Following are some of the mathematics books that I would recommend reading (not cover to cover!). The inspiration for this list came from the Chicago undergraduate mathematics bibliography.

One might need a university library membership to access some of the out-of-print books (or use the power of the internet). I have arranged the topics and books in the order of increasing requirements for mathematical maturity.

• Popular Maths and Fiction
• Biographies and Autobiographies
• Olympiad Maths
• High School Maths
• Undergraduate Maths
  • Rudimentary Mathematics
  • Discrete Mathematics
  • Algebra
  • Analysis
  • Topology
  • Geometry
• Number Theory
  • Elementary
  • Algebraic
  • Analytic
  • Topological
  • Geometric
  • Miscellaneous
• Scary Maths
  • Complex Geometry
  • Enumerative Geometry

Popular Maths and Fiction

• Fun and Fundamentals of Mathematics by J. V. Narlikar and M. Narlikar: This was the first book I bought through Flipkart (when it used to sell only books). Have read it many times since April 2011.
• Euclid’s Window by Leonard Mlodinow: I loved reading this book (Mlodinow has co-authored “The Grand Design” by Stephen Hawking).
• Euler’s Gem by David Richeson: 2+E=F+V was my favorite formula in school and this was the first book I read in college.
• Mathematics: A very Short Introduction by T. Gowers: A broad overview of research career in mathematics.
• What is Mathematics ? by Richard Courant and Herbert Robbins: This is must read book for all students serious about mathematics. This book will be your lifelong friend.
• Fermat’s Last Theorem by Simon Singh: A very detailed account of Fermat’s Theorem with a good number of references.
• The Poincare Conjecture by Donal O’Shea: It was nice to see how the different branches of topology were developed to solve this conjecture.
• Prime Obsession by John Derbyshire: I finally understood the meaning of “analytic continuation” from this book.
• The Einstein Theory of Relativity by Lillian R. Lieber: A very clear exposition of the mathematical contribution of Albert Einstein.
• Journey Through Genius by William Dunham: Wonderful discussions about mathematical works of great mathematicians.
• The Code Book by Simon Singh: My first introduction to the world of cryptography.
• The Codebreakers by David Kahn: One of the best references for the history of cryptography. (Nowadays, there are nice cryptography playlists by Khan Academy, PBS Infinte Series, and Numberphile+Computerphile.)
• The Book of Numbers by John H. Conway and Richard K. Guy: Clear explanation of complicated concepts, especially Bernoulli numbers and cardinal numbers (counting cards). Plus it’s a colorful maths book!
• Magical Mathematics by Persi Diaconis and Ron Graham: Very interesting mathematical card tricks, a great treat for people who love the collection of Scientific American articles by Martin Gardner.
• The Penguin Dictionary of Curious and Interesting Geometry by David Wells: My favourite entry is “pursuit curves”.
• Flatland by Abbott: It’s a short and sweet book.
• The Number Devil by Hans Magnus Enzensberge: Though the author is not a mathematician, he is a well-known and respected European intellectual and author with wide-ranging interests. This book is similar to George Gamow’s book “Mr. Tompkins”. Plus it’s a colorful maths book!
• The Curious Incident of the Dog in the Night-Time by Mark Haddon: The protagonist is an autistic boy who loves prime numbers and aloo gobi.
• The Devotion of Suspect X by Keigo Higashino: The protagonist is a brilliant but lonely mathematician.
• A Man Called Ove by Fredrik Backman: The protagonist is an autistic old man who liked maths when he was a kid.

Biographies and Autobiographies

• Surely You’re Joking, Mr. Feynman! by Richard Feynman: MUST READ! It changed my perspective on life, science, and maths. Fun fact: Feynman was a Putnam fellow and he wrote the exam just for fun, like most of his research (fooling around).
• The Man Who Knew Infinity by Robert Kanigel: The finest biography of Srinivasa Ramanujan Iyenger.
• The Man Who Loved Only Numbers by Paul Hoffmann: The biography of Paul Erdos which motivated me to become a mathematician.
• A Beautiful Mind by Sylvia Nasar: This is the biography of John Nash, a mathematician whose personality is quite the opposite of mine.
• The Strangest Man by Graham Farmelo: This is the biography of Paul Dirac, whose life I wanted to know about because he is believed to be autistic.
• Julia, a Life in Mathematics by Constance Reid: This is a nice biography by Julia Robinson’s sister. I came to know about Hilbert’s Tenth Problem from this book, which had a deep impact on my taste in mathematics. It is a quick read but can be particularly helpful for young women aspiring to a career in mathematics.
• Rosalind Franklin, The Dark Lady of DNA by Brenda Maddox: I came to know about her in college. She is referred to as “The Dark Lady of DNA” because her contributions to the discovery of the structure of DNA were largely unrecognized during her life. (To read: Dorothy Wrinch and Dorothy Crowfoot)

Eventually, I should also read the biographies of Sofya Kovalevskaya, Sophie Germain, and Emmy Noether.

Olympiad Maths

These books should be accessible to the students in class VIII (final year of Primary education in India), i.e. children aged 13+.

• Little Mathematics Library by MIR Publishers: These books include short introductory material (without much detail) on various topics that can help students to get an idea of different fields of mathematics, written especially for high school students preparing for Olympiad. All of these can be downloaded legally from mirtitles.org.
• Mathematical Circles (Russian Experience) by Fomin, Genkin, and Itenberg : A thought provoking book for students of class VIII & IX
• Non-Routine Problems in Mathematics by AMTI (Editor: V. K. Krishnan): Lovely book but has a few wrong solutions for problems posed in exercises
• Challenge and Thrill of Pre-College Mathematics by Krishnamurthy, Pranesachar, Ranganathan and Venkatachala: I learned geometry and combinatorics from this book, was fascinated by the Friendship theorem and the work of Erdos.
• A primer on Number Sequences by Shailesh Shirali : A plethora of sequences!
• A Mathematical Mosaic: Patterns & Problem-Solving by Ravi Vakil: One of my favorite books (but sadly not available in India, bought it from the USA).
• Hands-on Geometry by Christopher M. Freeman: Lovely book! A step-by-step guide to learning constructions
• Problem-Solving Strategies by Arthur Engel: This book is more than enough for someone preparing for olympiads. I found the problems in this book to be quite hard but satisfying.
• The Art and Craft of Problem Solving by Paul Zeitz: The first four chapters of this book taught me how to streamline my thinking process.

