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Groupes algébriques et corps de classes. 2e ed. revue et corrigee
... This universal isogeny is given by the Artin-Schreier map ℘ (= Frobenius minus identity), and the abelianized fundamental group is given by the kernel of ℘, which is the group of k-valued points of Gu ab X (cf. [Se,VI,[6][7][8][9][10][11][12]in part. No. 11,Prop. ...
... For the Sec. 6 purpose of Galois coverings, it is sufficient to work with a universal affine group Lu ab X (see [Se,VI,No. 8]). Then the central problem is the construction of a Sec. 5 non-commutative universal affine pro-algebraic group Lu X , i.e., a group satisfying the universal factorization property even for non-commutative groups. ...
... This is possible, since the Artin-Schreier map (the commutative version of the Lang map ℘ : x −→ x −1 F(x), where F is the Frobenius) is a homomorphism of groups, providing a universal isogeny (cf. [Se,VI,No. 6,Prop. 6]). ...
The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \'etale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a projective variety X a (non-commutative) universal affine pro-algebraic group Lu(X), such that for any given affine subvariety U of X any finite and \'etale Galois covering of U over F is a pull-back of a Galois covering of a quotient Lu(X,U) of Lu(X). Then the geometric fundamental group of U is a completion of the k-points of Lu(X,U), where k is an algebraic closure of F. We obtain explicit descriptions of the universal affine groups Lu(X,U) for U the affine line and the punctured affine line over F.
... There is a classical theory of 0-cycles on a smooth complete curve C with a modulus condition at a finite set of points S , due to Rosenlicht and Serre [109,III]. The idea is quite simple, instead of relations coming from divisors (zeros minus poles) of an arbitrary rational function f , f is required to have a power series expansion at each point p 2 S , with leading term 1 and the next nonzero term of the form ut n p p , with u.p/ ¤ 0, t p a local coordinate at p and the integer n p > 0 being the "modulus." ...
... The idea is quite simple, instead of relations coming from divisors (zeros minus poles) of an arbitrary rational function f , f is required to have a power series expansion at each point p 2 S , with leading term 1 and the next nonzero term of the form ut n p p , with u.p/ ¤ 0, t p a local coordinate at p and the integer n p > 0 being the "modulus." This is applied to the class field theory of a smooth open curve U C over a finite field [109,Theorem 4], that identifies the inverse limit of the groups of degree 0 cycle classes on U , with modulus supported in C n U , with the kernel of the map ét 1 .U / ab ! Gal. ...
... The notion of a reciprocity sheaf and its relation to motives with modulus goes back to the theorem of Rosenlicht-Serre. In our discussion of reciprocity sheaves, we work over a fixed perfect field k. [109,III]). Let k be a perfect field, let C be a smooth complete curve over k, let G be an smooth commutative algebraic group over k, and let f W C Ü G be a rational map over k. ...
... Note that in the product (10) only finitely many factors differ from 1. The last fact can be generalized to the case of smooth projective algebraic curves over arbitrary fields (see, for example, [62], § III.4). This is how the famous Weil reciprocity law arises, which is a geometric analogue of the Gauss quadratic reciprocity law and the Artin reciprocity law in algebraic number theory. ...
... When A is a field, the invariance of the residue with respect to continuous automorphisms of the field L n (A) over A is a classical statement for n = 1 (see, for example, [62], Chapter 2, § 11), and is also known for n ⩾ 2 from works of Parshin, Lomadze, and Yekutieli (see [57], Proposition 1.1, [43], Lemma 6, (VIII), and [67], Theorem 2.4.3). ...
... is satisfied if for any closed point x ∈ Spec(S), the identity component of the algebraic group G κ(x) is an extension of an Abelian variety by a unipotent group. Indeed, by the corollary in § 8 of Chapter III in [62], any rational map from a projective line to an Abelian variety is constant, and by Corollary IV.2.2.4 in [14] any homomorphism from the multiplicative group to the unipotent group is trivial. Remark 7.1.8. ...
This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol.
The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case 1$> , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described.
As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol.
Bibliography: 67 titles.
