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Journal articles on the topic 'Finite fields (Algebra) Geometry, Projective'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

RAFIE-RAD, M. "SPECIAL PROJECTIVE ALGEBRA OF RANDERS METRICS OF CONSTANT S-CURVATURE." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1250034. http://dx.doi.org/10.1142/s021988781250034x.

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The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most [Formula: see text]. Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

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2

González-Avilés, Cristian D. "On K2 of varieties over number fields." Journal of K-Theory 1, no. 1 (January 7, 2008): 175–83. http://dx.doi.org/10.1017/is007011012jkt004.

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AbstractLet k be a number field and let X be a smooth, projective and geometrically integral k-variety. We show that, if the geometric Néron-Severi group of X is torsion-free, then the Galois cohomology group is finite. Previously this group was only known to have a finite exponent. We also obtain a vanishing theorem for this group, showing in particular that it is trivial if X belongs to a certain class of abelian varieties with complex multiplication. The interest in the above cohomology group stems from its connection to the torsion subgroup of the Chow group CH2(X) of codimension 2 cycles on X. In the last section of the paper we record certain results on curves which must be familiar to all specialists in this area but which we have not formerly seen in print.

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3

Miyatani, Kazuaki, and Makoto Sano. "An exponential sum and higher-codimensional subvarieties of projective spaces over finite fields." Hiroshima Mathematical Journal 44, no. 3 (November 2014): 327–40. http://dx.doi.org/10.32917/hmj/1419619750.

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4

Antieau, Benjamin, and Ben Williams. "Godeaux–Serre varieties and the étale index." Journal of K-Theory 11, no. 2 (April 2013): 283–95. http://dx.doi.org/10.1017/is013003003jkt220.

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AbstractWe use Godeaux–Serre varieties of finite groups, projective representation theory, the twisted Atiyah–Segal completion theorem, and our previous work on the topological period-index problem to compute the étale index of Brauer classes α ∈ Brét(X) in some specific examples. In particular, these computations show that the étale index of α differs from the period of α in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Godeaux–Serre varieties in terms of projective representation theory.

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5

Hu, Wenchuan. "On Additive invariants of actions of additive and multiplicative groups." Journal of K-theory 12, no. 3 (May 1, 2013): 551–68. http://dx.doi.org/10.1017/is013003003jkt219.

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AbstractLet X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

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6

Stebletsova, Vera, and Yde Venema. "Undecidable theories of Lyndon algebras." Journal of Symbolic Logic 66, no. 1 (March 2001): 207–24. http://dx.doi.org/10.2307/2694918.

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AbstractWith each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski's axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In our proof we develop and use a connection between projective geometries and diagonal-free cylindric algebras.

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7

Shangdi, Chen, Zhang Xiaollian, and Ma Hao. "Two constructions of A3-codes from projective geometry in finite fields." Journal of China Universities of Posts and Telecommunications 22, no. 2 (April 2015): 52–59. http://dx.doi.org/10.1016/s1005-8885(15)60639-2.

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8

Wildberger, Norman. "Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative." Universe 4, no. 1 (January 1, 2018): 3. http://dx.doi.org/10.3390/universe4010003.

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9

Chen, Shangdi, and Xiaolian Zhang. "Three constructions of perfect authentication codes from projective geometry over finite fields." Applied Mathematics and Computation 253 (February 2015): 308–17. http://dx.doi.org/10.1016/j.amc.2014.12.088.

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10

Larose, Benoit. "Finite projective ordered sets." Order 8, no. 1 (1991): 33–40. http://dx.doi.org/10.1007/bf00385812.

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11

Givant, Steven. "Inequivalent representations of geometric relation algebras." Journal of Symbolic Logic 68, no. 1 (March 2003): 267–310. http://dx.doi.org/10.2178/jsl/1045861514.

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AbstractIt is shown that the automorphism group of a relation algebra constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of , for finite geometries P, is the sum of the numberswhere D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

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12

Lorenzo, Elisa, Giulio Meleleo, Piermarco Milione, and Alina Bucur. "Statistics for biquadratic covers of the projective line over finite fields." Journal of Number Theory 173 (April 2017): 448–77. http://dx.doi.org/10.1016/j.jnt.2016.09.007.

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13

YANG, RONGWEI. "PROJECTIVE SPECTRUM IN BANACH ALGEBRAS." Journal of Topology and Analysis 01, no. 03 (September 2009): 289–306. http://dx.doi.org/10.1142/s1793525309000126.

