18.785F16 Number Theory I Lecture 26 Notes: Global Class Field Theory, Chebotarev Density

18.785F17 Number Theory I Lecture 28 Notes: Global Class Field Theory, Chebotarev Density

the unit group of AK but has a finer topology (using the restricted product topology ensures that a 7→ a−1 is continuous, which is not true of the subspace topology). As a topological group, IK is locally compact and Hausdorff. The multiplicative group K× is canonically embedded as a discrete subgroup of IK via the diagonal map x 7→ (x, x, x, . . .), and the idele class group is the quotient CK...

18.785F16 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions

18.785F16 Number Theory I Lecture 21 Notes: Class Field Theory

In the Lecture 20 we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We would like to make the correspondence between H and L explicit, and then ge...

Notes on (0,2) Superconformal Field Theories

In these lecture notes, I review the " linear σ-model " approach to (0,2) string vacua. My aim is to provide the reader with a toolkit for studying a very broad class of (0,2) superconformal field theories with the requisite properties to be candidate string vacua.

Notes on Class Field Theory , Postech Summer School 2010

Preface 1 Preliminaries: Galois theory, algebraic number theory 2 Lecture 1. CFT of Q: classical (Mo. 19/7/10, 9:40–10:40) 4 Lecture 2. CFT of Q: via adeles (Mo. 19/7/10, 11:00–12:00) 6 Lecture 3. Local CFT, local-global compatibility (Tu. 20/7/10, 9:40–10:40) 8 Lecture 4. Global CFT, l-adic characters (Tu. 20/7/10, 11:00–12:00) 10 Appendix A. More on GLC for GL1: algebraic Hecke characters 12 ...