Ha! Who do you think I am? For my first explanation of why I do not understand Langlands, take Prof. G. Harder of Mathematisches Institut der Universitat Bonn's "The Langlands Program (An Overview)" where the first section is titled "A Simple Example"
[...]The first object is a pair of integral, positive definite, quaternary quadratic forms:
\( P(x, y, u, v) = x^2 + xy + 3y^2 + u^2 + uv + 3v^2 \)
\( Q(x, y, u, v) = 2(x^2 + y^2 + u^2 + v^2) + 2xu + xv + yu - 2yv \)
These forms have discriminant \(11^2\), and I mention that these two quadratic forms \(Q\) and \(P\) are the only integral, positive definite quaternary forms with discriminant \(11^2\). This may not be so easy to verify, but it is true. (Rainer Schulze-Pillot pointed out that this is actually not true; there is a third form \(S(x, y, u, v) = x^2 + 4(y^2 + u^2 + v^2) + xu + 4yu + 3yv + 7uv\), but the two forms above are sufficient for the following considerations.
This pair will give us automorphic forms; we come to this point later.
The second object is an elliptic curve \(E\), for us, this is simply a polynomial:
\(G(x, y) = y^2 + y - x^3 + x^2 + 10x + 20\).
This object is a diophantine equation. For any commutative ring \(R\) with identity, we can consider the set of solutions:
\(\{(a, b) \in R^2 | G(a, b) = 0\}\)
[...]