 3. The Order of Higher Brauer Groups 33 3.1. Bilinear form . 34 3.2. Alternating form . 37 3.3. Steenrod squares and omega . 40 3.4. Proof of the theorem . 43 A. Hypercohomology 47 Zusammenfassung 2 Die klassische kohomologische Brauergruppe Br(X) = Het(X, Gm) einer glatten algebraischen Varietät X ist in verschiedenen Kontexten der algebraischen Geo- metrie von Interesse. Beispielsweise steht die kohomologische Brauergruppe in Zusammenhang mit Werten von Zeta-Funktionen und einer Vermutung von Tate über die Surjektivität gewisser Zyklenabbildungen. Für eine glatte projektive Fläche X über einem endlichen Körper gilt die Tate Vermutung für Divisoren an der Primzahl ` ungleich der Charakteristik genau dann, wenn die `-primär Gruppe Br(X)(`) endlich ist. Weiter hat die Zeta-Funktion von X eine Darstel- lung als eine rationale Funktion, wobei nach einer Vermutung von Tate und Artin sich bestimmte Werte einer der auftretenden Faktoren durch eine Formel approxi- mieren lassen, die die Ordnung jBr(X)j der Brauergruppe involviert.
    [Show full text]
  • Arxiv:1606.01771V2 [Math.AG]
    THE PICARD RANK OF AN ENRIQUES SURFACE CHRISTIAN LIEDTKE ABSTRACT. In this note, we use crystalline methods and the Tate-conjecture to give a short proof that the Picard rank of an Enriques surface is equal to its second Betti number. 1. INTRODUCTION Enriques surfaces are one of the four classes of minimal, smooth, and proper surfaces of Kodaira dimension zero. The following fundamental result relates the Picard rank ρ to the second Betti number b2 of these surfaces. Theorem 1.1 (Bombieri–Mumford [BM76]). Let X be an Enriques surface over an algebraically closed field k. Then, ρ(X)= b2(X) = 10. Using this result, it is not difficult to show that the N´eron–Severi lattice of an Enriques surface is even, unimodular, of signature (1, 9), and of discriminant −1, see [Il79, Corollaire II.7.3.7]. Thus, it is isometric to U ⊥ E8 by lattice theory, see [CDL, Chapter I.5]. In particular, there exist non-zero isotropic vectors, which implies that every Enriques surface carries a genus-one fibration. Moreover, this result is also essential for the analysis of linear systems [Co85], projective models [Co83], [Li15], automorphism groups [BP83], and moduli spaces [GH16] of these surfaces. 2 If k = C, then Theorem 1.1 is an easy consequence of H (OX ) = 0 and the Lefschetz theorem on (1, 1) classes. On the other hand, the known proofs of this result if char(k) > 0 are rather delicate and complicated. (1) The first proof is due to Bombieri and Mumford [BM76], where they first arXiv:1606.01771v2 [math.AG] 5 Aug 2016 establish with some effort the existence of a genus-one fibration f : X → P1.
    [Show full text]
  • Problems from the AIM Tate Conjecture Workshop July 23 to July 27, 2007 Transcribed by Christopher Lyons
    Problems from the AIM Tate Conjecture Workshop July 23 to July 27, 2007 Transcribed by Christopher Lyons 1. (C. Hall) We know how to construct points on an elliptic curve E over Fq(T ) when the analytic rank is 1 by using (Drinfel'd) Heegner points. What if X is a hyperelliptic curve over Fq(T ) of higher genus? Is there any way of constructing the points on the Jacobian of X that ought to be there? Comments: 1) We have a lot of explicit examples when the analytic rank is 1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a 2 curve over Fq(T ). Typically (since the rank of H is even), one has two Tate classes. One of these is given by hyperplane sections; can we \see" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen) 2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on X.
