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awesome-number-theory

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A list of awesome number theory resources

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# awesome-number-theory
A list of awesome number theory resources

## Elementary Number Theory

<details>
<summary>Richard E. Borcherds - <a href="https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8">Introduction to number theory</a></summary>
Detailed and insightful introduction.
</details>

<details>
<summary>Michael Penn - <a href="https://www.youtube.com/playlist?list=PL22w63XsKjqwn2V9CiP7cuSGv9plj71vv">Number Theory v2</a></summary>
Easy to digest and fast introduction.
</details>

## Algebraic Number Theory

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<summary>Jürgen Neukirch - Algebraic Number Theory</summary>
The most famous algebraic number theory textbook.
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<details>
<summary>Daniel Marcus - Number Fields</summary>
A lot of exercises.
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<summary>Jean-Pierre Serre - Local Fields</summary>
Concentrates on local theory. Towards local class field theory.
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<details>
<summary>James Milne - Algebraic Number Theory</summary>
Easy to read. Include solutions to exercises.
</details>

## Analytic Number Theory

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<summary>Tom Apostol - Introduction to Analytic Number Theory</summary>
Introductory book. Towards a proof of Dirichlet's theorem on arithmetic progressions using Dirichlet $L$-functions.
</details>

## Class Field Theory

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<summary>James Milne - Class Field Theory</summary>
Includes proof of local/global class field theory.
</details>

<details>
<summary>David Cox - Primes of the form $x^2 + ny^2$ (Fermat, Class Field Theory, and Complex Multiplication) </summary>
Classicial approach rather than modern adelic approach. It is less formal, but provides motivation.
</details>

<details>
<summary>Jürgen Neukirch - Class Field Theory </summary>
Cohomology of finite fields. Relatively comprehensive. 
</details>

## Automorphic Forms and Representations

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<summary>Daniel Bump - Automorphic Forms and Representations</summary>
Concentrates on $\mathrm{GL}_2$ theory over $\mathbb{Q}$.
</details>

<details>
<summary>Jan Bruinier et al. - 1-2-3 of Modular Forms</summary>
Great book on elliptic/Hilbert/Siegel modular forms with tons of applications. 
</details>

<details>
<summary>Jayce Getz, Heekyoung Hahn - An Introduction to Automorphic Forms (with a view toward Trace Formulae)</summary>
Modern aspects of automorphic forms and representations, beyond $\mathrm{GL}_2$.
Last chapters are devoted to (simple, relative) trace formulae with related topics.
</details>

<details>
<summary>Neal Koblitz - Introduction to Elliptic Curves and Modular Forms</summary>
Towards the theory of half-integral weight modular forms, Shimura correspondence, Waldspurger's formula and Tunnell's theorem on congruent numbers.
</details>

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<summary>Fred Diamond, Jerry Shurman - A First Course in Modular Forms</summary>
Goal of the book is to understand the statement of Wiles' modularity theorem.
</details>

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<summary>Larry Rolen - <a href="https://math.vanderbilt.edu/rolenl/ModularForms.html">Modular Forms: Theory and Applications</a></summary>
Lectures at Vanderbilt University. Covers various topics related to modular forms.
</details>

## Arithmetic Geometry

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<summary>Joseph Silverman - The Arithmetic of Elliptic Curves</summary>
The most famous introductory textbook on elliptic curves. A bit of applications on cryptography included.
</details>

<details>
<summary>Joseph Silverman - Advanced Topics in the Arithmetic of Elliptic Curves</summary>
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.
</details>

<details>
<summary>Joseph Silverman, Marc Hindry - Diophantine Geometry</summary>
Towards the proof of Faltings' theorem.
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<summary>James Milne - Elliptic Curves</summary>
Nicely written. Introductory textbook.
</details>

## Galois Representations

## Iwasawa Theory

## Misc

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<summary>Cornell et al. - Modular Forms and Fermat's Last Theorem</summary>
Series of articles explaning the details of the proof of Fermat's Last Theorem.
</details>

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<summary><a href=https://kconrad.math.uconn.edu/blurbs/>Keith Conrad's expository papers</a></summary>
Tons of notes on various topics. Especially there are many useful notes on number theory written explicitly.
</details>

<details>
<summary>Anthony Vasaturo - <a href="https://youtube.com/playlist?list=PLK-gFjlmrFy7kbHJ7uOmawNTPz9Hea1vp">Fermat's Last Theorem</a></summary>
Ongoing youtube series that aims to explain the proof of FLT.
</details>

<details>
<summary><a href="https://ctnt-summer.math.uconn.edu/schedules-and-abstracts-2020-online/">Connecticut Summer School in Number Theory</a></summary>
Provide nice lectures on various topics in number theory.
</details>

<details>
<summary><a href="https://swc-math.github.io/">Arizona Winter School</a></summary>
Annual winter school held at University of Arizona. All the lecture notes, videos, and exercises for the previous schools can be found.
</details>

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