I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials

Symmetric functions and Hall polynomials

Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

A Course in Computational Algebraic Number Theory

Propriétés galoisiennes des points d'ordre fini des courbes elliptiques

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Complex multiplication of Abelian varieties and its applications to number theory

Introduction Automorphic forms and families of abelian varieties Arithmeticity of automorphic forms Arithmetic of differential operators and nearly holomorphic functions Eisenstein series of simpler types Zeta functions associated with Hecke eigenforms Analytic continuation and near holomorphy of Eisenstein series of general types Arithmeticity of the critical values of zeta functions and Eisenstein series of general types Appendix References Index.

Arithmeticity in the theory of automorphic forms

Preliminaries on Modular Forms and Dirichlet Series.- Critical Values of Dirichlet L-functions.- The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms.- Eisenstein Series.- Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields.- Supplementary Results.

Elementary Dirichlet series and modular forms

Automorphic functions on the upper half plane, especially modular functions- Elliptic curves and the fundamental theorems of the classical theory of complex multiplication- Relation between the points of finite order on an elliptic curve and the modular functions of higher level- Abelian varieties and siegel modular functions- The endomorphism-ring of an abelian variety the field of moduli of an abelian variety with many complex multiplications- The class-field-theoretical characterization of K' (?(z))- A further method of constructing class fields- The hasse zeta function of an algebraic curve- Infinite galois extensions with l-adic representations- Further generalization and concluding remarks

五郎志村

Papers

Introduction to the arithmetic theory of automorphic functions

Complex multiplication of Abelian varieties and its applications to number theory

Arithmeticity in the theory of automorphic forms

Elementary Dirichlet series and modular forms

Automorphic functions and number theory

五郎 志村

Papers

Introduction to the arithmetic theory of automorphic functions

Complex multiplication of Abelian varieties and its applications to number theory

Arithmeticity in the theory of automorphic forms

Elementary Dirichlet series and modular forms

Automorphic functions and number theory

五郎志村