Here is an outdated list of olympiad preparation books.

High School Maths

The High School consists of class IX-XII (Secondary and Senior Secondary education in India), i.e. students aged 14 to 18.

• NCERT textbooks: These books were written before 2005. They still follow the old notation to represent natural numbers, integers etc., for example: using N instead of ℕ to represent the set of natural numbers. Since, only after 2003, bold font (like N) was completely replaced by blackboard bold font (like ℕ ) to represent some familiar systems of numbers (like “set of natural numbers”) in print. All of these can be downloaded legally from NCERT e-books website.
  • NCERT Mathematics Textbook for Class IX [NCF – 2005] is a fantastic book as it touches nearly all topics (like geometry, polynomials, Number Theory (rational - irrational numbers), Introduction to mathematical modeling).FOR EXAMPLE: I was spellbound by the chapter - “Introduction to Euclid’s Axioms” and I ended up reading “Euclid’s Window by Leonard Mlodinow “ & “Fun & Fundamentals of Mathematics by Narlikar”
  • NCERT Mathematics Textbook for Class X [NCF – 2005] also consists of basics of “Number Theory” topics like “Euclid’s division algorithm”. Moreover the appendices on “Proofs in Mathematics” & “Mathematical Modelling” are worth reading even at later stages. FOR EXAMPLE: The discussion on ‘Proof by Contradiction’ is awesome.
  • NCERT Mathematics Textbook for Class XI [NCF – 2005] includes some of the most fundamental mathematics concepts like “Set Theory”, “Principle of Mathematical Induction”, “Summation of Series”, “Binomial Theorem” & “Permutation & Combinations”. Also appendices on “Infinite Series” & “Mathematical Modelling” are worth reading.
  • NCERT Mathematics Textbook for Class XII [NCF – 2005] major focus on Calculus, but still its appendices on “Proofs in Mathematics” & “Mathematical Modelling” are worth reading. FOR EXAMPLE: In Appendix – 1 (Proofs in Mathematics), there are proofs for two beautiful theorems of Number theory (a) Prime Numbers are infinite & (b) Fermat Numbers are NOT Prime
• MATH!: Encounters with High School Students by Serge Lang: A collection of conversations involving high-school geometry.
• Higher Algebra by Barnard and Child: A classic book in college algebra and discrete maths, contains my favorite proof of Wilson’s theorem. Freely available on Internet Archive.
• The Cartoon Guide to Calculus by Larry Gonick: A Fantastic book! Everyone will love calculus after reading this. An illustrative guide to elementary calculus.
• Calculus: Basic Concepts for High Schools by Lev Tarasov: Among the many calculus textbooks I read, this was the one that gave me the most clarity. Re-written in LaTeX by The Mitr.

Undergraduate Maths

The Undergraduate constitutes of 4-year bachelor’s degree program in mathematics (tertiary education in India), i.e. students aged 18 to 22.

Rudimentary Mathematics

The name of this subsection is based on the book titled “Rudimentary Mathematics for Economists and Statisticians.”

• Calculus I, II, III by Jerrold E. Marsden and Alan Weinstein: These books are old-fashioned versions of Calculus I, II and II (US Education System). The books are available online for free as part of the American Institute of Mathematics’ Open Textbook Initiative. This topic is an amalgamation of analysis, topology and geometry. We use geometric and topological properties of Euclidean space to find some logic behind analytic facts about functions defined on Euclidean space. This is the “maths” needed for most of the undergraduate Physics courses.
• Tea Time Numerical Analysis by Leon Q. Brin: An introductory textbook for numerical calculus with algorithms illustrated using GNU Octave. It is available online for free and is a part of the American Institute of Mathematics’ Open Textbook Initiative.
• Introduction to Probability by J. K. Blitzstein and J. Hwang: It contains lots of examples, motivation and intuition. Calculus is the only prerequisite. Each chapter ends with a section showing how to perform relevant simulations and calculations in R. The softcopy of the book and the lecture videos are available online for free. After learning the basics, one can learn about stochastic processes and the related applications in other fields like finance. Recommended by TPR.
• Mathematical Statistics with Applications in R by K.M. Ramachandran and C. Tsokos: This is a very well-written calculus-based introduction for someone really new to statistics. After a bit of experience, one can move to the book by N. Mukhopadhyay and then in the end to the book by Rohatgi and Saleh. Recommended by TPR.

Discrete Mathematics

• A Friendly Introduction to Number Theory by J. H. Silverman: A modern introduction to elementary number theory. There are many other similar books published by AMS and MAA.
• Introductory Combinatorics by R. A. Brualdi: There are some typos, but there are some really interesting discussions like the theorem by Erdős and Szekeres (p. 76, 5th ed).
• A First Course in Graph Theory by G. Chartrand and P. Zhang: Good introductions to chapters, lots of examples and also has historical remarks in some places. Recommended by SSH.
• Introduction to Algorithms – A Creative Approach by U. Manber: Unlike the standard algorithms textbook duo CLRS and ADM, this book utilizes the known mathematical proof techniques for designing algorithms (note that it is not a universal approach). Moreover, the exercises are divided into two types, namely, drill exercises and creative exercises.
  • Problem Solving with Algorithms and Data Structures using Python by B. M. Miller and D. L. Ranum: This is an introductory text on algorithms which uses Python 3.2 (2nd edition) for illustrating concepts, helpful for learning to translate the pseudocode to an actual programming language. This book is also available online, however, you will need to use an adblocker or host it yourself.
• Introduction to the Theory of Computation by M. Sipser: This is an introductory text on automata theory, computability theory, and complexity theory. A complete course based on this book is available via MIT OCW (videos also available on YouTube and Internet Archive).

Algebra

• Linear Algebra by Friedberg, Insel, and Spence: To avoid typos, use the 1999 reprint of the third edition, or more recent editions. It is a good book to learn basic stuff used in algebra (modules over PID, representation of finite groups, etc.), discrete mathematics (incidence matrix etc.), analysis (Banach space, numerical analysis, differential equations, etc.), and geometry (coordinate change, orientation, multilinear maps, etc.).
• Abstract Algebra by Dummit and Foote: For an introduction to group theory, ring theory, modules over PID, field theory and Galois theory (answers to the straight edge & compass construction questions we encountered in high school). Has lots of solved examples. Beginners might find the books by Fraleigh or Gallian more useful for learning basic group theory. For a more concise introduction, one can read the book by Grove.
• Introduction to Commutative Algebra by Atiyah and MacDonald: For those who want to learn commutative algebra for algebraic geometry. Really helpful for someone planning to read Hartshorne’s book “Algebraic Geometry”. For a gentler introduction, one can read the book by N S Gopalakrishnan.
• Representation and Characters of Groups by G. James and M. Liebeck: This is an introductory text for the representation theory of finite groups. It could have been an ideal introductory text for undergraduates but they chose bad notations. However, it is more accessible as compared to the standard textbook by Serre.
  • Representation Theory of Finite Groups by M. Burrow: Must read the introduction to the first chapter. Notations are a bit outdated.

