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Fernando Pablos Romo

Publications and source records attributed to Fernando Pablos Romo.

12 recordsLinked to original sources
arXiv2021

On Bounded Finite Potent Operators on arbitrary Hilbert Spaces

The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate's trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.

math.FA↗
arXiv2020

On G-Drazin inverses of Finite Potent Endomorphisms and Arbitrary Square Matrices

The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix.

math.RA↗
arXiv2019

An idelic quotient related to Weil reciprocity and the Picard group

This paper studies the function field of an algebraic curve over an arbitrary perfect field by using the Weil reciprocity law and topologies on the adele ring. A topological subgroup of the idele class group is introduced and it is shown how it encodes arithmetic properties of the base field and of the Picard group of the curve. These results are applied to study extensions of the function field.

math.AG↗
arXiv2013

Classification of Finite Potent Endomorphisms

The aim of this work is to offer a family of invariants that allows us to classify finite potent endomorphisms on arbitrary vector spaces, generalizing the classification of endomorphisms on finite-dimensional vector spaces. As a particular case we classify nilpotent endomorphisms on infinite-dimensional vector spaces.

math.RA↗
arXiv2013

A General Reciprocity Law for Symbols on Arbitrary Vector Spaces

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces, and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov - M. Kerr and the author -together with D. Hernández Serrano-, can also be deduced from this general expression.

math.NT↗
arXiv2013

Determinants of finite potent endomorphisms, symbols and reciprocity laws

The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck's determinant for finite rank endomorphisms and is equivalent to the classic analytic definitions. The theory can be interpreted as a multiplicative analogue to Tate's formalism of abstract residues in terms of traces of finite potent linear operators on infinite-dimensional vector spaces, and allows us to relate Tate's theory to the Segal-Wilson pairing in the context of loop groups.

math.RA↗
arXiv2006

On the Steinberg property of the Contou-Carrere symbol

The aim of this work is to show that the Contou-Carrère symbol satisfies the property of Steinberg symbols: $ _{A((t))^\times} = 1$ for all elements $f,1-f \in A((t))^\times$. Moreover, we offer a cohomological characterization of the Contou-Carrère symbol from the commutator of a central extension of groups.

math.AC↗
arXiv2004

Geometric Construction of Generalized Local Symbols on Algebraic Curves

The aim of this work is to provide a construction of generalized local symbols on algebraic curves as morphisms of group schemes. From a closed point of a complete, irreducible and non-singular 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.

math.AG↗
arXiv2002

Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring

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-DeConcini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

math.NT↗