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.
being 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.
Review of automorphic representation: ring of adeles is defined as and we equip with the topology an open subset of (last coordinate being (how ideles are topologized).
The space of cuspidal automorphic forms is the set of all functions such that (i) is smooth, i.e. In other words, is locally constant in the finite places and smooth in the infinite places. (ii) is -finite where is if is real and if is complex (right translates by this subspace is finite-dimensional over .) (iii) is -finite (Here is the center of the universal enveloping algebra of the and acts by . Harish-Chandra showed that .
John Thompson showed that monster group (largest sporadic simple group) can be realized as a quotient of in infinitely many ways. (Journal of Algebra 89)
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 .
Conclusion:
a nonreduced ring if
, if doesn’t divide and, either is a q.r. over if is odd, or if .
, remaining case
In case 1, ; case 2: ; case 3: is a prime.
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.
Completion is often integral domain
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.
(Big conjecture) If we fix an field-isomorphism , then there is a bijection from the set of
Irreeducible algebraic cuspidal automorphic representations of
to that of
Irreducible algebraic continuous representation of
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: if we look at be a primitive root of unity, the cyclotomic extension is ramified only at as well, and if we take the field corresponding to the unique index 2 subgroup of , it will be a quadratic extension ramified only at , so it is . Note that is in (index 2 subgroup) iff is a quadratic residue . Equivalently, we can look at the character which is the quadratic residue , which 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: Hecke eigenvalue of at is ; on the other hand, this is the Frobenius eigenvalue or , which is if splits in (since -th power map is trivial on ) and otherwise. By what is said last time, 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.