Analysis

• Foundations of Analysis by E. Landau: Fantastic short book. I learned about Peano’s axioms from this book. Terry Tao’s book “Analysis I” also starts with a similar discussion.
• Introduction to Real Analysis by Bartle and Sherbert: A very nice introduction to concepts like countability and epsilon-delta method for functions defined over $\mathbb{R}$. The focus is to rigorously prove the stuff learned in Calculus I and II. Lots of examples and good exercises. Also compares improper Riemann integration with Lebesgue integration (pros and cons).
• Topology of Metric Spaces by S. Kumaresan: Discusses various useful inequalities needed in the analysis and helps us see the importance of metric space structure of $\mathbb{R}^n$ in real analysis. Also have a look at the book “The Cauchy-Schwarz Master Class” by J. Michael Steele.
• Lebesgue Integration On Euclidean Space by F. Jones: An introductory text on measure theory and Fourier analysis over $\mathbb{R}^n$. We learn that Lebesgue measure space is the completion of the Borel measure space and the space of Lebesgue integrable functions is the completion of the space of Riemann integrable functions.
  • Real Analysis: Modern Techniques and Their Applications by Folland: The first three chapters of this book complement the content of the above book.
• Complex Analysis by Bak and Newman: A comprehensive introduction to analysis of functions defined over $\mathbb{C}$. We study things like holomorphic functions (Fundamental Theorem of Algebra), contour integration (Residue theorem), Mobius transformation etc. This book also contains a brand-new proof of the prime number theorem. For a more gentle introduction, see the book by Sasane and Sasane, “A Friendly Approach to Complex Analysis” or the book by Brown and Churchill, “Complex Variables and Applications”. Also, for GeoGebra animations see this online textbook by Juan Carlos Ponce Campuzano: https://complex-analysis.com/.
• Introduction to Topology and Modern Analysis by G. F. Simmons: This book should be renamed as “An Introduction to Functional Analysis”. Here we talk about normed linear spaces over $\mathbb{R}$ and $\mathbb{C}$.
• Differential Equations by S. L. Ross: Nothing special, any other text like Simmons and Krantz’s book “Differential Equations: Theory, Technique and Practice” is also good. We will see the proofs of facts learned in Calculus II.

Topology

• Topology by J. Munkres: Standard textbook for point-set topology (first part of the book). The focus is on generalizing the properties of continuous functions learned in Calculus I and III.
• An Introduction to Manifolds by L. Tu: A good introduction to differential topology (de Rham cohomology). Can skip the Lie group, Lie algebra and Lie derivative discussions during the first reading. Some of its content is similar to the one found in Hubbard and Hubbard’s book (eg: Stokes theorem).
  • Analysis on Manifolds by J. Munkres: A rigorous introduction to manifolds in $\mathbb{R}^n$ (de Rham cohomology). This book provides better notations for differentiation and integration, hence clearing the confusion caused due to Calculus notations like “dx” etc.
  • Differential Topology by A. Mukherjee: This text nicely complements the study of smooth manifolds. A good reference for Sard’s theorem, Whitney’s embedding theorem, and Morse theory.
• An Introduction to Algebraic Topology by J. Rotman: A good introductory text with a good number of examples about fundamental groups and homology theory. Even though it is part of a graduate text series, it is quite accessible as compared to the more advanced books by Massey or Hatcher.
  • Elements of Algebraic Topology by J. Munkres: Provides a more detailed discussion about both homology and cohomology.

Geometry

• Geometry from a Differentiable Viewpoint by J. McCleary: The Part B of this book provides a good introduction to the geometry of curves and surfaces lying in $\mathbb{R}^3$ (Theorema Egregium and Gauss-Bonnet Theorem). The other two parts are expositions which connect high school geometry with differential geometry (Part A) and introduce Riemannian geometry (Part C).
  • Vector Calculus, Linear Algebra, And Differential Forms: A unified approach by J. H. Hubbard and B. B. Hubbard: A visual introduction to multivariable calculus. It also briefly discusses the geometry of curves and surfaces in $\mathbb{R}^3$. The focus is on rigorously proving the stuff learned in Calculus III and extending those ideas. Though “differential forms” are not needed for understanding curves and surfaces in $\mathbb{R}^3$, they will be useful when learning about Riemannian manifolds (tensor calculus).
• An Introduction to Differentiable Manifolds and Riemannian Geometry by W. Boothby: This book provides a good introduction to Riemannian manifolds. The content of the first four chapters overlaps with some of the stuff covered in Tu’s book on manifolds. Roughly speaking, geometry has a local structure (eg: Gaussian curvature), while topology only has a global structure (eg: Euler characteristic).
  • Riemannian Geometry: A Beginner’s Guide by Frank Morgan: It is a nice reference for visual illustrations needed for a better understanding of Riemannian geometry.
• Projective Geometry by P. Samuel (translated from the French by S. Levy): The purpose of this book is to give an algebraic introduction to projective spaces. Keep in mind that the “projective space” we encounter in algebraic topology/differential geometry and projective geometry/algebraic geometry is different because of the underlying topology (related discussion). For those interested in combinatorics, a good follow-up text would be the book “An Introduction to Incidence Geometry” by Bart De Bruyn.
  • Geometry by M. Audin: This book will help to bridge the gap between high-school geometry (Euclidean plane, Cartesian 2D and 3D, and Argand plane) and university geometry (differential geometry and algebraic geometry). Also, the chapters V and VI of this book supplement the contents of Samuel’s book.
• An Invitation to Algebraic Geometry by K. E. Smith, et. al.: This book nicely complements the first chapter of Hartshorne (the rest of that book is about scheme theory). Lots of solved examples. Note that, though the “genus” of a curve (Riemann surface) is defined in algebraic topology via Euler characteristic and in (complex) algebraic geometry via Riemann-Roch theorem, there is a close correspondence between “divisors” (algebraic geometry) and holomorphic “line bundles” (algebraic topology) on a Riemann surface. You can find more about this in the last chapter (Chapter 8) of this book. A good follow-up read would be the book “Algebraic Curves” by Fulton (available online) where a proof of Riemann-Roch theorem for curves is discussed.

Number Theory

Number Theory is a vast field of study and it is easy to feel overwhelmed. Therefore, I would suggest using the encyclopedia Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories by Yuri I. Manin and Alexei A. Panchishkin as the guide (MathOverflow). Moreover, those without the knowledge of undergraduate mathematics can use the book Number Theory in Science and Communication by Manfred R. Schroeder as the guide.

Elementary

• An Introduction to Diophantine Equations by D. Andrica, I. Cucurezeanu, and T. Andreescu: A really nice place to learn elementary methods for solving various types of Diophantine equations.
  • Hilbert’s Tenth Problem: An Introduction to Logic, Number Theory, and Computability by M. Ram Murty and B. Fodden: The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. In 1970, it was shown that no such algorithm exists.
• The Higher Arithmetic by H. Davenport: It is written in prose form, many interesting things to learn.
• An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery: Covers all flavors of number theory in adequate detail. Requires some mathematical maturity. The video recordings of the course taught based on this book by Richard Borcherds are also available.
• Prime Numbers: A Computational Perspective by R. Crandall and C. Pomerance: It is a survey of the most important methods for proving primality and factoring (a kind of expanded version of the books by Bressoud and Wagstaff).

Algebraic

• Number Fields by D. Marcus: A very good introductory text on algebraic number theory (chapters 2 to 5), where most big proofs are in the form of exercises. When I read this book in 2016, it was available only in old typewriter font but fortunately in 2018 Springer released its LaTeX edition! For supplementary reading, it is worth looking into the expository articles by Keith Conrad.
  • Algebraic number theory and Fermat’s last theorem by I. Stewart and D. Tall: It nicely illustrates the calculation of class numbers using examples.
  • Algebraic Number Theory, a Computational Approach by W. Stein: The chapters 2 to 8 complement Marcus’ book by providing many examples solved using SageMath. It is freely available online.
• Algebraic Number Theory by J. Neukirch (translated from German by N. Schappacher): A really nice second introduction to algebraic number theory for someone interested in arithmetic geometry. It serves the purpose of standard reference for global fields and local fields. I have read parts of Chapters I, II, III, and IV. It has a much better introduction to the geometry of numbers (sections I.4 and I.5) than Marcus.
  • Number Theory 1: Fermat’s Dream by K. Kato, N. Kurokawa, and T. Saito: Offers a unifying perspective for algebraic number theory by drawing attention to various interesting problems. A nice segue from Marcus to Neukirch.
  • p-adic Numbers: An Introduction by F. Q. Gouvêa: A good supplementary text for basic facts about $p$-adic numbers.
  • A Course in Computational Algebraic Number Theory by H. Cohen: The encyclopedic reference (no math/complexity proofs) for algorithmic number theory, written by the founder of PARI/GP. One can ignore the word “algebraic” in the title since it discusses all types of number theoretic algorithms. The chapters 4-6 of Cohen nicely complement the chapters I and II of Neukirch.
  • Number Theory in Function Fields by M. Rosen: This text nicely complements the study of function fields (chapter III of Neukirch). In particular, Chapter 5 introduces the machinery of divisors needed to state the Riemann-Roch theorem. A good reference for Artin’s primitive root conjecture for function fields, Brumer-Stark theory, the ABC Conjecture, results on class numbers, …
• Primes of the Form $x^2 + ny^2$ by D. Cox: Provides a working knowledge of the important results of (global) class field theory and ability to apply them to relevant situations (MathOverflow; MAA).
  • Number Theory 2: Introduction to Class Field Theory by K. Kato, N. Kurokawa, and T. Saito: It uses a similar motivation as in Cox’s book but the proofs are terse. The focus is on helping readers see the full picture instead of getting lost in details.
  • Advanced Topics in Computational Number Theory by H. Cohen: It focuses primarily on algebraic number theory, and should have “algebraic” in its title. The chapters 3 to 6 contain the algorithms concerning class field theory over number fields.
  • Elliptic functions by S. Lang: Examples of explicit class field theory in the global case with an imaginary quadratic base field via the theory of elliptic curves with complex multiplication. Uses Deuring’s formulation and complements the chapter 7 of Cox’s text.
• Class Field Theory by J.S. Milne: This is good for learning local class field theory from the perspective of Galois cohomology (unlike Neukirch’s text above). I have read parts of Chapter I, II, III and V. It is freely available online. Also see the article by Keith Conrad which complements Milne’s Dramatis Personæ and Introduction.
  • Notes on class field theory by K. Kedlaya: It starts with the Kronecker-Weber theorem (where Marcus left us) and then builds on Neukirch’s first three chapters, following Milne’s notes. That is, it follows Milne up through local class field theory, then recasts local class field theory in an Artin-Tate-style abstract framework following Neukirch, and then use that framework to develop global class field theory. It is available online.
  • Local Fields by P. Serre: Understanding of the structure and basic properties of local fields and adelic/idelic stuff.
• Automorphic Representations and L-Functions for the General Linear Group: Vol 1 and 2 by D. Goldfeld and J. Hundley: The purpose of this book is to provide an elementary yet extremely rigorous exposition of the theory of cuspidal automorphic representations and L-functions for the general linear group in a textbook form that can be understood by the beginning graduate student with minimal background in representation theory. Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program.
  • Lie Groups, Lie Algebras, and Representations by B. C. Hall: The Part I of the text develops the theory of (matrix) Lie groups and their Lie algebras using only linear algebra, without requiring any knowledge of manifold theory. The Part II of the text covers semisimple Lie algebras and their representations. Finally, the Part III of the book presents the compact-group approach to representation theory. Moreover, the results in Parts II and III of the book have been explained and motivated using figures whenever possible. I have only read some topics from the Part I of this book.
  • An Introduction to the Langlands Program: Proceedings of the school held at the Hebrew University of Jerusalem (March 12-16, 2001) edited by J. Bernstein and S. Gelbart: Gives a formulation of the Langlands program, painting a very broad picture connecting automorphic forms and L-functions. Also see the article by Solomon Friedberg.
  • Number Theory 3: Iwasawa Theory and Modular Forms by N. Kurokawa, M. Kurihara and T. Saito: This provides a high-level overview of Iwasawa theory and Langlands conjectures for automorphic representations. Both of these involve the intertwinement of algebra and analysis. Algebraic entities are Galois groups and algebraic-geometric objects, and analytic entities are $\zeta$ functions, modular forms, and automorphic representations.
  • Number theory with applications by W. C. Winnie Li: This book is about constructing Ramanujan graphs. It discusses Weil Conjectures (Riemann hypothesis for curves over finite fields), Ramanujan-Petersson conjecture (estimates of Fourier coefficients of modular forms), and Jacquet-Langlands theory (representation theory of GL(2)). Also see the notes by Winnie Li in IAS/PCMI 2002 and CBMS 2014.

Analytic

• An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright: It is a classic textbook with lot of interesting stuff using elementary analytic methods.
• Making Transcendence Transparent by E.B. Burger and R. Tubbs: Very nice introductory text for learning the techniques used in transcendental number theory. Also see the short lecture series by Michel Waldschmidt.
  • Numbers: Rational and Irrational by Ivan Niven: A beautiful little book (MAA).
• Multiplicative Number Theory by H. Davenport: This is a tough book for introducing analytic number theory (the analytic theory of L-functions), but worth the hard work since you get to prove Prime Number Theorem with a good estimate (chapters 1–22). However, anyone interested in analytic number theory must read chapter 8 about Riemann’s contribution (he wrote only one paper on this topic!).
  • The Theory of Functions by E. C. Titchmarsh. A good reference for the complex analysis used by Davenport. Might also be worth looking at Titchmarsh’s other book “The Theory of the Riemann Zeta-Function.”
  • An Introduction to the Analytic Theory of Numbers by R. Ayoub: A good supplementary text for the methods and estimates used by Davenport.
  • A Classical Introduction to Modern Number Theory by K. Ireland and M. Rosen: Provides a straightforward proof of Dirichlet’s theorem on infinite primes arithmetic progressions in chapter 16. Also see chapters 10 and 11 for a nice exposition on Gauss Sums, Jacobi Sums, and Weil Conjectures.
• Fourier Analysis on Number Fields by D. Ramakrishnan and R. J. Valenza: It is an introduction to the adelic approach to number theory organized around John Tate’s Ph.D. thesis which uses representation theory for locally compact groups (review). The last chapter contains the proof of Dirichlet’s class number formula. A nice exposition about the connection between classical methods (Davenport’s book above) and idelic methods (Tate’s thesis) can be found in this article by Gelbart and Miller.
  • Algebraic Number Theory: Proceedings of the instructional conference held at the University of Sussex (Sep 1-17, 1965) edited by J. W. S. Cassels and A. Fröhlich: It uses local fields and adeles heavily. Also contains Tate’s doctoral thesis that, in some sense, laid the foundations for the Langlands program (Math.SE). Also, see the notes by Bjorn Poonen and Kudla’s articles from Bernstein and Gelbart’s book above about Langland’s Program.
  • Algebraic Number Theory by S. Lang: The last part is a nice source for some important analytic aspects of algebraic number theory, similar to chapter 7 of Marcus’ book and chapter VII of Neukirch’s book mentioned above. It also contains the proof of Brauer-Siegel theorem (for imaginary quadratic fields, i.e. discriminant D<0, the class number is approximately of size √|D|).
  • Basic Number Theory by A. Weil: It presents algebraic number theory from the adelic perspective, showing how adelic methods could provide simple and unified proofs of all the results proved in algebraic number theory (MathOverflow).
• Additive Number Theory: The Classical Bases by M. B. Nathanson: This book gives an introduction to the circle method and the sieve method. Nowadays, this field is called “arithmetic combinatorics” and uses ergodic theory and harmonic analysis.
  • Elementary Methods in Number Theory by M. B. Nathanson: The part 3 of this book gives a good introduction to additive number theory (Waring’s problem, sums of squares, and asymptotics of partition functions).
  • Introduction to Analytic and Probabilistic Number Theory by G. Tenenbaum (translated from the French by P. D. F. Ion): Probabilistic methods (Part III of this book) grew out of Erdős’ investigations of additive number theory problems. For more details see this lecture by Rényi at ICM 1958. (COVID-19 era video lectures by Emmanuel Kowalski based on his smaller textbook; lecture videos of Kaneenika Sinha)
  • Sieves in Number Theory by G. Greaves: Discusses the “small” sieve method of V. Brun, A.Selberg, H. Iwaniec and others.
  • Three Pearls of Number Theory by A. Y. Khinchin: A beautiful little book (MAA).
• Number Theory, Fourier Analysis and Geometric Discrepancy by G. Travaglini: This book consists of two parts. The first part is concerned with classical topics and results from elementary number theory, while the second part is concerned with the particular subject of uniform distribution theory and discrepancy theory. Many of the results are shown using Fourier-analytic arguments, and a basic introduction to Fourier analysis is incorporated into the text at the necessary positions (MAA and MathOverflow)
  • Fourier Analysis: An Introduction by E. M. Stein and R. Shakarchi: It is a good reference for “Weyl’s Equidistribution Theorem” (section 4.2) which is closely related to “Kronecker’s Theorem” (Diophantine approximation) discussed in Hardy & Wright’s book (section 23.3, latest edition).
  • Uniform distribution of sequences by L. Kuipers and Niederreiter: The first-ever survey in this field. Also see the commentary by Pete Clark involving “Equidistribution in Number Theory, An Introduction” SMS–NATO ASI Summer School Montreal 2005 notes edited by Granville and Rudnick.
  • Sequences, Discrepancies and Applications by M. Drmota and R. F. Tichy: This book discusses special developments and methods of the theory of uniform distribution since Kuipers-Niederreiter’s book.
  • Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis by H. L. Montgomery: A standard reference for number theorists to learn about exponential sums.
  • Topics in Ergodic Theory by W. Parry: The result “Uniform distribution (mod 1)” of H. Weyl can be regarded as the first ergodic theorem to be discovered.
  • Introduction to Quasi-Monte Carlo Integration and Applications by Leobacher and Pillichshammer: The theory of Uniform Distribution Modulo One is used for numerical integration based on QMC rules.
  • Applied Number Theory by H. Niederreiter and A. Winterhof: The chapters 4 and 5 nicely complement the main text. One can supplement it with the classic text “Random Number Generation and Quasi-Monte Carlo Methods” by Niederreiter.
• p-adic Numbers, p-adic Analysis, and Zeta-Functions by N. Koblitz: It discusses the p-adic Mazur-Mellin transform and Dwork’s proof of the rationality of zeta-function (Weil conjecture). Never read.
  • p-adic Analysis Compared with Real by S. Katok: A gentle undergraduate-level exposition, like Gouvêa’s textbook cited above. It also talks about the notion of differentiability for functions from $\mathbb{Q}_p$ into itself.
• Introduction to Elliptic Curves and Modular Forms by N. Koblitz: A nice introduction to modular forms, which basically adds details and exercises to Serre’s condensed one chapter introduction in his classic book “A Course in Arithmetic”. The study of modular forms is motivated via congruent number problem, and a brief discussion of analytic theory of elliptic curves. In particular, it contains an explanation of Tunnell’s solution to the “congruence number problem’’ (modulo the weak Birch-Swinnerton-Dyer conjecture). The lecture series by Richard Borcherds nicely complements this book (a more modern introduction).
  • Complex Analysis by E. M. Stein and R. Shakarchi: A good reference for the facts from complex analysis needed for understanding the proofs involving modular forms.
  • Fuchsian groups by S. Katok: A good reference for understanding the shapes of fundamental domains.
  • Topics in Classical Automorphic Forms by Iwaniec: An introduction to the study of the analytic approach to modular forms.
  • Some Applications of Modular Forms by Sarnak: Discusses the solution of three problems (Ruziewicz’s problem from measure theory, constructing Ramanujan graphs, and Lennik’s conjecture about the sum of three squares) using modular forms.
  • The 1-2-3 of Modular Forms by Zagier et al.: It has lecture notes by four authors. The first lecture discusses the applications of classical (elliptic) modular forms to questions of number theory, combinatorics, differential equations, geometry, and mathematical physics. Furthermore, there is a bonus chapter about the congruence between a Siegel and an elliptic modular form. (Bonus: Dimension formulas for spaces of Siegel modular forms of degree 2)
  • Lattices and Codes by W. Ebeling: Lattices are studied in number theory and in the geometry of numbers. Many problems about codes have their counterpart in problems about lattices and sphere packings via the associated modular forms (theta functions). In this book the connections between weight enumerators of codes and theta functions of lattices, the classification of even unimodular 24-dimensional lattices, and the Leech lattice are discussed.

Topological

• The Sensual (quadratic) Form by J. H. Conway: Introduces the concept of “topograph” to be able to see quadratic forms. I came to know about its usage in number theory from this paper by James Rickards.
  • An Illustrated Theory of Numbers by M. H. Weissman: The Part III of this book introduces the topograph of binary quadratic forms following Chapter 1 of J. H. Conway’s book.
  • Topology of Numbers by Hatcher: This book studies quadratic forms in two variables with integer coefficients from a perspective that emphasizes Conway’s notion of the topograph of a quadratic form. However, it uses different names of terms than the ones defined by Conway. The book draft is available on author’s website.
  • Rigid cocycles and singular moduli for real quadratic fields by Jan Vonk: Lectures notes from PCMI 2022 will be posted here.

Geometric

• Diophantine Geometry: An Introduction by M. Hindry and J. H. Silverman: The study of Diophantine equations using tools from algebraic number theory and “classical” algebraic geometry. It contains the proof of the four fundamental finiteness theorems: Mordell-Weil Theorem, Roth’s theorem, Siegel’s theorem, and Faltings’ theorem (Bombieri’s proof based on Vojta’s). The Part A is a great algebraic geometry reference for number theorists. Also see the short lecture series based on this book by Elisa Lorenzo García.
  • Rational Points on Elliptic Curves by J. H. Silverman and J. Tate: One of the most accessible introductions to the world of elliptic curves, especially the theory of heights (a major part of Diophantine geometry). I read chapters I to III of this book a while ago for Mordell’s theorem (the special case of elliptic curves with a rational point of order two). Its Section 4.4 contains a nice exposition of Lenstra’s elliptic curve factorization algorithm. The appendix on projective geometry is also great.
  • Diophantine Approximation by W. M. Schmidt: Standard reference for the general theory. Also contains proofs of theorems belonging to “Geometry of Numbers.”
  • Exploring the Number Jungle: A Journey into Diophantine Analysis by E. B. Burger: A beautiful little book (MAA).
  • Algebraic functions and projective curves by D. Goldschmidt: There are a number of formal similarities between number fields and global function fields (function field of an irreducible algebraic curve over a finite field). However, it is usually easier to work in the global function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory (section B.10 of Hindry-Silverman) and its exploitation by Faltings (as well as Vojta) in the proof of the Mordell conjecture is a dramatic example. Furthermore, in the last chapter of this book, Weil bound is proved following the work of Stöhr and Voloch and Bombieri. Also see the lecture notes by Pete Clark.
• The Arithmetic of Elliptic Curves by J. H. Silverman: The standard introductory text about elliptic curves using techniques from algebraic geometry (varieties in projective spaces). Make sure to use the 2nd edition corrected in 2016 printing, since earlier versions have lot of typos. It contains the proof of Weil’s conjectures for elliptic curves, Mordell-Weil theorem, and Siegel’s theorem, along with a discussion on Szpiro’s conjecture and the ABC conjecture. The recordings of a course based on this book taught by Álvaro Lozano-Robledo are also available.
  • Basic Algebraic Geometry by I. Shafarevich (translated from the Russian by K. A. Hirsch): This is a reference for many proofs skipped in the first two chapters by Silverman. Note that this is quiet different, in terms of notations and organization, from the newer expanded translation by Miles Reid.
  • Galois Representations by Gabor Wiese: These notes complement the discussion in section III.7 of Silverman’s text. They are available online. For an overview, see the article by Mark Kisin.
  • Quaternion Algebras by J. Voight: The endomorphism ring of supersingular elliptic curves is an order of quaternion algebra. The chapter 42 of Voight complements section V.3 of Silverman. The ebook is available here.
  • Topics in Number Theory (Volumes I and II) by W. J. LeVeque: The best place to understand the proof of Roth’s theorem and Delaunay-Nagell theorem (discussed in the volume 2 part). The volume 1 part can be used as an introduction to elementary number theory.
  • Elliptic Curves by D. Husemöller: Its Chapter 13 complements Chapter V of Silverman. Moreover, the last two chapters contain remarks about BSD, FLT and TSW.
  • Elliptic curves: A computational approach by S. Schmitt, and H. Zimmer: It nicely complements Silverman’s book, providing a path into the computational theory of elliptic curves. It includes the determination of torsion groups, computations concerning the Mordell–Weil group, especially concerning the rank and basis of that group, height calculations, and the determination of integral and, more generally, S-integral points. Also see the lecture notes by Sutherland.
  • Drinfeld Modules by M. Papikian: This text develops the theory of Drinfeld modules in parallel with the theory of elliptic curves developed in Silverman’s textbook. Note that Drinfeld modules are essentially linear algebra objects, hence their theory can be presented without any appeal to algebraic geometry, which is impossible to do for elliptic curves. Furthermore, Drinfeld modules were introduced to prove the function field analogue of Langlands conjectures for GL(2).
• A First Course in Modular Forms by F. Diamond and J. Shurman: Don’t get fooled by the title of the book (Koblitz’s book above is better suited for a first course). This book explains the Modularity Theorem (Taniyama–Shimura-Weil conjecture), “All rational elliptic curves arise from modular forms.” The modularity theorem is, in some sense, a special case of the Langlands program.
  • Algebraic Curves and Riemann Surfaces by R. Miranda: It is a nice introductory text. Its Chapter IX is a good source for learning about Cech cohomology. This complements chapter 2 of Diamond-Shurman. Also see the article by James Milne.
  • Advanced Topics in the Arithmetic of Elliptic Curves by J.H. Silverman: The chapters I and II nicely complement Diamond-Shurman, especially the discussion on modular polynomials. Also see the above books by Lang (Elliptic Functions) and Cox (primes of the form…).
  • Arithmetic Geometry: Proceedings of the CMI summer school held at Göttingen, Germany (Jul 17 - Aug 11, 2006) edited by H. Darmon, D. A. Ellwood, B. Hassett, and Y. Tschinkel: It contains the articles “Rational points on curves” by Darmon and “Merel’s theorem on the boundedness of the torsion of elliptic curves” by Rebolledo which discuss Mazur’s torsion theorem, proved using modular curves. The proceedings are available online. One can supplement it with the lecture notes by Snowden and Weston.
  • Modular Forms and Fermat’s Last Theorem: Proceedings of the instructional conference held at Boston University (Aug 9-18, 1995) edited by G. Cornell, J.H. Silverman and G. Stevens: The proof of Fermat’s last theorem led to the proof of Modularity theorem. This text sheds light on its relation with Serre’s conjectures and Langlands program. Also see the article by Darmon, Diamond and Taylor to see how Mazur’s proof of Torsion conjecture essentially gave a complete answer to the analogue of Fermat’s Last Theorem for modular curves. Also see the last chapter of Milne’s textbook.
  • Fermat’s Last Theorem (Basic Tools and The Proof) by T. Saito (translated from Japanese by M. Kuwata): The treatment of modular forms in Chapter 2 of the first volume “Basic Tools” emphasizes the algebro-geometric point of view of modular curves, serving as a companion to Diamond and Shurman. Also see the review by B. Sury.
  • Modular Forms: A Computational Approach by W. Stein: This is a graduate-level textbook about algorithms for computing with modular forms, written by the founder of SageMath. The ebook copy is available online. Also see Cremona’s textbook and LMFDB.
  • Lectures on Modular Forms and Hecke Operators by K. Ribet and W. Stein: This is draft of a modern textbook on modular forms and modular curves that takes into account the full modularity theorem and full Serre conjecture, along with the available computational techniques. Also see the survey article by Diamond and Im and the handouts from B. Conrad’s course.
  • Algebraic Curves over Finite Fields by C. Moreno: This book explains the Tsfasman-Vlǎduţ-Zink theorem involving modular curves. It includes a discussion on L-functions (Bombieri’s elementary proof of the Weil conjectures for curves), estimates for exponential sums, Goppa codes and modular curves. Also see the notes by Voight.
  • Algebraic Geometric Codes (Basic Notions and Advanced Chapters) by Tsfasman, Vlǎduţ, and Nogin: The first volume “Basic Notions” serves as a textbook and the second volume “Advanced Chapters” serves as a reference text. It gives a nice exposition on elliptic curves (Basic Notions) and modular curves (Advanced Chapters) over finite fields. It also ventures into using class field theory for constructing curves with many points (https://manypoints.org/) and sphere packings. Also see the lecture notes by Schoof. (MAA).
• Rational Points on Varieties by B. Poonen: It provides a good introduction to the theory of obstructions to rational points on a variety, while briefly discussing other useful tools like étale cohomology and its relation to Weil conjectures. However, I have only read the first four chapters which discuss basic stuff. The book draft is available online.
  • Central Simple Algebras and Galois Cohomology by P. Gille and T. Szamuely: This book discusses the proofs and exercises given in the first and fourth chapters of Poonen’s book. Central Simple Algebras (CSAs) over a field K are a non-commutative analog to extension fields over K – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a division algebra). This is of particular interest in “noncommutative number theory” as generalizations of number fields. A detailed discussion of Brauer group examples related to class field theory is available in chapter IV and section VIII.4 of Milne’s Class Field Theory notes.
  • Algebraic Geometry and Arithmetic Curves by Q. Liu (translated from the French by R. Erné): This book discusses the proofs and exercises given in the second and third chapters of Poonen’s book. An introduction to the language of schemes (chapters 1 to 4), without assuming the base field to be algebraically closed, or of characteristic zero. Some prior knowledge of complex geometry might be helpful when reading this book.
• Étale Cohomology and the Weil Conjecture by E. Freitag and R. Kiehl (translated from the German by B. S. Waterhouse and W. C. Waterhouse): The only book I am aware of which discusses Deligne’s proof of Weil conjectures (especially, Riemann hypothesis in char p and Ramanujan-Petersson conjecture). Also look at the lecture notes by Chao Li, Katz, Milne and Szamuely. (Math.SE + ALGANT thesis)
  • Algebraic Geometry by B. Edixhoven, D. Holmes, A. Kret, and L. Taelman: These lecture notes discuss Weil’s proof of the Riemann Hypothesis for curves over finite fields using intersection theory on surfaces. No etale cohomology needed. Also see the notes by Hansen.
  • The Riemann hypothesis in characteristic p in historical perspective by P. Roquette: It contains the story of the Riemann hypothesis for function fields, or curves, of characteristic p starting with Artin’s thesis in the year 1921, covering Hasse’s work in the 1930s on elliptic fields and more, until Weil’s final proof in 1948. Also see the expository article by Milne for story after Weil’s proof.
• Arithmetic Geometry: Proceedings of the instructional conference held at the University of Connecticut (Jul 30 - Aug 10, 1984) edited by G. Cornell and J. H. Silverman: The reference to learn basics of advanced topics like Jacobian varieties, Siegel modular variety, and Néron Models. It also contains the article by Milne which discusses abelian varieties without assuming ground field to be algebraically closed. Most of the articles are intended as expositions of the tools used in the original proof of Faltings’ theorem.
  • Algebraic Geometry by R. Hartshorne: The chapter II and III of this book provide an introduction to algebraic geometry via the theory of schemes over algebraically closed field. Good reference for the cohomology of schemes.
  • Abelian Varieties by D. Mumford: A scheme-theoretic treatment of most of the basic theory of abelian varieties. However, all ground fields are assumed to be algebraically closed, which limits the usefulness of the book as a reference for arithmetic geometers. Also see the handouts from B. Conrad’s course.
  • Abelian varieties over finite fields by F. Oort: Hond-Tate theory of abelian varieties over finite fields gives their classification up to isogeny. This is a step towards learning Falting’s proof of Mordell’s conjecture. These notes are available online. Also see the lecture notes by Papikian for the elliptic curves case.
  • Introduction to Shimura Varieties by J.S. Milne: The section 6 is about the Siegel modular variety, the most basic Shimura variety (available online). In fact, the main idea of Faltings’ proof is the comparison of Faltings heights and naive heights via Siegel modular varieties. For more information, see this article by Bloch. Furthermore, there exists proof for geometric Jacquet–Langlands correspondence for some Siegel modular forms.
  • Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2 by J. W. S. Cassels and E. V. Flynn: By Falting’s theorem, the number of rational points on a curve of genus 2 is finite. However, the proof doesn’t provide any algorithm for finding those points. The method of Chaubauty was later extended to get feasible approaches for determining the set of rational points on a given curve of genus at least 2. This is also a good reference to learn about Richelot isogeny (chapters 9 and 10). The MAPLE code used in this book is available on Flynn’s website.

