Math

Wiles, Andrew

“Modular elliptic curves and Fermat’s Last Theorem”

This paper does a lot of the work to prove that all elliptic curves are modular. A gap remained concerning Hecke algebras.

Taylor, Richard, and Andrew Wiles

“Ring-theoretic properties of certain Hecke algebras”

This paper fills in the gap in Wiles’s proof, completing the proof that all elliptic curves are modular.

Ribet, Ken

“On modular representations of Gal(Q/Q) arising from modular forms”

This paper proves Fermat’s Last Theorem, by connecting the fact that all elliptic curves are modular to FLT.

Ahlfors, Lars V.

Complex Analysis

This is a classic textbook on complex analysis.

Buck, R. Creighton

Advanced Calculus

This is the classic text on calculus. Every advanced analysis book that needs a citation for some basic concept from calculus cites this book. It is difficult; far more advanced than a regular calculus textbook.

Lebesgue, Henri

Doctoral Thesis on Integration

Here is Lebesgue’s actual doctoral thesis (in French!) in which he first introduces the Lebesgue integral.

Rudin, Walter

Principles of Mathematical Analysis

The most famous/classic textbook on real analysis.

Spivak, Michael

Calculus

A classic undergraduate textbook. It’s really more of an introduction to real analysis than calculus.

Artin, Michael

Algebra

A popular book on algebra. It’s at the level of advanced undergraduate or introductory graduate. It starts in an odd place (for me), matrix operations, and apparently emphasizes linear algebra throughout. I don’t think I like that as much, but it would probably be good to get comfortable with nonetheless.

Fraleigh, John

A First Course in Abstract Algebra

This was my first textbook in abstract algebra.

Stewart, Ian

Galois Theory

When I was learning Galois Theory, we used the first edition of this book, and it was utterly FILLED with errors. It drove me up the wall, and I was always complaining to my teacher Ken Ribet about it, which made him more annoyed with me than I was with Stewart. Not my finest hour. Nonetheless, this book is intended to be an easy-to-follow introduction specifically to Galois Theory, without having to go all the way through abstract algebra to get there. Here is the third edition; maybe the errors have been corrected.

Stewart, James

Calculus – Early Transcendentals

This is my original calculus textbook from way back in college!

Serre, Jean-Pierre

A Course in Arithmetic

Serre was a big figure in the proof of Fermat’s Last Theorem.

Silverman, Joseph

The Arithmetic of Elliptic Curves

I don’t know anything about this book, but I believe it is a book about the arithmetic of elliptic curves.

Miyake, Toshitsune

Modular Forms

A classic book on modular forms.

Shimura, Goro

Introduction to the Arithmetic Theory of Automorphic Functions

This is the classic textbook on modular forms. This book is probably where most mathematicians in the field first learned the topic. Fermat’s Last Theorem was proven by observing that all elliptic curves are modular, then showing that any solution to Fermat’s Last Theorem can be associated to an elliptic curve, and that that curve would not be modular. Hence, if it is true that all elliptic curves are modular, then it can not be true that there are any solutions to aⁿ+bⁿ=cⁿ when n > 2. It was in 1957 that Yutaka Taniyama and Goro Shimura first conjectured that all elliptic curves are modular, which became known as the Taniyama-Shimura conjecture. It is this conjecture that Andrew Wiles proved in 1995 (for “semi-stable” curves—that’s a technical but unimportant detail). This is the textbook that Goro Shimura wrote on the subject in 1971.

Munkres, James

Topology

This was my textbook in topology (when I should have been studying for law school!).

Wasserman, Larry

All of Statistics: A Concise Course in Statistical Inference.

Nothing whatever to do with Fermat’s Last Theorem, this is a popular textbook on statistics.