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Apologies if this question is too vague, or if it's been asked before. I'm wondering if there is, as the title might imply, a more advanced analogue of Ireland and Rosen's A Classical Introduction to Modern Number Theory. More explicitly, books that collect various (if somewhat unrelated) nonstandard topics in advanced number theory, not hesitating to assume algebraic number theory (like the first one or two chapters of Neukirch, or even all of class field theory). I'm aware Ireland and Rosen does touch on some advanced topics, but I'm looking for a book more directly focused on advanced topics. I'm also aware of Cox's book, but it treats one topic (primes of the form $x^2+ny^2$) in huge depth. In contrast, I'm looking for a book that introduces various interesting topics. Is there any such book?

littleman
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What do you consider the "nonstandard" topics in Ireland & Rosen?

A suggested book: Manin and Panchishkin's Introduction to Modern Number Theory

KCd
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  • It might not actually be nonstandard, but while writing this question I was considering the inclusion of higher reciprocity laws, equations over finite fields, zeta functions, and Bernoulli numbers to be unusual. I'm not quite sure what a better word to use would be, but there's a specific feeling from Ireland & Rosen that I was referring to. – littleman May 29 '23 at 09:22
  • And about Manin and Panchishkin: What level would you say the book is at, and what would you advise a reader to know beforehand? I know it technically is self-contained but with the huge amount of material covered, I worry I won't be able to keep up (having only read the first two chapters of Neukirch, though I am still working through the rest). – littleman May 29 '23 at 09:29
  • The chapters of M&P are at very different levels, beginning with elementary number theory and ending with chapters on the proof of FLT and Arakelov geometry. Unlike I&R, there are almost no complete proofs in M&P. Although M&P has chapters on algebraic number theory, I think a reader should already have read about that and (global and local) class field theory elsewhere first, along with algebraic geometry. Look at the table of contents online. – KCd May 29 '23 at 14:58