... The Riemann-Roch theorem (see Proposition IV.4, p. 75 in [22]) says ...
... In fact, let g i be the geometric genus of A i , that is, the genus of the desingularization of A i . Then the adjunction formula (see Proposition IV, 5, p. 75 in [22]) says ...
Let be a normal surface singularity. Let be a minimal good resolution of V. The weighted dual graphs associated with A completely describes the topology and differentiable structure of the embedding of A in M. In this paper, we classify all the weighted dual graphs of such that one of the curves is a -curve, and all the remaining ones are -curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.
... (1) In case (i), c F L agrees with the Rosenlicht-Serre conductor [47] if L has perfect residue field. If ch(k) = p is positive and F = W n is the group scheme of p-typical Witt vectors of length n, then c F L agrees with a conductor defined by Kato and Russell in [32] for any L. ...
... symbol (−,−) L sh ι ,σ from Section 4.37, with σ :k ֒→ O sh L,ι the unique coefficient field; in this case this is the symbol defined by Rosenlicht and Serre (see[47, III,§1]). If we choose a different k-embedding ι ′ : κ ֒→k, then we find an automorphism τ :k →k34 K. Rülling and S. Saito with τ • ι = ι ′ inducing a unique isomorphism of O L -algebras τ : ...
We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on F(L) , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X , then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank 1 connections (in characteristic 0 ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.
... Then X is a compact Hausdorff space. We define a modulus with support S, according to Serre([16]). Definition 2.1. A modulus m with support S is the data of an integer m(P ) > 0 for each point P ∈ S. A modulus m with support S is also considered as a positive divisor on X. ...
... The toroidal group G 0 has the structure of principal (C * ) r−s -bundle σ : G 0 −→ T over an sdimensional complex torus T. Replacing fibres (C * ) r−s with (P 1 ) r−s , we obtain the associated (P 1 ) r−s -bundle σ : G 0 −→ T. We call G = (P 1 ) p+q × G 0 the standard compactification of G. The following theorem is considered as an analytic version of Théorème 1 in [16]. Theorem 5.20. ...
We study singular curves from analytic point of view. We give completely analytic proofs for the Serre duality and a generalized Abel's theorem. We also reconsider Picard varieties, Albanese varieties and generalized Jacobi varieties of singular curves analytically. We call an Albanese variety considered as a complex Lie group an analytic Albanese variety. We investigate them in detail. For a non-singular curve (a compact Riemann surface) X, there is the relation between the meromorphic function fields on X and on its Jacobi variety J(X). We extend this relation to the case of singular curves.
... This last equality follows from the Gorenstein property [21]. Therefore, we have q −1 c = (ϕ * (OX (−F − Z K ))) 0 . ...
... The map λ G is injective, and its image is the set of isomorphism classes of principal G-bundles admitting a rigidification (see [Ser59,Théorème 15.5]). ...
Let A be an abelian variety over a complete non-Archimedean field K. The universal cover of the Berkovich space attached to A reflects the reduction behaviour of A. In this paper the universal cover of the universal vector extension E(A) of A is described. In a forthcoming paper ( arXiv:2007.04659), this will be one of the crucial tools to show that rigid analytic functions on E(A) are all constant.
Comment: This is the first part of arXiv:2007.04659 which is now split into two
... [BLR,Proposition 9.2.9] treats the case where K is perfect and states that Ker (Pic C ′ /K → Pic C/K ) is unipotent. In the proof of [BLR, Proposition 9.2.9], results of [Ser1] are used and we can see that Ker (Pic C ′ /K → Pic C/K ) is also connected and smooth. ...
Let X be a regular geometrically integral variety over an imperfect field K. Unlike the case of characteristic 0, X':=X\times_{\Spec K}\Spec K' may have singular points for a (necessarily inseparable) field extension . In this paper, we define new invariants of the local rings of codimension 1 points of , and use these invariants for the calculation of -invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and [PW,Theorem 1.2]. Moreover, when X is a curve, we show that the Jacobian number of X is times of the genus change by using the above calculation. In this case, we also relate the structure of the Picard scheme of X to invariants of singular points of X. To prove such a relation, we give a characterization of geometrical normality of algebras over fields of positive characteristic.