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For a tuple A = (A1, A2, …, An) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.

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14

Dabrowski, Ludwik, Thomas Krajewski, and Giovanni Landi. "Non-linear σ-models in noncommutative geometry: fields with values in finite spaces." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2371–79. http://dx.doi.org/10.1142/s0217732303012593.

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We study σ-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space [Formula: see text].

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15

Wennink, Thomas. "Counting the number of trigonal curves of genus 5 over finite fields." Geometriae Dedicata 208, no. 1 (January 9, 2020): 31–48. http://dx.doi.org/10.1007/s10711-019-00508-3.

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AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

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16

VAY, CRISTIAN. "ON PROJECTIVE MODULES OVER FINITE QUANTUM GROUPS." Transformation Groups 24, no. 1 (November 27, 2017): 279–99. http://dx.doi.org/10.1007/s00031-017-9469-y.

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17

Eastwood, Michael, and Hubert Goldschmidt. "Zero-energy fields on complex projective space." Journal of Differential Geometry 94, no. 1 (May 2013): 129–57. http://dx.doi.org/10.4310/jdg/1361889063.

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18

Bruin, Peter. "Computing in Picard groups of projective curves over finite fields." Mathematics of Computation 82, no. 283 (September 14, 2012): 1711–56. http://dx.doi.org/10.1090/s0025-5718-2012-02650-0.

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19

Rémy, Bertrand, Amaury Thuillier, and Annette Werner. "Automorphisms of Drinfeld half-spaces over a finite field." Compositio Mathematica 149, no. 7 (April 26, 2013): 1211–24. http://dx.doi.org/10.1112/s0010437x12000905.

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AbstractWe show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

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20

Beil, Charlie. "Nonnoetherian geometry." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650176. http://dx.doi.org/10.1142/s0219498816501760.

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We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

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21

Bani-Ata, Mashhour, and Mariam Al-Rashed. "On certain finite dimensional algebras over finite fields." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 58, no. 1 (August 19, 2016): 195–200. http://dx.doi.org/10.1007/s13366-016-0312-8.

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22

Shirvani, M. "The finite inner automorphism groups of division rings." Mathematical Proceedings of the Cambridge Philosophical Society 118, no. 2 (September 1995): 207–13. http://dx.doi.org/10.1017/s030500410007359x.

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Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

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23

Nesin, Ali. "On bad groups, bad fields, and pseudoplanes." Journal of Symbolic Logic 56, no. 3 (September 1991): 915–31. http://dx.doi.org/10.2307/2275061.

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Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and1. K = A − A,2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.If K is finite this is equivalent to 1 and2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.Finite sharp-fields are special cases of difference sets [De]

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24

Feng, Rongquan, Liwei Zeng, and Yang Zhang. "Constructions of 112-Designs from Unitary Geometry over Finite Fields." Algebra Colloquium 24, no. 03 (September 2017): 381–92. http://dx.doi.org/10.1142/s1005386717000232.

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In this paper, we construct some [Formula: see text]-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the unitary space. Furthermore, these [Formula: see text]-designs yield six infinite families of directed strongly regular graphs.

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25

Lo, Catharine Wing Kwan, and Matilde Marcolli. "𝔽ζ-geometry, Tate motives, and the Habiro ring." International Journal of Number Theory 11, no. 02 (March 2015): 311–39. http://dx.doi.org/10.1142/s1793042115500189.

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In this paper, we propose different notions of 𝔽ζ-geometry, for ζ a root of unity, generalizing notions of 𝔽1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate 𝔽ζ-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of 𝔽ζ-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.

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26

Lachaud, Gilles, Isabelle Lucien, Dany-Jack Mercier, and Robert Rolland. "Group Structure on Projective Spaces and Cyclic Codes over Finite Fields." Finite Fields and Their Applications 6, no. 2 (April 2000): 119–29. http://dx.doi.org/10.1006/ffta.1999.0268.

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27

Liu, Weijun, and Jinglei Li. "Finite projective planes admitting a projective linear group PSL (2,q)." Linear Algebra and its Applications 413, no. 1 (February 2006): 121–30. http://dx.doi.org/10.1016/j.laa.2005.08.026.

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28

Ertaş, Nil Orhan. "On Almost Projective Modules." Axioms 10, no. 1 (February 16, 2021): 21. http://dx.doi.org/10.3390/axioms10010021.

