The plan, which may be modified, is to develop the standard algebraic theory of numbers to a point where students can understand the recent solution by Dasgupta and Kakde of Hilbert's 12th problem for totally real fields. In its original formulation, Hilbert's problem asked whether all abelian extensions of a general number field K could be generated by special values of transcendental functions. When K = Q or an imaginary quadratic field, it was known that this could be done by taking as transcendental functions the complex exponential or elliptic functions, respectively. When K is a totally imaginary quadratic extension of a totally real field, a partial solution was given by Shimura and Taniyama in their theory of complex multiplication; this in turn led to Shimura's study of the arithmetic properties of complex algebraic varieties that are uniformized by hermitian symmetric spaces, now known as Shimura varieties, that have been central in the development over the last 50 years of the Langlands program.
Dasgupta and Kakde built on several decades of work to solve a version of Hilbert's problem, where the transcendental functions are p-adic rather than complex. While their work is undoubtedly too technical to be treated in full detail in the space of a single semester, the hope is that students will be able to read and understand their papers by the end.
Here is a tentative schedule:
Adèles and idèles of number fields, and the basic theory of p-adic fields, including a proof of finiteness of class number and Dirichlet's unit theorem (3 weeks)
Tate's thesis (functional equations of L-functions of Hecke characters), including the analytic class number formula (2 weeks)
Review of class field theory, mainly without proofs (2 weeks)
Introduction to modular forms (2 weeks)
Introduction to p-adic modular forms (2 weeks)
The Brumer-Stark and Gross-Stark conjectures (3 weeks)