Miscellaneous

These are some topics that can’t be put into one of the above categories. A good reference for such topics is the Arizona Winter Schools.

• Arithmetic Dynamics: Silverman’s list of resources.
• Arithmetic Statistics: Lozano-Robledo’s list of resources
• Applied Model Theory: MathOverflow discussion
• p-adic Hodge Theory: Gulotta’s list of resources
• Shimura curves: MathOverflow discussion
• Anabelian geometry: nLab references

Scary Maths

I studied these topics for my master’s thesis (2018-19).

Complex Geometry

• Analytic Functions of Several Complex Variables by Gunning and Rossi: The necessary background about several variable complex-valued functions, and a detailed proof of $\overline{\partial}$-Poincare lemma.
  • Functions of One Complex Variable by J. B. Conway: I really like the proofs given in this book (Runge’s theorem, Mittag-Leffler’s theorem and Weierstrass factorization theorem) which are critical for understanding Riemann surfaces.
  • From Holomorphic Functions to Complex Manifolds by Fritzsche and Grauert: For analytic subvariety properties. Didn’t spend much time on it.
  • Function Theory of Several Complex Variables by S. G. Krantz: The proof of Dolbeault theorem for $\mathbb{C}^n$ is discussed in section 6.3.
  • Differential Analysis on Complex Manifolds by R. O. Wells: For understanding vector bundles and more.
• Principles of Algebraic Geometry by J. Harris and P. Griffiths: The only available introductory text on complex algebraic geometry. Not at all a good book for self study, my whole master’s thesis was an attempt to understand 14 pages of this book (pp. 34-47): “Any analytic hypersurface in $\mathbb{C}^n$ is the zero-locus of a global holomorphic function.” However, trying to understand the given collection of ideas is worth the effort.
  • Complex Geometry by D. Huybrechts: The first two chapters of this book cover the necessary complex analysis and complex manifolds background needed for Griffiths-Harris. Especially the proof of $\overline{\partial}$-Poincare lemma. The proof of Poincare lemma can be found in Munkres’ “Analysis on Manifolds”.
  • Foundations of Differentiable Manifolds and Lie Groups by W. F. Warner: Chapter 5 where the equivalence of Cech cohomology to sheaf cohomology is discussed.
  • Topological Methods in Algebraic Geometry by F. Hirzebruch (translated from the German by R. L. E. Schwarzenberger): Section 2 for the relation between Cech and de Rham cohomology.

Initially the plan was to learn Hodge Theory, but I found this preliminary stuff to be too difficult to grasp. A good reference for that would be the book Hodge Theory and Complex Algebraic Geometry by Claire Voisin.

Enumerative Geometry

• Enumerative Geometry and String Theory by S. Katz: Standard introductory text. Can be supplemented with the articles: (1) “Enumerative Algebraic Geometry of Conics” by Bashelor, Ksir, and Traves (MAA), (2) “Introduction to Moduli Spaces: A cycle of lectures at Universidad de Costa Rica” by Renzo Cavalieri (combined pdf)
• Mirror Symmetry and Algebraic Geometry by D. A. Cox and S. Katz: Standard reference for learning about Gromov-Witten Invariants (chapter 7). Not an easy read.
  • Algebraic Geometry by A. Gathmann: A gentle introduction to algebraic geometry with lots of computational exercises oriented towards the useful “intersection theory”. There are many editions available online, but I have read only the 2002 edition.
  • Differential Forms in Algebraic Topology by Bott and Tu: A good reference for learning to compute the first Chern class of a complex line bundle (Proposition 6.4).
  • Mirror Symmetry: Proceedings of the CMI summer school held at Pine Manor College, Massachusetts (May 30 - June 25, 2000) by C. Vafa, et al.: For a more gentle introduction to Gromov-Witten Invariants, try reading Chapter 21 to 23 of this book. It is available online on Clay Mathematics Institute website.
• An Invitation to Quantum Cohomology: Kontsevich’s Formula for Rational Plane Curves by Kock and Vainsencher: A good place to learn teachiniques involved in calculations (haven’t read it myself). You can supplement it with the article “Counting Plane Rational Curves: Old and New Approaches” by Aleksey Zinger (arXiv) and the Ph.D. thesis “Gromov-Witten and degeneration invariants: computation and enumerative significance” by Andreas Gathmann.

I made an attempt to learn this topic so that I could build upon the skills gained while learning complex geometry. However I didn’t make much progress due to my lack of interest.