Fernando Pablos Romo

Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring

Abstract The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

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 .

GENERALIZED RECIPROCITY LAWS

The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank n vector bundles.

A generalization of the Contou-Carrère symbol

Using techniques of Algebraic Geometry, the aim of this paper is to give a generalized definition of the Contou-Carrère symbol as a morphism of schemes. In fact, from formal schemes and Heisenberg groups, we provide a new definition of the Contou-Carrère symbol and a generalization of it associated with a separable extension. Moreover, a reciprocity law is proved and classical explicit reciprocity laws are recovered from it.

A negative answer to the question of the linearity of Tate’s Trace for the sum of two endomorphisms

The aim of this note is to solve a problem proposed by J. Tate in 1968 by offering a counter-example of the linearity of the trace for the sum of two finite potent operators on an infinite-dimensional vector space.

On the linearity property of Tate's trace

The aim of this article is to show that Tates trace of finite-potent endomorphisms of infinite-dimensional vector spaces is not linear.The aim of this article is to show that Tates trace of finite-potent endomorphisms of infinite-dimensional vector spaces is not linear.

A Note on Steinberg Symbols on Algebraic Curves

Abstract The aim of this paper is to provide a method for defining Steinberg symbols on a complete algebraic curve over a perfect field k from the commutator of a certain extension of groups. This extension is associated with a group morphism ϕ : k* → G. With this definition the reciprocity law is a consequence of the finiteness of the cohomology groups H 0(C, 𝒪 C ) and H 1(C, 𝒪 C ). Using this method, Hilberts norm residue symbol on an algebraic curve and the symbol (a, b) v for the field ℚ p (n = 2) can be defined.

On n-Dimensional Steinberg Symbols

The aim of this work is to provide a new approach for constructing n-dimensionalSteinberg symbols on discrete valuation fields from (n + 1)-cocycles and to study reciprocity laws on curves related to these symbols.

An Explicit Reciprocity Law Associated to Some Finite Coverings of Algebraic Curves

We provide a new reciprocity law associated with finite coverings of algebraic curves. Moreover, we give explicit examples of this new reciprocity law that are not trivial consequences of the Weil reciprocity law over the base curve.

On the Drazin inverse of finite potent endomorphisms

ABSTRACT The aim of this work is to prove the existence of Drazin inverses for matrices associated with finite potent endomorphisms on arbitrary vector spaces. The result offered coincides with the classical Drazin inverse matrix for finite-dimensional vector spaces over .