TAILIEUCHUNG - Đề tài " Kloosterman identities over a quadratic extension "

We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands’ principle of functoriality is the quadratic base change. Namely, let E/F be a quadratic extension of global fields and z → z the corresponding Galois conjugation. The base change associates to every automorphic representation π of GL(n, F) an automorphic representation Π of GL(n,E). If n = 1 then π is an id`ele class character and Π(z) = π(zz) | Annals of Mathematics Kloosterman identities over a quadratic extension By Herv re Jacquet Annals of Mathematics 160 2004 755 779 Kloosterman identities over a quadratic extension By Hervé Jacquet Contents 1. Introduction 2. Proof of Proposition 1 3. The Kloosterman transform 4. Key lemmas 5. Proof of Proposition 3 6. Complement Abstract We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands principle of functoriality is the quadratic base change. Namely let E F be a quadratic extension of global fields and z z the corresponding Galois conjugation. The base change associates to every automorphic representation n of GL n F an automorphic representation n of GL n E . If n 1 then n is an idele class character and n z n zz . An automorphic representation n of GL n E is a base change if and only if it is invariant under the Galois action. The existence of the base change is established by the twisted trace formula 3 . Formally if f and f are smooth functions of compact support on G EA and G FA respectively then one defines Kf x y 52 f x-1 Kf x y 52 f x i y The author was partially supported by NSF grant DMS-9619766. 756 HERVE JACQUET The identity of the twisted trace formula is that ịKf x x dx 1K . for many pairs of functions f f . The existence of such an identity depends on a simple relation between orbital integrals of the form y f xYx-i dx Ị f xY x 1 dx. In turn to establish such a relation one needs to compare at almost all places v of F inert in E the orbital integrals of specific functions. This is the fundamental lemma 9 . There is another possible characterization of the base change. Indeed in the case n 1 n is a base change if and only if it is trivial on the group of elements of norm 1 that is on the unitary group in one variable. Thus it is natural to conjecture that a representation n is a base change if .

• Thạc sĩ - Tiến sĩ - Cao học
• Kloosterman identities
• subjects of mathematics
• Annals of Mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center