Tame Symbol Research Articles

Bounding generators for the kernel and cokernel of the tame symbol for curves

Let C be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of C for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.

Hilbert reciprocity using $K$-theory localization

Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol visible as a boundary map.

K2 of families of curves with non-torsion differences in divisorial support

We construct g independent (integral) elements in the kernel of the tame symbol on several families of curves with genus g = 1 , 2 , 4 , 7 . Furthermore, we prove that there exist non-torsion divisors P − Q with P , Q in the divisorial support of these K 2 elements when g = 1 , 2 , which is potentially different from previous constructions.

Tate tame symbol and the joint torsion of commuting operators

We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.

Orthogonal to Principal Ideles

We describe the orthogonal to the group of principal ideles with respect to the global tame symbol pairing on the group of ideles of a smooth projective algebraic curve over a field.

𝐾₂ of certain families of plane quartic curves

We construct three elements in the kernel of the tame symbol on families of quartic curves. We show that these elements are integral under certain conditions on the parameters. Moreover, we prove that these elements are in general linearly independent by calculating the limit of the regulator.

The index map in algebraic K-theory

For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of $K$-theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor relates higher loop groups with higher $K$-theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the $K$-theory torsor to previously studied dimension and determinant torsors.

Functional Calculus and Joint Torsion of Pairs of Almost Commuting Operators

This paper investigates the transformation of determinants of pairs of Fredholm operators with trace class commutators. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we extend some results of Carey and Pincus on determinants of Toeplitz operators and Tate tame symbols. Additionally, we obtain variational formulas for joint torsion.

The two-dimensional Contou-Carrère symbol and reciprocity laws

We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dimensional Contou-Carrère symbol to the two-dimensional class field theory.

Higher adeles and non-abelian Riemann–Roch

We show a Riemann–Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson–Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann–Roch formula as in Kapranov–Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P1. Our set-up provides an extension of several aspects of the classical Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces.

OnK2of Certain Families of Curves

We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have nonhyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol. We also give families of elliptic curves over real quadratic fields with two independent integral elements.

Jacobienne locale d'une courbe formelle relative

This article is devoted to the proof of a relative duality formula on a noetherian scheme S , giving rise on the spectrum of a field S=\operatorname{Spec}\,k to local symbols of class field theory. Relative local symbols are obtained in terms of the universal property of a couple (\Im,f) , of a S -group functor \Im , associated to a S -formal curve \mathfrak X locally of the form \mathfrak X=\operatorname{ Spf}\, A[[T]] ( S=\operatorname{Spec}\,A) . \Im is a S -group extension of the completion \check{W} of the universal S -Witt vectors group W , by the group of units {\cal O}_{S}[[T]]^{*} . We associate an S -functor \Im_{\text omb} to \Im , and we define an Abel–Jacobi morphism f:\mathfrak A=\operatorname{Spec}\ A[[T]][T^{-1}],\longrightarrow \,\Im_{\text omb} , setting up a group isomorphism: \operatorname{Hom}_{S-gr}(\Im,G)\simeq G(\mathfrak A), where G denotes a commutative smooth S -group scheme. We define an S -bihomomorphism \Im \times \Im \longrightarrow \mathbb G_m which is a local symbol (The Tame Symbol), identifying \Im to its own Cartier dual group \check\Im=\underline{\operatorname{Hom}} (\Im;\mathbb G_m) , and inducing the above isomorphism for G=\mathbb G_m . It follows that \Im may be interpreted as the relative Loop Group: \mathbb G_m(\mathfrak A) : S' \longrightarrow \mathbb G_m(\mathfrak A_{\{S'\}}) , S'=\operatorname{Spec} A' denotes a S -scheme, and we write \mathfrak A_{\{S'\}}=\operatorname{Spec} A'[[T]][T^{-1}] , and as the A -universal group of Witt-Bivectors. The couple (\Im,f) may be seen as the local analogue of the relative Rosenlicht Jacobian (Generalized Jacobian) defined by a S -smooth curve X .

A categorical proof of the Parshin reciprocity laws on algebraic surfaces

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

Transfer maps and nonexistence of joint determinant

Transfer Maps, sometimes called norm maps, for Milnor’s K -theory were first defined by Bass and Tate [H. Bass, J. Tate, The Milnor ring of a global field, in: Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic, Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972, Lecture Notes in Math., vol. 342, Springer, Berlin, 1973, pp. 349–446] for simple extensions of fields via tame symbol and Weil’s reciprocity law, but their functoriality had not been settled until Kato [Kazuya Kato, A generalization of local class field theory by using K-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (3) (1980) 603–683]. On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor’s K-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, Q , R and C into the respective group of units of given field.

Geometry of polysymbols

We introduce a multiple generalization of the tame symbol, called {\em polysymbols}, associated to meromorphic functions on a Riemann surface as the Massey products in Deligne cohomology, and also give a geometric construction of polysymbols using Chen's iterated integrals. We then deduce some basic properties of polysymbols using our holonomy formula, and show that trivializations of polysymbols give variations of mixed Hodge structure.

Hermitian-holomorphic (2)-gerbes and tame symbols

The tame symbol of two invertible holomorphic functions can be obtained by computing their cup product in Deligne cohomology, and it is geometrically interpreted as a holomorphic line bundle with connection. In a similar vein, certain higher tame symbols later considered by Brylinski and McLaughlin are geometrically interpreted as holomorphic gerbes and 2-gerbes with abelian band and a suitable connective structure. In this paper we observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit “reduction of the structure group”. Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well. We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitian-holomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.

Residues and tame symbols on toroidal varieties

We introduce a new approach to the study of a system of algebraic equations in whose Newton polytopes have sufficiently general relative positions. Our method is based on the theory of Parshin's residues and tame symbols on toroidal varieties. It provides a uniform algebraic explanation of the recent result of Khovanskii on the product of the roots of such systems and the Gel'fond–Khovanskii result on the sum of the values of a Laurent polynomial over the roots of such systems, and extends them to the case of an algebraically closed field of arbitrary characteristic.

An Algebraic-Geometric Method for Constructing Generalized Local Symbols on Curves

The aim of this work is to provide an algebraic-geometric method to construct generalized local symbols on curves as morphisms of group schemes. From a closed point of a complete, irreducible, and nonsingular curve C over a perfect field k as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol.

Algebraic construction of the tame symbol and the Parshin symbol on a surface

The aim of this work is to offer a new characterization of the tame symbol associated with a discrete valuation from the commutator of a certain central extension of groups. This construction allows us to generalize recent constructions of the tame symbol of an algebraic curve to the arithmetical case, and to offer an easy algebraic proof of a Parshin Reciprocity Law on an algebraic surface with the same method as Tate's proof of the Residue Theorem.

ON THE TAME SYMBOL OF AN ALGEBRAIC CURVE

ABSTRACT The aim of this work is to give a new definition of the tame symbol of an algebraic curve from the commutator of a certain central extension of groups. The definition offered is valid for each closed point of an irreducible non singular curve over a perfect field. Moreover, when the curve is complete, the reciprocity law can be deduced from the finiteness of the cohomology groups and .