Holomorphic affine vector fields on weil bundles

We obtain necessary and sufficient conditions for a holomorphic vector field to be affine for a holomorphic linear connection defined on aWeil bundle. We also prove that the Lie algebra over R of holomorphic affine vector fields for the natural prolongation of a linear connection from the base to theWeil bundle is isomorphic to the tensor product of theWeil algebra by the Lie algebra of affine vector fields on the base.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Groups definable in local fields and pseudo-finite fields

Using model-theoretic methods we prove: Theorem A If G is a Nash group over the real or p-adic field, then there is a Nash isomorphism between neighbourhoods of the identity of G and of the set of F-rational points of an algebraic group defined over F. Theorem B Let G be a connected affine Nash group over ℝ. Then G is Nash isogeneous with the (real) connected component of the set of real points of an algebraic group defined over ℝ. Theorem C Let G be a group definable in a pseudo-finite field F. Then G is definably virtually isogeneous with the set of F-rational points of an algebraic group defined over F. Both authors supported by NSF grants.     

Sur le produit tensoriel des groupes affines

Let A and B be two commutative affine group schemes over a field. There exists an affine group A?B such that Hom(A?B,C)?Bil(A×B,C) for any affine group C. We use technics of the commutative algebraic groups theory, in order to compute these tensor products and to characterize “flat” groups in the unipotent case. The tensor product of commutative affine groups has most properties of the usual tensor product but it is not always associative. As an application we prove a structure theorem of the category of modules over some affine connected prosmooth rings.     

Relations Between Algebras and Their Subalgebras of Invariants

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.     

On Weierstraß semigroups at one and two points and their corresponding Poincaré series

The aim of this paper is to introduce and investigate the Poincaré series associated with the Weierstraß semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.     

Principal bundles on the affine line

We prove that any principal bundle on the affine line over a perfect field with a reductive group as structure group comes from the base field by base change.     

Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-N degree-one vector bundle over an elliptic curve with n marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For N = 2 and n = 1, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure defined on the direct sum of n copies of sl(N). The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 163–183, August, 2008.     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

On the ample vector bundles over curves in positive characteristic

Let E be an ample vector bundle over a smooth projective curve defined over an algebraically closed field of positive characteristic. We construct a family of curves in the total space of E, parametrized by an affine space, that surjects onto the total space of E and give a deformation of (nonreduced) zero section of E. To cite this article: I. Biswas, A.J. Parameswaran, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Quantum Field Theory on Affine Bundles

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.     

Affine complete varieties are congruence distributive

An algebra is affine complete iff its polynomial operations are the same as all the operations over its universe that are compatible with all its congruences. A variety is affine complete iff all its algebras are. We prove that every affine complete variety is congruence distributive, and give a useful characterization of all arithmetical, affine complete varieties of countable type. We show that affine complete varieties with finite residual bound have enough injectives. We also construct an example of an affine complete variety without finite residual bound.? We prove several results concerning residually finite varieties whose finite algebras are congruence distributive, while leaving open the question whether every such variety must be congruence distributive. Received February 28, 1997; accepted in final form December 9, 1997.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Duality between bent functions and affine functions

A Boolean function in an even number of variables is called bent if it is at the maximal possible Hamming distance from the class of all affine Boolean functions. We prove that there is a duality between bent functions and affine functions. Namely, we show that affine function can be defined as a Boolean function that is at the maximal possible distance from the set of all bent functions.     

On low weight codewords of generalized affine and projective Reed–Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed–Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given.     

Torsion points on elliptic curves over a global field

Let C be an elliptic curve defined over a global field K and denote by C_K the group of rational points of C over K. The classical Nagell-Lutz-Cassels theorem states, in the case of an algebraic number field K as groud field, a necessary condition for a point in C_K to be a torsion point, i.e. a point of finite order. We shall prove here two generalized and strongthened versions of this classical result, one in the case where K is an algebraic number field and another one in the case where K is an algebraic function field. The theorem in the number field case turns out to be particularly useful for actually computing torsion points on given families of elliptic curves.     

On Artin′s Conjecture over Function Fields

We prove an unconditional analog of Artin′s conjecture for the function field of a curve over a finite field.