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In this note, we investigate the relationship between almost projective modules and generalized projective modules. These concepts are useful for the study on the finite direct sum of lifting modules. It is proved that; if M is generalized N-projective for any modules M and N, then M is almost N-projective. We also show that if M is almost N-projective and N is lifting, then M is im-small N-projective. We also discuss the question of when the finite direct sum of lifting modules is again lifting.

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29

Mallick, Vivek Mohan. "Roitman's theorem for singular projective varieties in arbitrary characteristic." Journal of K-theory 3, no. 3 (November 14, 2008): 501–31. http://dx.doi.org/10.1017/is008007021jkt054.

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AbstractIn this paper, we prove Roitman's theorem regarding torsion 0-cycles for singular projective varieties over algebraically closed fields of arbitrary characteristic, for torsion which is of exponent prime to the characteristic. This generalizes earlier results for complex projective varieties. Our proof even in that case is different from the earlier ones.

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30

Gao, Suogang, Zengti Li, Weili Wu, Panos M. Pardalos, and Dingzhu Du. "Group testing with geometry of classical groups over finite fields." Journal of Algebraic Combinatorics 49, no. 4 (June 6, 2018): 381–400. http://dx.doi.org/10.1007/s10801-018-0828-0.

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31

Camenga, Kristin A., Brandon Collins, Gage Hoefer, Jonny Quezada, Patrick X. Rault, James Willson, and Rebekah B. Johnson Yates. "On the geometry of numerical ranges over finite fields." Linear Algebra and its Applications 628 (November 2021): 182–201. http://dx.doi.org/10.1016/j.laa.2021.07.008.

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32

Koike, Masao. "Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields." Hiroshima Mathematical Journal 25, no. 1 (1995): 43–52. http://dx.doi.org/10.32917/hmj/1206127824.

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33

Biswas, Indranil, Ajneet Dhillon, and Norbert Hoffmann. "On the essential dimension of coherent sheaves." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 735 (February 1, 2018): 265–85. http://dx.doi.org/10.1515/crelle-2015-0028.

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AbstractWe characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curveC, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne–Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curveCis elliptic.

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34

Haution, Olivier. "Fixed point theorems involving numerical invariants." Compositio Mathematica 155, no. 2 (January 29, 2019): 260–88. http://dx.doi.org/10.1112/s0010437x18007911.

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We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo $p$ prevents the existence of an action without fixed points of certain finite $p$-groups. The case of base fields of characteristic $p$ is included. Counterexamples are systematically provided to test the sharpness of our results.

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35

Johnson, F. E. A. "Homotopy classification and the generalized Swan homomorphism." Journal of K-theory 4, no. 3 (January 7, 2009): 491–536. http://dx.doi.org/10.1017/is008012013jkt072.

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AbstractIn his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

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36

Biswas, Indranil. "On principal bundles over a projective variety defined over a finite field." Journal of K-Theory 4, no. 2 (October 2009): 209–21. http://dx.doi.org/10.1017/is009010006jkt077.

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AbstractLet M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

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37

Korchmáros, Gábor, and Tamás Szőnyi. "Fermat Curves over Finite Fields and Cyclic Subsets in High-Dimensional Projective Spaces." Finite Fields and Their Applications 5, no. 2 (April 1999): 206–17. http://dx.doi.org/10.1006/ffta.1998.0242.

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38

Foxby, Hans-Bjørn, and Esben Bistrup Halvorsen. "Grothendieck groups for categories of complexes." Journal of K-Theory 3, no. 1 (January 9, 2008): 165–203. http://dx.doi.org/10.1017/is008001002jkt023.

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AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

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39

Evans, Ron, and John Greene. "Evaluations of hypergeometric functions over finite fields." Hiroshima Mathematical Journal 39, no. 2 (July 2009): 217–35. http://dx.doi.org/10.32917/hmj/1249046338.

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40

MIGNACO, J. A., C. SIGAUD, F. J. VANHECKE, and A. R. DA SILVA. "CONNES–LOTT MODEL BUILDING ON THE TWO-SPHERE." Reviews in Mathematical Physics 13, no. 01 (January 2001): 1–28. http://dx.doi.org/10.1142/s0129055x01000582.

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In this work we examine generalized Connes–Lott models, with C⊕C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over S2. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.

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41

Colliot-Thélène, Jean-Louis, and Bruno Kahn. "Cycles de codimension 2 et H3 non ramifié pour les variétés sur les corps finis." Journal of K-Theory 11, no. 1 (February 2013): 1–53. http://dx.doi.org/10.1017/is012009001jkt194.

