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I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do something involving Shimura varieties, Galois representation, modular forms). I have been reading (but not doing the exercises to save time) various books, papers, going to conferences, seminars, etc but that has not seemed to be helping me retain any information or develop the skills to prove anything. I’ve decided I just need to start writing up solutions to a bunch of exercises. (I confess I didn’t do enough exercises in graduate school and that could be why I have been a failure as a researcher, now I have the luxury of time to go back and fill in gaps in my knowledge).

Here are some possibly relevant books that I’ve found that have exercises, and my question is what are some others?:

  • Childress Class field theory

  • Neukirch Algebraic Number Theory

  • Neukirch Schmidt Winberg Cohomology of Number Fields


  • Silverman Elliptic Curves (both volumes)
  • Helgason Differential Geometry, Lie Groups, and Symmetric Spaces

  • Washington Cyclotomic Fields

  • Liu Algebraic Geometry and Arithmetic Curves

  • Ramakrishnan Valenza Fourier Analysis on Number Fields

  • Diamond Shurman First Course in Modular Forms

  • Cox Primes of the form x^2 + ny^2

  • Gille Szamuely Central Simple Algebras and Galois Cohomology
  • Bump Automorphic Forms and Representations
  • Bourbaki Lie Groups and Lie Algebras
  • Vakil Foundations of Algebraic Geometry
  • Hartshorne Algebraic Geometry

What are some others?

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This is perhaps not the type of answer you were looking for (in which case I apologize), but it isn't clear to me that doing all of the exercises in an enormous number of books is the most efficient way to get to your goal of contributing to some aspect of the Langlands program, Shimura varieties, Galois representations, modular forms, etc.

The issue, I think, is that (1) the approach you outline is bound to take an enormous amount of time, and (2) while you are reading all of this material it will never be clear which topics are going to be relevant to your future research and which aspects won't be. In general I find that learning from a book is easiest when I have a clear goal (e.g., I want to generalize a known result from quaternion algebras to arbitrary central simple algebras, and this book concerns the structure of central simple algebras). Absent such a goal I find it very hard to figure out what aspects of the relevant theory are most important and wind up retaining very little of what I read.

In light of the above, I would recommend that you begin by choosing a single topic from your list, say modular forms. Obviously you can't work on problems about modular forms if you don't know anything about them, so I would read through Diamond and Shurman's First Course in Modular Forms. (Some of the other topics on your list require more in the way of background knowledge.) My goal would not be to memorize the book or even complete all of the exercises. I would just focus on trying to get to the point that you are familiar with the basic objects of study, definitions and fundamental results (i.e., congruence subgroups, Hecke operators, newforms, etc).

Once I've gotten to this level I would start looking for a paper about modular forms that looked interesting to me (i.e., the statement of the main theorem makes sense to me, the proof doesn't look too technical, it isn't too long, etc) and start reading the paper. You'll of course need to do a lot of supplementary reading (from textbooks) while working your way through the paper, but I believe that you will retain much more of what you read because you have a clear goal (understand what the author of the paper wrote). Keep in mind that it might take a few iterations of this process before you land on a paper that you like / are able to work through.

Once you've read this paper you could try to find a different paper to read, or even try to decide if it would be interesting to generalize the paper (e.g., This paper proved something about integral weight modular forms. I remember vaguely reading something about half-integral weight modular forms; perhaps it would be interesting to learn more about them and see if I can generalize the paper to the setting of half-integral weight modular forms). Also, these topics all interact in a variety of ways, so (to continue the example with modular forms) your work on modular forms could very well lead you to learn a bit about elliptic curves.

None of this will get you from 0 to writing important papers about Shimura varieties of course, but hopefully it will help you start learning about whatever area interests you in an efficient manner and build up some confidence in your ability to do research in this area.

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