Trends in Number Theory.

Cover -- Title page -- Contents -- Preface -- Automorphic Galois Representations and the Inverse Galois Problem -- 1. Introduction -- 2. Some classical cases -- 3. Compatible systems and the inverse Galois problem -- 4. Galois representations attached to automorphic forms -- 5. Main statements and ingredients of proof -- References -- Two Mayer-Vietoris Spectral Sequences for -modules -- 1. Introduction -- 2. Basics on spectral sequences -- 3. Mayer-Vietoris spectral sequence -- 4. Relative Mayer-Vietoris spectral sequence -- 5. Arrangements of hyperplanes -- References -- From Galois to Hopf Galois: Theory and Practice -- 1. Introduction -- 2. Classifying Hopf Galois structures -- 3. The lattice of sub-Hopf algebras and the main theorem -- 4. Hopf Galois in small degree -- 5. Intermediate extensions -- 6. Further developments -- References -- Elliptic Curves with Torsion Group ℤ/8ℤ or ℤ/2ℤ×ℤ/6ℤ -- 1. Introduction -- 2. Torsion group -- 3. Search for new rational points -- 4. Rank 1 families -- 5. Rank 2 families for the torsion group \Z/8\Z -- 6. Rank 2 families for the torsion group \Z/2\Z×\Z/6\Z -- 7. Examples of curves with high rank -- References -- Equidistribution, -Functions, and Sato-Tate Groups -- 1. Introduction -- 2. Equidistribution and -functions -- 3. The Sato-Tate group and the generalized Sato-Tate Conjecture -- 4. Group-theoretic classification results -- References -- Covering Techniques and Rational Points on Some Genus 5 Curves -- 1. Introduction -- 2. An algorithm -- 3. Examples -- Appendix A. On quartic elliptic curves -- Acknowledgements -- References -- On Quantum Integers and Rationals -- Introduction -- 1. Quantum integers -- 2. Quantum binomial coefficients -- 3. Quantum rational numbers -- 4. Twisted powers -- References -- References -- A Space of Weight 1 Modular Forms Attached to Totally Real Cubic Number Fields.

1. Introduction -- 2. Weight 1 forms attached to non-totally real cubic fields. -- 3. Totally real cubic fields -- 4. Explicit Scholz reflection principle -- Acknowledgements -- References -- Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases -- 1. Numerical semigroups -- 2. A characterization of elements and gaps in terms of Groebner bases -- 3. A first application: a bound "á la Wilf" -- 4. Acknowledgments -- References -- Classification des représentations modulaires de _{ }( ) en caractéristique non naturelle -- 1. Introduction -- Remerciements -- Notations et conventions -- 2. Le groupe \GL_{ } sur un corps fini -- 2.1. Foncteurs d'induction et restriction paraboliques -- 2.2. Décomposition de Mackey -- 2.3. Unicité du support cuspidal -- 2.4. Classification des représentations cuspidales en fonction des supercuspidales -- 2.5. Unicité du support supercuspidal -- 2.6. Réduction modulo ℓ -- 2.7. Fonctions de partition -- 3. Classification axiomatique -- 3.1. Axiomes -- 3.2. Classification -- 3.3. Preuve de (1) -- 3.4. Preuve de (2) -- 3.5. Preuve de (3) -- 3.6. Preuve de (4) -- 4. Les représentations (\s, ) et (\s, ) -- 4.1. Représentations presque projectives -- 4.2. Définition des représentations (\s, ) et (\s, ) -- 5. Les classifications de James et Dipper-James -- 5.1. La classification de Dipper-James -- 5.2. Classification de James -- 5.3. Sur le morphisme de réduction modulo ℓ -- 6. Impossibilité d'une classification duale -- References -- Fractional -adic Differentiability Under the Amice Transform -- Introduction -- Part 1. A resume of non-Archimedean r-fold differentiability -- 1. Cp-functions -- 2. Cv-functions or iterated non-Archimedean differentiability -- The Archimedean setting -- The non-Archimedean setting -- 3. Cr-functions.

Part 2. Comparison of the fractionally differentiable and locally analytic topologies -- 4. Density of locally polynomial functions -- 5. The van der Put-basis -- 6. Comparison between Differentiability and Analyticity -- Part 3. The Amice transform -- 7. Local analyticity via the Amice transform -- 8. Duality -- Schikhof Duality -- The Cr-function space -- References -- Reciprocity Laws Related to Finite Potent Endomorphisms -- 1. Introduction -- 2. Finite Potent Endomorphisms -- 3. Abstract Residue and Segal-Wilson Pairing -- 4. Reciprocity Laws from Finite Potent Endomorphisms -- References -- When the Modular World Becomes Non-Holomorphic -- Introduction -- 1. -functions, functional equations and summation formulas -- Journey to a less known world -- 2. Preliminary concepts -- 3. Non-holomorphic automorphic forms -- 4. The spectral expansion -- 5. The Selberg trace formula -- 6. The Kuznetsov formula -- 7. Arithmetic applications -- 8. Beyond the (2) theory -- References -- Back Cover.

This volume contains the proceedings of the Fifth Spanish Meeting on Number Theory, held from July 8-12, 2013, at the Universidad de Sevilla, Sevilla, Spain. The articles contained in this book give a panoramic vision of the current research in number theory, both in Spain and abroad. Some of the topics covered in this volume are classical algebraic number theory, arithmetic geometry, and analytic number theory.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.