The great arithmetician Kazuya Kato visited twice over the last few weeks, so I thought I’d use the occasion to recommend some writings. An undergraduate level textbook on number theory is
Number Theory I: Fermat’s Dream
published by the American Mathematical Society. It is short and covers fairly standard material, but contains many unusual insights. A research article that represents quite well Kato’s vision of number theory is
available online if your institution subscribes to Springer. The background required for a genuine reading of the paper is rather extensive, but even without it, you can enjoy the introduction, the first few sections, and the closing remarks.
A main theme of the work is that the somewhat mysterious p-adic zeta functions and L-functions are the objects with direct relevance to the important problems of arithmetic geometry, while the usual complex functions are a sort of intermediary. It took me a long time to come to terms with this view, especially since I still don’t understand these functions (actually elements of some non-commutative algebra of measures) at all well, but it is eventually an essential component of my own thoughts about Diophantine geometry.