First we review some preliminary materials on algebraic number theory.
There are two isomorphisms (of -algebras) between . They are Galois conjugates and no reason to prefer one over another. We can view as living inside , or any other algebraic closures of , the point is that it can be equipped with different topologies.
The absolute Galois group is defined to be the inverse limit of Galois group of for finite Galois over , so has a profinite topology. If we compare different constructions of sitting inside different algebraic closures, we see that is only pinned down up to conjugation.
is a very complicated group. For example, John Thompson showed that monster group (largest sporadic simple group) can be realized as a quotient of in infinitely many ways.
We next review the splitting of primes in quadratic extensions. For each prime , . If , can be taken to be ; otherwise it is . The discriminant is in the former and in the latter. If is odd and , then has double root and the same holds for . If doesn’t divide and odd, then is irreducible iff is not a quadratic residue mod . If , then and . Since , doesn’t have double root. It is irreducible iff .
The conclusion is there are three cases for the isomorphism types of the residue ring :
a nonreduced ring if ;
, if doesn’t divide and, either if is odd, or if ;
, remaining case
In case 1, ; case 2: ; case 3: is a prime.
We can reinterpret this in the language of schemes: is etale morphism, but not very interesting, so we spread it out to one-dimension: is not etale. The fiber (base change to ) over or split or inert primes is etale.
For more general , we want to understand what happens over each prime systematically. To do this we look at the completion with the -adic metric (or the valuation) and the unit disk which allows us to do analysis on it.
Over , and uniquely extends but not longer discrete; (if we take this idea further we get to Newton polygon). The open disk is the set of topologically nilpotent elements. The quotient . This is not a Noetherian ring.
Fixing an embedding gives a compatible choice of primes above in each finite extension . More precisely, it determines an extension to known as a place of above . The values of tells us about how ramified is in . This also gives by restriction, and we can further look at its image in every finite quotient . The image of in is trivial iff is unramified in . The image is the decomposition group.
Thus to understand ramification and splitting of in a finite Galois extension we can always complete at . We don’t just want to understand the structure of we should understand the whole collection of subgroups , as well as the archimedean place (different ways of embedding into and the absolute value it inherits), and also how acts. All of these subgroups are defined up to conjugacy by .
When studying actions of on topological spaces, we can study the restrictions of the action to these subgroups. The action is unramified at if it is trivial on and most of the time it is unramified at almost all primes.
The goal of the course is to understand what the following big conjecture is saying:
Big conjecture
Fix an field-isomorphism , then there is a bijection from the set of
to that of
where algebraic on the left means has Harish-Chandra parameters in ; algebraic on the right means 1. unramified at all but finitely many (nonarchimedean) places 2. de Rham at (condition from -adic Hodge theory). This bijection is characterised by the ‘local-global compatibility’ at almost all unramified places. More precisely, at unramified places, we want to match Hecke eigenvalues (Satake paramters) to Frobenius eigenvalues (via ). This match the -factors.
We can also talk about this conjecture over , then we need to replace by and by .
This specializes to quadratic reciprocity: Take , suppose is an odd prime, set , note that . Note that is ramified only at , and it is the unique quadratic extension of with this property. We can write down another quadratic extension with this property: Let be a primitive -th root of unity, the cyclotomic extension is ramified only at as well, and if we take the subfield corresponding to the unique index 2 subgroup of , it will be a quadratic extension ramified only at , so it is . Note that is in (the index 2 subgroup) iff . Equivalently, we can look at the character which comes from restriction, and the above essentially says that this is . Viewing it as a Dirichlet character, it induces an automorphic representation of . This match with just by checking on cyclotomic extensions (Kronecker-Weber). The local-global compatibility is the only nontrivial part: The Hecke eigenvalue of at is ; on the other hand, this is the Frobenius eigenvalue or at , which is just . Recalling , so for (unramified places) and , it will be if splits in (since -th power map is trivial on ) and otherwise. Whether splits in the quadratic extension depends on whether is a quadratic residue mod if and something else for . From this we can get the quadratic reciprocity.