This Study Text has been reviewed by the examiner and concentrates on the key areas of the syllabus, taking into account the examiner's guidance on how topics will be examined. The Study Text has a step-by-step approach to topics and lots of exercises in which you can practise the calculations. We provide a detailed chapter on spreadsheets and a basic maths appendix, for those who need some revision in that area.
The area of automorphic representations is a natural continuation of studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called “liftings”. This book concentrates on two initial examples: the symmetric square lifting from SL(2) to PGL(3), reflecting the 3-dimensional representation of PGL(2) in SL(3); and basechange from the unitary group U(3, E/F) to GL(3, E), [E : F] = 2.The book develops the technique of comparison of twisted and stabilized trace formulae and considers the “Fundamental Lemma” on orbital integrals of spherical functions. Comparison of trace formulae is simplified using “regular” functions and the “lifting” is stated and proved by means of character relations.This permits an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), a proof of multiplicity one theorem and rigidity theorem for SL(2) and for U(3), a determination of the self-contragredient representations of PGL(3) and those on GL(3, E) fixed by transpose-inverse-bar. In particular, the multiplicity one theorem is new and recent.There are applications to construction of Galois representations by explicit decomposition of the cohomology of Shimura varieties of U(3) using Deligne's (proven) conjecture on the fixed point formula.
The 2010 edition has been written in conjunction with the examiner to fully reflect what could be tested in the exam. Fully revised with additional readings and examples, it provides complete study material for the May and November 2010 exams.
Your guide to annual exams 2023 is now “Simplified”! The first-of-its-kind sample paper booklet, i.e., one incorporating not only the practice papers but also the basic concepts for each chapter, is here. Some salient features of this book are as follows: 1. This sample paper booklet begins with basic concepts about each chapter, providing a snapshot of the entire chapter. It hence facilitates the purpose of last-minute revisionary notes needed by the students. 2. To help students practice and evaluate their understanding, the booklet and a total of 15 sample papers. It has -: CBSE Sample Paper-2023 (Solved) 5 Sample Papers( Solved) detailed step-by-step solutions 10 Sample Papers for Practice (answers for objective questions are included). 3. A blueprint based on the specimen paper released by the CBSE Board has also been included in this booklet to enable the students to gauge the unit-wise weightage and the marking scheme of the paper. 4. Special emphasis has been laid to include the new typology of questions in each paper i.e. objective-type (mcq), assertion and reason based and case study based questions etc. 5. This book is indeed a one stop destination for all the subject matter required for the final revision to ace in the annual exam of mathematics.
