# Number Theory Lecture 1

This lecture was given on August 27th 2009

### Reciporcity

Thm (Gauss): Let be odd primes. Then and are either both soluble or insoluble except when p,q are both congruent to 3 mod 4, in which case exactly one is soluble.

Perspective: fix q and let p vary. Then to determine ( for ) if it suffices (in most cases) to determine if

### frob p and S(E)

Now consider some finite Galois extensions. In let be the splitting field of a monic of degree n. Let be its roots (notice at this point we’ve picked a way to order the roots). The Galois group permutes to roots and this action in fact gives an injection into .

Say in there is a factorization into distinct irreducible factors (note this is an additional assumption on f(x)) of degree . So for each prime p we get a partition of n . Denote this partition as . Recall in conjugacy classes are determined by cycle type = partition of n. Its possible that two different conjugacy classes in G end up having the same cycle type when considered in the larger group . However when (descriminant) its a thm that there is a unique conjugacy class in G that has cycle type . In other words, for such p, we get a unique conjugacy we’ll call . In fact its also a (hard) thm that every conjugacy class of G is realized as some .

A prime p splits completely in E if it does not divide the descriminant of f(x) and . This says in particular that spits into linear factors in . Define

Thm: The map from Galois number fields to subsets of primes given by injective ( meaning if are not isomorphic then as sets but could have nonempty intersection).

### Langlands Stuff

Consider now faithful representations (injections) . For p not dividing the discriminant, gives a well defined semi-simple (meaning diagonalizable) conj. class. For such p define (s is the complex variable)

For example when (so is trivial ) plugging everything in as above yields

In particular taking the product over all primes give . This motivates the global L function

Fact: this converges to a mermorphic function on . A theorem of Dirichlet says the can be recovered from the global L function. Notice when , is a power of so in some sense, the global L function “knows” when .

So from a finite Galois extension we’ve obtained , a rather analytic object. Langlands said this meromorphic function should in fact be associated to something else: an automorphic representation.

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- August 28, 2009 / 10:44 am

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- Number Theory (course)

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