... For further details on quotient varieties see for instance [16]. In [12] and [4], the authors studied a particular class of Togliatti systems arising from actions of the cyclic group over R. In particular, let d ≥ 3 and 1 ≤ a < b ≤ d − 1 be integers such that GCD(a, b, d) = 1. ...
In this note, we study Togliatti systems generated by invariants of the dihedral group acting on . This leads to the first family of non monomial Togliatti systems, which we call systems with group . We study their associated varieties , called surfaces with group . We prove that they are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, , is minimally generated by quadrics and we find a minimal free resolution of .
... Proof. See [20,Section 12,Proposition 19]. ...
Given any diagonal cyclic subgroup of order d, let be the ideal generated by all monomials of degree d which are invariants of . is a monomial Togliatti system, provided , and in this case the projective toric variety parameterized by is called a GT-variety with group . We prove that all these GT-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case n=2, we compute explicitly the Hilbert function, polynomial and series of . We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.
... where δ = P δ(P ) is the sum of so-called δ-invariants of singular points of C. The analytic invariant δ(P ) is positive if and only if the point P is singular (see [31,§1,Prop. 1]). The δ-invariant admits several definitions. ...
The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are . Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for μ ≠ 0 this happens exactly when and the parameters (λ, μ) lie on an algebraic curve called the l-spectral curve and defined as zero locus of determinant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dynamical systems on 2-torus depending on real parameters (B, A; ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every . We also calculate its genus for and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in . We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.
... This theorem reduces to the case where Λ is finite, which was originally proved by Serre and Lang, cf. ( [La56], 6) and [Se59], by using the Albanese property of Rosenlicht's generalized Picard schemes [Ro54]. Deligne gave another proof in the tamely ramified case. ...
Inspired by the work of Laumon on -factors and by Deligne's 1974 letter to Serre, we give an explicit cohomological definition of -factors for -adic Galois representations over henselian discrete valuation fields of positive equicharacteristic , with (not necessarily finite) perfect residue fields. These geometric local -factors are completely characterized by an explicit list of purely local properties, such as an induction formula and the compatibility with geometric class field theory in rank 1, and satisfy a product formula for -adic sheaves on a curve over a perfect field of characteristic p.
... It is clear that X is singular provided Z is non-empty and that X ν −→ X is the normalization map. It always exists, as follows from [50]. ...
In this paper, we develop a geometric approach to study derived tame finite dimensional associative algebras, based on the theory of non-commutative nodal curves.
... The case where S is the spectrum of a perfect field is originally due Serre and Lang, cf. ( [La56], 6) and [Se59]. Their proof relies on the Albanese property of Rosenlicht's generalized Jacobians [Ro54]. ...
We generalize Deligne's approach to tame geometric class field theory to the case of a relative curve, with arbitrary ramification.
... It was developped by Rosenlicht and Lang [Ser75]. Here we consider only the unramified case. ...
We discuss recent developments in the Langlands program for function fields, and in the geometric Langlands program. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G over a function field, indexed by global Langlands parameters. The proof uses the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence.
... This result will be used in the proof of Proposition 3.3. Let J D (X) be the generalized Jacobian variety of X with respect to the divisor D [29]. Note that since D is reduced, J D (X) is a semi-abelian variety over k. ...
Let U be a smooth curve over a complete discrete valuation ring R with algebraically closed residue field k of characteristic together with a smooth compactification X with ¥'etale over R. Let denote its special fiber.
Grothendieck's specialization theorem for the prime-to-p ¥'etale fundamental group includes the fact that any prime-to-p Galois covering of can be uniquely lifted to a prime-to-p covering of U.
In this paper, we will ask such a lifting problem for finite linearly reductive torsors and prove that the existence of a lifting is true if one allows a finite extension of R. We will also observe that the unicity of a lifting cannot be valid in general even in the equal characteristic case and discuss on a moduli-dependency of the corresponding fundamental group scheme as a related topic.
