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Intro; Contents; Overview of the Work of Kumar Murty; 1 Introduction; 2 Algebraic Cycles; 2.1 The Hodge Conjecture; 2.2 The Tate Conjecture; 2.3 Shimura Varieties and Period Relations; 2.4 Reduction in Tate Cycles Modulo a Prime; 3 L-Functions; 3.1 Sato-Tate Conjecture; 3.2 Artin L-Functions; 3.3 Chebotarev Density Theorem and its Applications; 3.4 Non-vanishing of L-Functions; 3.5 Distribution of Euler-Kronecker Constants; 3.6 Spaces of L-Functions; 4 Cryptography and Further Applications; 4.1 Koblitz's Conjecture; 4.2 Factorization and Modular Forms
4.3 Explicit Arithmetic on Abelian VarietiesReferences; On the Average Value of a Function of the Residual Index; 1 Introduction; 2 The Case of Additive Arithmetic Functions; 3 The Case of General Arithmetic Functions; 3.1 Evaluation of the Main Term of (3.7); 3.2 Evaluation of the Error Term of (3.7); 4 Average of Local Average Values; References; Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals; 1 Introduction and Statement of the Main Theorem; 2 Square-Sieve Approach
3 Application of Chebotarev Density Theorem Under the GRH4 Counting; 5 Evaluation of Character Sums; 6 Proof of Theorem2; 7 Proof of Theorem3; References; R-Group and Multiplicity in Restriction for Unitary Principal Series of GSpin and Spin; 1 Introduction; 2 Preliminaries; 2.1 Basic Notation; 2.2 Knapp-Stein R-Group; 2.3 Multiplicity in Restriction; 2.4 Group Structure for Unitary Principal Series of GSpin and Spin; 3 Main Result
The Relation Between Knapp-Stein R-Group and Multiplicity; References; The 2-Class Tower of mathbbQ(sqrt-5460); 1 Introduction
2 2-Tower Groups of Imaginary Quadratic Fields3 Ingredients Regarding p-Group Theory; 4 Beginning the Search; 5 Abelianizations of Larger Index Subgroups; 6 Capitulation; 7 Nilpotency Class 3 Quotient of GK; 8 Moribund p-Groups; 9 Nilpotency Class >3 Quotients of GK; References; On the Bad Reduction of Certain U(2, 1) Shimura Varieties; 1 Three Integral Models with Parahoric Level Structure; 1.1 Shimura Varieties; 1.2 The Moduli Problems; 1.3 Translation into the Language of Rapoport and Zink; 1.4 Modular Curves on the Picard Modular Surface; 2 The Structure of the Special Fiber of mathscrS
4.3 Explicit Arithmetic on Abelian VarietiesReferences; On the Average Value of a Function of the Residual Index; 1 Introduction; 2 The Case of Additive Arithmetic Functions; 3 The Case of General Arithmetic Functions; 3.1 Evaluation of the Main Term of (3.7); 3.2 Evaluation of the Error Term of (3.7); 4 Average of Local Average Values; References; Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals; 1 Introduction and Statement of the Main Theorem; 2 Square-Sieve Approach
3 Application of Chebotarev Density Theorem Under the GRH4 Counting; 5 Evaluation of Character Sums; 6 Proof of Theorem2; 7 Proof of Theorem3; References; R-Group and Multiplicity in Restriction for Unitary Principal Series of GSpin and Spin; 1 Introduction; 2 Preliminaries; 2.1 Basic Notation; 2.2 Knapp-Stein R-Group; 2.3 Multiplicity in Restriction; 2.4 Group Structure for Unitary Principal Series of GSpin and Spin; 3 Main Result
The Relation Between Knapp-Stein R-Group and Multiplicity; References; The 2-Class Tower of mathbbQ(sqrt-5460); 1 Introduction
2 2-Tower Groups of Imaginary Quadratic Fields3 Ingredients Regarding p-Group Theory; 4 Beginning the Search; 5 Abelianizations of Larger Index Subgroups; 6 Capitulation; 7 Nilpotency Class 3 Quotient of GK; 8 Moribund p-Groups; 9 Nilpotency Class >3 Quotients of GK; References; On the Bad Reduction of Certain U(2, 1) Shimura Varieties; 1 Three Integral Models with Parahoric Level Structure; 1.1 Shimura Varieties; 1.2 The Moduli Problems; 1.3 Translation into the Language of Rapoport and Zink; 1.4 Modular Curves on the Picard Modular Surface; 2 The Structure of the Special Fiber of mathscrS