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A series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches.

Number theory. --- Topological groups. --- Lie groups. --- Algebraic geometry. --- Number Theory. --- Topological Groups, Lie Groups. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Number study --- Numbers, Theory of --- Algebra --- Trace formulas. --- Formulas, Trace --- Automorphic forms --- Discontinuous groups --- Representations of groups --- Fórmules de traça --- Fórmules de traces --- Fórmules traça --- Formes automorfes --- Grups discontinus --- Representacions de grups --- Fórmula de traça de Selberg

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This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory. For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitions and theorems with complete and self-contained proofs. Written in an engaging and educational style, the book also includes exercises and provides their solution.

Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Teoria de nombres --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics

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Teoria de nombres --- Dones matemàtiques --- Matemàtiques --- Científiques --- Matemàtics --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Number theory

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Number theory. --- Geometria algebraica aritmètica --- Teoria de nombres --- Number study --- Numbers, Theory of --- Algebra --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Geometria algèbrica aritmètica --- Geometria diofàntica --- Geometria algebraica --- Punts racionals (Geometria) --- Varietats de Shimura

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This book gives its readers a unique opportunity to get acquainted with new aspects of the fruitful interactions between Analysis, Geometry, Quantum Mechanics and Number Theory. The present book contains a number of contributions by specialists in these areas as an homage to the memory of the mathematician Erik Balslev and, at the same time, advancing a fascinating interdisciplinary area still full of potential. Erik Balslev has made original and important contributions to several areas of Mathematics and its applications. He belongs to the founders of complex scaling, one of the most important methods in the mathematical and physical study of eigenvalues and resonances of Schrödinger operators, which has been very essential in advancing the solution of fundamental problems in Quantum Mechanics and related areas. He was also a pioneer in making available and developing spectral methods in the study of important problems in Analytic Number Theory. .

Number theory. --- Quantum physics. --- Operator theory. --- Physics. --- Environmental monitoring. --- Number Theory. --- Quantum Physics. --- Operator Theory. --- Mathematical Methods in Physics. --- Monitoring/Environmental Analysis. --- Biomonitoring (Ecology) --- Ecological monitoring --- Environmental quality --- Monitoring, Environmental --- Applied ecology --- Environmental engineering --- Pollution --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Functional analysis --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Number study --- Numbers, Theory of --- Algebra --- Measurement --- Monitoring --- Spectral theory (Mathematics) --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Teoria espectral (Matemàtica) --- Teoria de nombres --- Teoria dels nombres --- Àlgebra --- Anàlisi diofàntica --- Arrels de la unitat --- Congruències i residus --- Conjectura de Catalan --- Darrer teorema de Fermat --- Formes automorfes --- Formes quadràtiques --- Fórmula de traça de Selberg --- Funcions aritmètiques --- Funcions L --- Funcions modulars --- Funcions recursives --- Funcions zeta --- Geometria algebraica aritmètica --- Geometria de nombres --- Grups modulars --- Lleis de reciprocitat --- Nombres de Fermat --- Nombres ordinals --- Nombres p-àdics --- Nombres transfinits --- Numeració --- Particions (Matemàtica) --- Quadrats màgics --- Sedàs (Matemàtica) --- Teorema de Fermat --- Teorema de Gödel --- Teoria algebraica de nombres --- Teoria de Galois --- Cossos algebraics --- Anàlisi funcional --- Equacions en derivades parcials --- Successions espectrals (Matemàtica) --- Espais de Hilbert