Having been inspired by this question I was wondering, what are some important papers in arithmetic geometry and number theory that should be read by students interested in these fields?
There is a related wikipedia article along these lines, but it doesn't mention some important papers such as Mazur's "Modular Curves and the Eisenstein ideal" paper or Ribet's Inventiones 100 paper.
On
Serre's Duke 54 paper on his famous conjecture about modularity of mod $p$ Galois representations (now a theorem of Khare--Wintenberger and Kisin, and one of the highlights of 21st century number theory so far).
Khare has written some expository papers on his argument with Wintenberger, and so you might like to read these in conjunction with Serre's original paper.
Also, there is an old paper of Tate where he proves the level $1$, $p = 2$ case of Serre's conjecture by algebraic number theory methods. It serves as the base-case of a very sophisticated induction (on both the level and the prime $p$!) in the Khare--Wintenberger argument, and is a good read.
There is also Serre's much older (early 70s) paper on weight one modular forms and the associated Galois representations. (It's from a Durham proceedings.) It is a kind of expository companion to his paper with Deligne where they construct the Galois representations for weight one forms. The Deligne--Serre paper itself is also fantastic (but a more technical read than Serre's Durham paper).
To add to the Galois representation-theoretic suggestions:
There is Shimura's article A non-solvable reciprocity law, in which he describes the elliptic curve $X_0(11)$ and the relationship to its points mod $p$ and the Hecke eigenvalues of the (unique normalized) cuspform of level $11$.
As a supplement and motivational guide to the paper, you could look at the paper What is a reciprocity law by Wyman. (I found it very helpful as an overview to what reciprocity laws were about when I first began trying to learn number theory.)
Answer posted by Zev Chonoles in the comments: