Serre Reciprocity Conjecture

These are notes from C. S. Dalawat's lecture on the Serre Reciprocity Conjecture at H.R.I and from the draft of an article written by him.

The conjecture relates representations of Gal(Qbar/Q) and modular forms. The "level 1" case of this has been proved by Chandrashekhar Khare, "by carrying out a strategy he had worked out in collaboration with Jean-Pierre Wintenberger, and using results by Luis Dieulefait and others". This apparently breaks " a psychological barrier".

In the following we shall attempt to present, somewhat concisely, the statement of this conjecture. For a pedagogical article suitable for (advanced) high school students, click here (postscript format).

Let us get a few terms defined: Everyone knows what a field is;

• the field G of Gaussian numbers is the set {a+bi, a,b rational} and that Aut(G)=Z/2Z (observe that since polynomials are inv under Aut, i-> -i)

• the set of algebraic numbers A, i.e., those that are roots of polynomials also forms a field and what about Aut(A)=? - this is the question. For instance, all finite order elements in Aut(A) are order 2 !
  

A (sub)field is called Galoisian if the only numbers invariant under all of Aut are the rationals. (there is apparently a conjecture that every finite group G appears as the Aut of some galoisian subfield of A)

Finite Fields and Local Fields