Mostrando recursos 1 - 20 de 572

  1. Correction to a proof in the article Patching and admissibility over two-dimensional complete local domains

    Neftin, Danny; Paran, Elad
    The proof of Lemma 1.8 of the article in the title is incorrect. We supply an alternate argument for Proposition 1.10, whose proof invoked that lemma.

  2. The basic geometry of Witt vectors, I The affine case

    Borger, James
    We give a concrete description of the category of étale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical [math] -typical and big Witt vector functors but also for certain analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for classical Witt vectors. Our larger purpose is to provide the affine foundations for the algebraic geometry...

  3. Elliptic nets and elliptic curves

    Stange, Katherine
    An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose [math] is an elliptic curve over a field [math] , and [math] are points on [math] defined over [math] . To this information we associate an [math] -dimensional array of values in [math] satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic...

  4. Local positivity, multiplier ideals, and syzygies of abelian varieties

    Lazarsfeld, Robert; Pareschi, Giuseppe; Popa, Mihnea
    We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

  5. On the Hom-form of Grothendieck's birational anabelian conjecture in positive characteristic

    Saïdi, Mohamed; Tamagawa, Akio
    We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arises from embeddings between the function fields.

  6. On the cluster category of a marked surface without punctures

    Brüstle, Thomas; Zhang, Jie
    We study the cluster category [math] of a marked surface [math] without punctures. We explicitly describe the objects in [math] as direct sums of homotopy classes of curves in [math] and one-parameter families related to noncontractible closed curves in [math] . Moreover, we describe the Auslander–Reiten structure of the category [math] in geometric terms and show that the objects without self-extensions in [math] correspond to curves in [math] without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

  7. Explicit CM theory for level 2-structures on abelian surfaces

    Bröker, Reinier; Gruenewald, David; Lauter, Kristin
    For a complex abelian surface [math] with endomorphism ring isomorphic to the maximal order in a quartic CM field [math] , the Igusa invariants [math] generate an unramified abelian extension of the reflex field of  [math] . In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on [math] . Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials,...

  8. Specializations of elliptic surfaces, and divisibility in the Mordell–Weil group

    Ingram, Patrick
    Let [math] be an elliptic surface defined over a number field [math] , let [math] be a section, and let [math] be a rational prime. We bound the number of points of low algebraic degree in the [math] -division hull of [math] at the fibre [math] . Specifically, for [math] with [math] such that [math] is nonsingular, we obtain a bound on the number of [math] such that [math] , and such that [math] for some [math] . This bound depends on [math] , [math] , [math] , [math] , and [math] , but is independent of [math] .

  9. Global descent obstructions for varieties

    Couveignes, Jean-Marc; Hallouin, Emmanuel
    We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having [math] as field of moduli, that admit models over every completion of  [math] , but have no model over  [math] .

  10. The number of 2$\times$2 integer matrices having a prescribed integer eigenvalue

    Martin, Greg; Wong, Erick
    What is the probability that an integer matrix chosen at random has a particular integer as an eigenvalue, or an integer eigenvalue at all? For a random real matrix, what is the probability of there being a real eigenvalue in a particular interval? This paper solves these questions for [math] matrices, after specifying the probability distribution suitably.

  11. On Oliver's $p$-group conjecture

    Green, David; Héthelyi, László; Lilienthal, Markus
    Let [math] be a [math] -group for an odd prime  [math] . B. Oliver conjectures that a certain characteristic subgroup [math] always contains the Thompson subgroup  [math] . We obtain a reformulation of the conjecture as a statement about modular representations of [math] -groups. Using this we verify Oliver’s conjecture for groups where [math] has nilpotence class at most two.

  12. Inner derivations of alternative algebras over commutative rings

    Loos, Ottmar; Petersson, Holger; Racine, Michel
    We define Lie multiplication derivations of an arbitrary non-associative algebra [math] over any commutative ring and, following an approach due to K. McCrimmon, describe them completely if [math] is alternative. Using this description, we propose a new definition of inner derivations for alternative algebras, among which Schafer’s standard derivations and McCrimmon’s associator derivations occupy a special place, the latter being particularly useful to resolve difficulties in characteristic [math] . We also show that octonion algebras over any commutative ring have only associator derivations.

  13. Smooth curves having a large automorphism $p$-group in characteristic $p\gt 0$

    Matignon, Michel; Rocher, Magali
    Let [math] be an algebraically closed field of characteristic [math] and [math] a connected nonsingular projective curve over [math] with genus [math] . This paper continues our study of big actions, that is, pairs [math] where [math] is a [math] -subgroup of the [math] -automorphism group of [math] such that [math] . If [math] denotes the second ramification group of [math] at the unique ramification point of the cover [math] , we display necessary conditions on [math] for [math] to be a big action, which allows us to pursue the classification of big actions. ¶ Our main source of examples comes...

  14. Integral points on hyperelliptic curves

    Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir; Stoll, Michael; Tengely, Szabolcs
    Let [math] be a hyperelliptic curve with the [math] rational integers, [math] , and the polynomial on the right-hand side irreducible. Let [math] be its Jacobian. We give a completely explicit upper bound for the integral points on the model [math] , provided we know at least one rational point on [math] and a Mordell–Weil basis for [math] . We also explain a powerful refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus [math] hyperelliptic models [math] and...

  15. Centers of graded fusion categories

    Gelaki, Shlomo; Naidu, Deepak; Nikshych, Dmitri
    Let [math] be a fusion category faithfully graded by a finite group  [math] and let [math] be the trivial component of this grading. The center [math] of [math] is shown to be canonically equivalent to a [math] -equivariantization of the relative center [math] . We use this result to obtain a criterion for [math] to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in [math] -matrices of weakly integral modular categories.

  16. Log minimal models according to Shokurov

    Birkar, Caucher
    Following Shokurov’s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension [math] implies that any klt pair of dimension [math] has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension [math] has a log minimal model or a Mori fibre space.

  17. F-adjunction

    Schwede, Karl
    In this paper we study singularities defined by the action of Frobenius in characteristic [math] . We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if [math] is a Gorenstein normal variety then to every normal center of sharp [math] -purity [math] such that [math] is [math] -pure at the generic point of [math] , there exists a canonically defined [math] -divisor [math] on [math] satisfying [math] . Furthermore, the singularities of [math] near [math] are “the same” as the singularities of [math] . As an application, we show that...

  18. Exponential sums nondegenerate relative to a lattice

    Adolphson, Alan; Sperber, Steven
    Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by [math] . We remedy that defect in this paper by systematically applying [math] -power reduction, making it possible to strengthen and extend our earlier results.

  19. On coproducts in varieties, quasivarieties and prevarieties

    Bergman, George
    If the free algebra [math] on one generator in a variety [math] of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if [math] generates the variety [math] . ¶ Generalizing the argument, it is shown that if we are given an algebra and subalgebras, [math] , in a prevariety ( [math] -closed class of algebras) [math] such that [math] generates [math] , and also subalgebras [math] [math] such that for each [math] the subalgebra of [math] generated by [math] and...

  20. A 2-block splitting in alternating groups

    Bessenrodt, Christine
    In 1956, Brauer showed that there is a partitioning of the [math] -regular conjugacy classes of a group according to the [math] -blocks of its irreducible characters with close connections to the block theoretical invariants. In a previous paper, the first explicit block splitting of regular classes for a family of groups was given for the 2-regular classes of the symmetric groups. Based on this work, the corresponding splitting problem is investigated here for the 2-regular classes of the alternating groups. As an application, an easy combinatorial formula for the elementary divisors of the Cartan matrix of the alternating groups...

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