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AbstractLet X be a smooth projective variety over a finite field $\mathbb{F}$. We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefold X equipped with a fibration onto a curve C, the generic fibre of which is a smooth projective surface V over the global field $\mathbb{F}$(C), the vanishing of H3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors on X implies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles on V. This sheds new light on work started thirty years ago.

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42

KHREBTOVA, EKATERINA S., and DMITRY MALININ. "ON FINITE GALOIS STABLE ARITHMETIC GROUPS AND THEIR APPLICATIONS." Journal of Algebra and Its Applications 07, no. 06 (December 2008): 773–83. http://dx.doi.org/10.1142/s0219498808003119.

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We prove the existence and finiteness theorems for integral representations stable under Galois operation. An explicit construction of the realization fields for representations of finite groups stable under the natural operation of the Galois group is given. We also compare the representations over fields and the rings of integers, and give a quantitative result on the rarity of integral Galois stable representations. There is a series of related conjectures and applications to arithmetic algebraic geometry, finite flat group schemes, positive definite quadratic lattices and Galois cohomology.

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43

Dadarlat, Marius. "Fiberwise KK-equivalence of continuous fields of C*-algebras." Journal of K-Theory 3, no. 2 (May 28, 2008): 205–19. http://dx.doi.org/10.1017/is008001012jkt041.

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AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

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44

Pumplün, S. "Diagonal forms of higher degree over function fields of p-adic curves." International Journal of Number Theory 16, no. 01 (September 5, 2019): 161–72. http://dx.doi.org/10.1142/s1793042120500098.

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We investigate diagonal forms of degree [Formula: see text] over the function field [Formula: see text] of a smooth projective [Formula: see text]-adic curve: if a form is isotropic over the completion of [Formula: see text] with respect to each discrete valuation of [Formula: see text], then it is isotropic over certain fields [Formula: see text], [Formula: see text] and [Formula: see text]. These fields appear naturally when applying the methodology of patching; [Formula: see text] is the inverse limit of the finite inverse system of fields [Formula: see text]. Our observations complement some known bounds on the higher [Formula: see text]-invariant of diagonal forms of degree [Formula: see text]. We only consider diagonal forms of degree [Formula: see text] over fields of characteristic not dividing [Formula: see text].

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45

Aubry, Yves, and Marc Perret. "On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields." Finite Fields and Their Applications 10, no. 3 (July 2004): 412–31. http://dx.doi.org/10.1016/j.ffa.2003.09.005.

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Cheong, Gilyoung. "Weighted distribution of points on cyclic covers of the projective line over finite fields." Finite Fields and Their Applications 57 (May 2019): 29–46. http://dx.doi.org/10.1016/j.ffa.2019.01.005.

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Koike, Masao. "Hypergeometric series over finite fields and Apéry numbers." Hiroshima Mathematical Journal 22, no. 3 (1992): 461–67. http://dx.doi.org/10.32917/hmj/1206128497.

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Coons, Jane Ivy, Jack Jenkins, Douglas Knowles, Rayanne A. Luke, and Patrick X. Rault. "Numerical ranges over finite fields." Linear Algebra and its Applications 501 (July 2016): 37–47. http://dx.doi.org/10.1016/j.laa.2016.03.024.

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MARION, JEAN. "ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY." International Journal of Mathematics 05, no. 03 (June 1994): 329–48. http://dx.doi.org/10.1142/s0129167x9400019x.

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Let M be a compact smooth manifold, let [Formula: see text] be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let [Formula: see text] be the unital involutive algebra [Formula: see text], let [Formula: see text] be an hermitian projective right [Formula: see text]-module of finite type, and let [Formula: see text] be the gauge group of unitary elements of the unital involutive algebra [Formula: see text] of right [Formula: see text]-linear endomorphisms of [Formula: see text]. We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations [Formula: see text] can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group [Formula: see text], [Formula: see text] being the group of unitary elements of [Formula: see text], associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over [Formula: see text]. In the case where [Formula: see text] is a von Neumann algebra of type II 1 a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over [Formula: see text] always exist, and that each of them induces an energy representation.

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Mok, Ngaiming. "Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball." Compositio Mathematica 155, no. 11 (September 19, 2019): 2129–49. http://dx.doi.org/10.1112/s0010437x19007577.

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We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

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