    [Show full text]
  • On the Tate Conjecture for Squares of K3 Surfaces Over Finite Fields
    On the Tate conjecture for Squares of K3 Surfaces over Finite Fields Ziquan Yang Abstract The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito–Ito– Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between K3 surfaces to link the Tate conjecture to finiteness results over finite fields, in the spirit of Tate. 1 Introduction In this note, we give a proof of the Tate conjecture for squares of K3 surfaces over finite fields. We will phrase and study the problem from the perspective of motives, so we first introduce some notation. Let ¯ k be a perfect field of characteristic p with a chosen algebraic closure k, MChow(k) be the category of Chow motives, which comes equipped with fiber functors !` (resp. !cris) given by `-adic cohomology for every prime ` 6= p (resp. crystalline cohomology with W[1=p] coefficients). Let Mhom(k) be the category MChow(k) but with Hom groups taken modulo the kernel of ! := !` × !cris. Let h(-) denote the usual functor from the category of varieties over k to Mhom(k) given by X 7 (X; id; 0). Theorem 1.1. Let X be a K3 surface over a finite field Fq of charactersitic p ≥ 5. Then natural morphism ! ∗ ! : End(h(X))⊗ End (H (X¯ ; )) ` Q` F ´et Fq Q` ∗ is an isomorphism for ` 6= p, where F denotes the Frobenius action on H (X¯ ; ). ! ´et Fq Q` When X is a K3 surface over k, h(X) is known to admit a Chow-Künneth decomposition h(X) = h0(X) ⊕ h2(X) ⊕ h4(X).
    [Show full text]
  • The Artin-Tate Pairing on the Brauer Group of a Surface
    THE ARTIN-TATE PAIRING ON THE BRAUER GROUP OF A SURFACE TONY FENG Abstract. There is a canonical pairing on the Brauer group of a surface over a finite field, and an old conjecture of Tate predicts that this pairing is alternating. In this talk I will present a resolution to Tate’s conjecture. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations. The talk will advertise these new tools (while assuming minimal background in algebraic topology). Contents 1. Introduction 1 2. Pairings on the cohomology of a surface 3 3. Steenrod squares 5 4. Plan of the proof 6 5. Characteristic classes 7 1. Introduction 1.1. Tate’s question. Let X be a smooth, projective, geometrically connected surface over Fq. Let ` be a prime distinct from the characteristic of Fq. Artin and Tate introduced a 1 skew-symmetric pairing h·; ·iAT on Br(X)[` ], which I will define a little later. I call this the Artin-Tate pairing. The question that concerns us is very easy to state: Conjecture 1.1 (Tate, 1966). The pairing h·; ·iAT is alternating. Let me remind you that skew-symmetric means hx; yiAT + hy; xiAT = 0 while alternating means hx; xiAT = 0: They are equivalent if 2 is invertible, but otherwise the second condition is stronger. The purpose of this talk is to present an answer to Conjecture 1.1. But before getting that I want to provide a little context and motivation for the problem. Notes for 60 minute talk on [?]. 1 2 TONY FENG 1.2.
    [Show full text]
  • The Tate Conjecture for K3 Surfaces Over Finite Fields
    Invent math DOI 10.1007/s00222-012-0443-y The Tate conjecture for K3 surfaces over finite fields François Charles Received: 6 July 2012 / Accepted: 29 November 2012 © Springer-Verlag Berlin Heidelberg 2012 Abstract Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p ≥ 5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p ≥ 5. Our results also yield the Tate conjec- ture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p ≥ 5. Mathematics Subject Classification (2010) 14C22 · 14C25 · 14G15 · 14J28 1 Introduction The goal of this paper is to study the Tate conjecture for varieties with h2,0 = 1 over finite fields. The main result is the following. Recall that Artin conjectured in [3] that the rank of the Néron-Severi group of a supersingular K3 surface over a finite field—in the sense of Artin, that is, a K3 surface whose formal Brauer group has infinite height—has the maximal possible value, that is, 22. Theorem 1 Artin’s conjecture holds for supersingular K3 surfaces over al- gebraically closed fields of characteristic p ≥ 5. F. Charles () IRMAR–UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: [email protected] F. Charles Let X be a smooth projective variety over a finite field k.Let be a prime number different from the characteristic of k.
    [Show full text]