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

Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

Given a continuous homomorphism 

$${G_{Q,S}}G{L_2}\left( {{Z_p}} \right)$$

where Gℚ,S is the Galois group of the maximal algebraic extension of ℚ unramified outside the finite set S of primes of ℚ, the motivating problem of this paper is to study, in a systematic way, the possible liftings of ρ to p-adic representations, 

$${G_{Q,S}}\mathop \to \limits^{{\rho _o}} G{L_2}\left( {{Z_p}} \right).$$

.

Deforming Galois Representations

The Arithmetic of Elliptic Curves.

Symmetries of Equations of Quantum Mechanics

Subrings of fields Complete fields Decomposition groups and the Artin map Analytic methods and ray classes Class field theory Quadratic fields Appendix References Index.

/pdf/algebraic-number-fields-2iqnqki62c.pdf

Algebraic Number Fields

Introduction. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Noncommutative separable algebras have been studied in [2]. Commutative separable algebras have been studied in [1] and in [2], [6] where the main ideas are based on the classical Galois theory of fields. This paper depends heavily on these three papers and the reader should consult them for relevant definitions and basic properties of separable algebras. We shall be concerned with commutative separable algebras in two situations. Let R be an arbitrary commutative ring with no idempotents except 0 and 1. We first consider separable P-algebras that are finitely generated and projective as an P-module. We later drop the assumption that the algebras are projective but place restrictions on P — e.g., P a local ring or a Noetherian integrally closed domain. In §1 we give a proof due to D. K. Harrison that any finitely generated, projective, separable P-algebra without proper idempotents can be imbedded in a Galois extension of P also without proper indempotents. We also give a number of preliminary results to be used in later sections. In §2 we generalize some of the results about polynomials over fields to the case of aground ring P with no proper idempotents. We show that certain polynomials (called \"separable\") admit \"splitting rings\" which are Galois extensions of the ground ring. We apply this to show that any finitely generated, projective, separable homomorphic image of R[X] has a kernel generated by a separable polynomial. In §3 we restrict our attention to separable algebras over a local ring. (The term \"local\" will not imply any finiteness conditions.) In certain cases every finitely generated, separable algebra is a homomorphic image of a finitely generated, projective, separable algebra. This gives an external characterization of the separable algebras. In §4 we consider the internal structure of seperable algebras over a Noetherian integrally closed domain. We show that a finitely generated separable algebra is the direct sum of projective, separable domains containing the ground ring and an algebra that is separable but not faithful over the ground ring. For the case of a Dedekind domain we can obtain specific infor-

Gerald J. Janusz

Papers

Algebraic Number Fields

Separable algebras over commutative rings

Linear Algebra and Group Representations.

Finite groups with an irreducible representation of large degree

Faithful representations of p groups at characteristic p, II