... We have tp*df = 0. Hence <P¡(gf%z) = -y'(JL), and since either rfa or |£ is not divisible by fi, it represents a meromorphic 1-form on A. We let Co(C, C¡) be the order of the pole at 0 of this meromorphic 1-form. Note that it is zero if and only if 0 is a By a similar computation of intersection numbers as in [7], p.218, it is shown that (1.9) co(C) = po(C) + r -I, where Po(C) denotes the Milnor number of C at 0. Especially we have co(C¡, C¡) = po(Ci), the Milnor number of C, at 0. We remark that the invariant cq(C) coincides with 2r5 ([4], Appendix I), where ô is the invariant defined in [10], Chapter IV, §1, and that the identity (1.9) coincides with the one proved in [8], Theorem 10.5. From Proposition (1.4) and the identities (1.8) and (1.9), we may write Let X be a complex analytic surface, i.e., a complex manifold of complex dimension two. ...
The local index of a holomorphic vector field relative to a (possibly singular) invariant complex analytic curve on a complex surface is defined and it is shown that, for a compact curve invariant by a one-dimensional singular foliation on a surface, the sum of the indices is equal to its self-intersection number. An interpretation of the indices in terms of holonomy is also given.
... 1.0.2. Let now X be a quasi-projective smooth k-variety equipped with an effective (and possibly non-reduced) Cartier divisor D. Let CH 0 (X|D) be the relative Kerz-Saito Chow group of 0-cycles on X with modulus D. This is defined as quotient of the free abelian group of closed points of X \ D modulo the subgroup generated by cycles of the form ν C, * (divf ), where C is an integral curve in X, properly meeting D, and f is a rational function on C that is congruent to 1 modulo the pull back to the normalization of C of the divisor D. When X is a projective curve over a k, the group CH 0 (X|D) is by definition the group of divisors on X − D having a fixed trivialization along D and agrees with the group of k-rational points of the Rosenlicht-Serre generalized Jacobian Jac(X|D) as defined in [13]. ...
In this note we show that given a smooth affine variety X over an algebraically closed field k and an effective (possibly non reduced) Cartier divisor D on it, the Kerz-Saito Chow group of zero cycles with modulus is torsion free, except possibly for p-torsion if the characteristic of k is . This generalizes to the relative setting classical theorems of Rojtman (for X smooth) and of Levine (for X singular). A stronger version of this result, that encompasses p-torsion as well, was proven with a different and more sophisticated method by A. Krishna and the author in another paper.
... One can see that the epimorphic map A 0 → A 1 , which splits the lower exact sequence, is induced by the morphism p 1 − p 2 : A A2 A → A. Finally, one can apply Serre's arguments [30] to obtain for any short exact ...
Suppose F = W(k)[1/p] where W (k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p not equal 2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of the Shafarevich Conjecture. If Y is a projective algebraic variety over Q with good reduction modulo all primes l not equal 3 and semi-stable reduction modulo 3 then for the Hodge numbers of Y-C = Y circle times C-Q, one has h(2)(Y-C) = h(1,1)(Y-C).
For a smooth and proper variety X over an algebraically closed field k of characteristic , the group is a direct sum of finitely many copies of and an abelian group of finite exponent. The latter is an extension of a finite group J by the group of k-points of a connected commutative unipotent algebraic group U. In this paper we show that (1) if X is ordinary, then ; (2) if X is a surface, then J is the Pontryagin dual of ; (3) if X is an abelian variety, then . Using Crew's formula, we compute for Enriques surfaces and abelian 3-folds. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree 2.
This introductory chapter has no new results and contains the main objects, results and tools that we shall use in the forthcoming chapters. The Cohen–Macaulay property is presented, the notions of a semigroup and a semigroup ring are introduced, the invariant theory of finite groups and the weak Lefschetz property are reviewed.
Max Noether’s Theorem asserts that if ω is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve, then the natural morphisms SymnH0(ω)→H0(ωn) are surjective for all n≥1. The result was extended for Gorenstein curves by many different authors in distinct ways. More recently, it was proved for curves with projectively normal canonical models, and curves whose non-Gorenstein points are bibranch at worse. Based on those works, we address the combinatorics of the general case and extend the result for any integral curve.
