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Explores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.

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Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. Its main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K.

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In a manner accessible to beginning undergraduates, this book introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, and more. Showing how experiments are used to test conjectures and prove theorems, it allows students to do original work on such problems.

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Explores an important aspect of number theory - the theory of exponential sums over finite fields and their Mellin transforms - from a categorical point of view. This book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.

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(Paperback)

Euler systems are special collections of cohomology classes attached to p-adic Galois representations. This book presents a development of the theory of Euler systems. It reviews and develops the necessary facts from Galois cohomology and then introduces Euler systems, states the main theorems, and develops examples and applications.

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Deals with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. This book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G).

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A study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers.

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Develops techniques based on two ingredients: the theory of perverse sheaves and Larsen's Alternative. These techniques are then used to calculate the geometric monodromy groups attached to some specific universal families of (L-functions attached to) character sums over finite fields.

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Seeks to describe the development in recent decades of sieve methods able to detect prime numbers. This work covers topics such as; primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander, and Iwaniec.

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Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras, this book introduces a class of non-associative algebras related to exceptional algebraic groups and their associated buildings.

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For hundreds of years, the study of elliptic curves has played a central role in mathematics. This book explores: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves It is suitable for those interested in number theory and algebraic geometry.

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(Paperback)

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. This book proves foundational results about these series and develops their combinatorics.

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(Paperback)

This outrageous graphic novel investigates key concepts in mathematics by taking readers on a voyage of forensic discovery, exploring some of the most fundamental ideas in mathematics within a thrilling murder mystery.

(Paperback)

(Paperback)

How Numbers Work is a tour of the mind-blowing but beautiful realm of numbers and the mathematical rules that connect them.