awesome-number-theory

A list of awesome number theory resources

Elementary Number Theory

Richard E. Borcherds - Introduction to number theory

Detailed and insightful introduction.

Michael Penn - Number Theory v2

Easy to digest and fast introduction.

Algebraic Number Theory

Jürgen Neukirch - Algebraic Number Theory

The most famous algebraic number theory textbook.

Daniel Marcus - Number Fields

A lot of exercises.

Jean-Pierre Serre - Local Fields

Concentrates on local theory. Towards local class field theory.

James Milne - Algebraic Number Theory

Easy to read. Include solutions to exercises.

Analytic Number Theory

Tom Apostol - Introduction to Analytic Number Theory

Introductory book. Towards a proof of Dirichlet's theorem on arithmetic progressions using Dirichlet $L$-functions.

Class Field Theory

James Milne - Class Field Theory

Includes proof of local/global class field theory.

David Cox - Primes of the form $x^2 + ny^2$ (Fermat, Class Field Theory, and Complex Multiplication)

Classicial approach rather than modern adelic approach. It is less formal, but provides motivation.

Jürgen Neukirch - Class Field Theory

Cohomology of finite fields. Relatively comprehensive.

Automorphic Forms and Representations

Daniel Bump - Automorphic Forms and Representations

Concentrates on $\mathrm{GL}_2$ theory over $\mathbb{Q}$.

Jan Bruinier et al. - 1-2-3 of Modular Forms

Great book on elliptic/Hilbert/Siegel modular forms with tons of applications.

Jayce Getz, Heekyoung Hahn - An Introduction to Automorphic Forms (with a view toward Trace Formulae)

Modern aspects of automorphic forms and representations, beyond $\mathrm{GL}_2$. Last chapters are devoted to (simple, relative) trace formulae with related topics.

Neal Koblitz - Introduction to Elliptic Curves and Modular Forms

Towards the theory of half-integral weight modular forms, Shimura correspondence, Waldspurger's formula and Tunnell's theorem on congruent numbers.

Fred Diamond, Jerry Shurman - A First Course in Modular Forms

Goal of the book is to understand the statement of Wiles' modularity theorem.

Larry Rolen - Modular Forms: Theory and Applications

Lectures at Vanderbilt University. Covers various topics related to modular forms.

Arithmetic Geometry

Joseph Silverman - The Arithmetic of Elliptic Curves

The most famous introductory textbook on elliptic curves. A bit of applications on cryptography included.

Joseph Silverman - Advanced Topics in the Arithmetic of Elliptic Curves

Volume 2 of Silverman's book. Include various topics that are not in the volume 1: elliptic and modular functions, CM elliptic curves, Neron models, Tate's algorithm, etc.

Joseph Silverman, Marc Hindry - Diophantine Geometry

Towards the proof of Faltings' theorem.

James Milne - Elliptic Curves

Nicely written. Introductory textbook.

Galois Representations

Iwasawa Theory

Misc

Cornell et al. - Modular Forms and Fermat's Last Theorem

Series of articles explaning the details of the proof of Fermat's Last Theorem.

Keith Conrad's expository papers

Tons of notes on various topics. Especially there are many useful notes on number theory written explicitly.

Anthony Vasaturo - Fermat's Last Theorem

Ongoing youtube series that aims to explain the proof of FLT.

Connecticut Summer School in Number Theory

Provide nice lectures on various topics in number theory.

Arizona Winter School

Annual winter school held at University of Arizona. All the lecture notes, videos, and exercises for the previous schools can be found.