In this note, we study Togliatti systems generated by invariants of the dihedral group D2d acting on k[x0, x1, x2]. This leads to the first family of non-monomial Togliatti systems, which we call GT-systems with group D2d. We study their associated varieties , called GT-surfaces with group D2d. We prove that there are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, , is minimally generated by quadrics and we find a minimal free resolution of .
Given any diagonal cyclic subgroup of order d, let be the ideal generated by all monomials of degree d which are invariants of . is a monomial Togliatti system, provided , and in this case the projective toric variety parameterized by is called a GT-variety with group . We prove that all these GT-varieties are arithmetically Cohen–Macaulay and we give a combinatorial expression of their Hilbert functions. In the case n=2, we compute explicitly the Hilbert function, polynomial and series of . We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.
Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber . We give a formula that relates the dimension of the first Arakelov-Chow vector space of X with the Mordell-Weil rank of the Albanese variety of and the rank of the N\'eron-Severi group of . This is a higher dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such analogy is strengthened by the fact we show that the only numerically trivial arithmetic -divisors on X are linear combinations of principal ones.
We prove that for any prime p there exists an algebraic action of the two-dimensional Witt group W2 (p) on an algebraic variety X such that the closure in X of the W2(p)-orbit of some point x ∈ X contains infinitely many W2(p)-orbits. This is related to the problem of extending, from the case of characteristic zero to the case of characteristic p, the classification of connected affine algebraic groups G such that every algebraic G-variety with a dense open G-orbit contains only finitely many G-orbits.
В работе доказано, что для всякого простого числа p существуют алгебраическое действие двумерной группы Витта W_2(p) на алгебраическом многообразии X и точка x ∈ X такие, что замыкание W_2(p)-орбиты точки x в X содержит бесконечно много W_2(p)-орбит. Это связано с задачей о распространении со случая нулевой характеристики на случай характеристики p
классификации таких связных аффинных алгебраических групп G, что всякое алгебраическое G-многообразие с открытой плотной G-орбитой содержит только конечное множество G-орбит.
Dans cette thèse nous étudions la filtration par le poids sur la cohomologie de De Rham d’une courbe hyperelliptique C définie sur une extension finie de Qp et à réduction semi-stable. L’objectif est de fournir des algorithmes calculant explicitement, étant donné une équation de C, les bases des crans de la filtration par le poids ainsi que la matrice de l’accouplement de Poincaré. Dans le premier chapitre, nous mettons en place des outils relatifs à la cohomologie de De Rham algébrique de la courbe hyperelliptique. Nous construisons une base adaptée de la cohomologie de De Rham de C, nous établissons une formule explicite pour le cup-produit et la trace, et enfin nous proposons un algorithme calculant la matrice de l’accouplement de Poincaré. Le deuxième chapitre est consacré à la description explicite de la flèche induite par l’inclusion du tube d’un point double sur les espaces de cohomologie. C’est l’ingrédient essentiel pour pouvoir décrire la filtration par le poids sur la cohomologie de De Rham de C. À cette fin nous nous plaçons dans le cadre de la géométrie analytique à la Berkovich et nous introduisons puis développons les notions de point résiduellement singulier standard et de forme apparente de l’équation de la courbe. Dans le troisième et dernier chapitre, nous faisons la synthèse des résultats obtenus et achevons la description de la filtration par le poids. Enfin, nous donnons les algorithmes calculant les bases de Fil0 et Fil1. Pour les algorithmes obtenus dans la thèse nous proposons une implémentation en sage, ainsi que des exemples concrets sur des courbes de genre un et deux.
Krall–Charlier polynomials (cna;F)n are orthogonal polynomials which are also eigenfunctions of a higher order difference operator. They are defined from a parameter a (associated to the Charlier polynomials) and a finite set F of positive integers. We study the algebra DaF formed by all difference operators with respect to which the family of Krall–Charlier polynomials (cna;F)n are eigenfunctions. Each operator D∈DaF is characterized by the so called eigenvalue polynomial λD: λD is the polynomial satisfying D(cna;F)=λD(n)cna;F. We characterize the algebra of difference operators DaF by means of the algebra of polynomials D̃aF={λ∈C[x]:λ(x)=λD(x),D∈DaF}. We associate to the family (cna;F)n a polynomial ΩFa and prove that, except for degenerate cases, the algebra D̃aF is formed by all polynomials λ(x) such that ΩFa divides λ(x)−λ(x−1). We prove that this is always the case for a segment F (i.e., the elements of F are consecutive positive integers), and conjecture that it is also the case when the Krall–Charlier polynomials (cna;F)n are orthogonal with respect to a positive measure.
Deninger et Werner ont développé un analogue pour les courbes p -adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré 0 et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellement à chaque fibré vectoriel sur une courbe p -adique, dont la réduction est fortement semi-stable de degré 0 , une représentation p -adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson p -adique développée par Faltings ? Nous répondons à ces questions dans cet article.
In this note we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Ostvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives.
We classify the pairs (C,G) where C is a seminormal curve over an arbitrary field k and G is a smooth connected algebraic group acting faithfully on C with a dense orbit, and we determine the equivariant Picard group of C. We also give a partial classification when C is no longer assumed to be seminormal.
Throughout this chapter G denotes a connected affine group, and all algebraic groups are understood to be affine.
In this section all algebraic groups are affine. The results here prepare the way for the construction of quotients in §6.
In this chapter, all algebraic groups are affine, unless the contrary is explicitly allowed. G is a k-group.
Consider an isolated two-dimensional complex analytic hypersurface singularity (V, p). A relation is given between the abstract topology of (V, p) and the multiplicity of (V, p), yielding an upper bound for the multiplicity. This relation is a necessary condition for a Gorenstein singularity to be a hypersurface.
In this brief survey on class field theory and related questions we mainly present the papers reviewed in the “Mathematics” section of Referativnyi Zhurnal during 1958–1967. Among the books published during this time we note those by Chevalley [20] (a systematic exposition and application of cohomology groups), Artin and Tate [12] (the most modern exposition of class field theory with the application of cohomology groups and of class systems; the ground field is either the field of algebraic numbers of finite degree or the field of functions of one variable over a finite field of constants), and Serre [99] (the connection of class field theory with algebraic curves). In 1963 there appeared a survey by Ribenboim [93] who set forth the results in the Hilbert — Takagi theory and the reciprocity law of Artin and who examined the problem of finding functions whose values could be generated by any Abelian extension of the field of algebraic numbers. On local class field theory we have the survey by Hochschild [43] (also see Samuel’s report [94] on Hochschild’s results: G. Hochschild, “Local class field theory,” Ann. Math., 51(2):331–347 (1950), where local class field theory is presented with the help of cohomologies).
Let S be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over S. We first compute a d\'evissage of the Picard group of a 1-motive M according to the weight filtration of M. This d\'evissage allows us to associate, to each line bundle L on M, a linear morphism from M to its Cartier dual. This yields a group homomorphism . We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism . Finally we prove that these two independent constructions of linear morphisms using line bundles on M coincide. However, the first construction, involving the d\'evissage of Pic(M), is more explicit and geometric and it furnishes the motivic origin of some linear morphisms between 1-motives. The second construction, involving the Theorem of the Cube, is more abstract but perhaps also more enlightening.
Consider the abelian category of commutative group schemes of finite type over a field k. By results of Serre and Oort, has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of by the full subcategory of finite k-group schemes. We show that has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for , which take a simpler form in positive characteristics.
Let X be a smooth projective curve over k. A divisor is a formal linear combination where the sum is over all closed points of X, the coefficients are integers and are almost always zero. We can add divisors formally and obtain a group: the group of divisors Div(X). A divisor is called effective if all nP are non-negative. The degree of a divisor is with deg(P) = [kv:k] the degree of P. (Recall that kv =k(P) is the residue field of P, see Lecture 1.) The subgroup of divisors of degree zero is denoted Div0(X).
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