Context for "Coronidis Loco" from Weil's Basic Number Theory

It is not hard to see that if $L/K$ is an extension of number fields, then the discriminant of $L/K$, which is an ideal of $K$, is a square in the ideal class group of $K$. Hecke's theorem lifts this fact to the different. (Recall that the discriminant is the norm of the different.)

If you recall that the inverse different $\mathcal D_{L/K}^{-1}$ is equal to $Hom_{\mathcal O_K}(\mathcal O_L,\mathcal O_K),$ you see that the inverse different is the relative dualizing sheaf of $\mathcal O_L$ over $\mathcal O_K$; it is analogous to the canonical bundle of a curve (which is the dualizing sheaf of the curve over the ground field). Saying that $\mathcal D_{L/K}$, or equivalently $\mathcal D_{L/K}^{-1}$, is a square is the same as saying that there is a rank 1 projective $\mathcal O_L$-module $\mathcal E$ such that $\mathcal E^{\otimes 2} \cong \mathcal D_{L/K}^{-1}$, i.e. it says that one can take a square root of the dualizing sheaf. In the case of curves, this is the existence of theta characteristics.

Thus, apart from anything else (and as indicated in the quotation given in the question), Hecke's theorem significantly strengthens the analogy between rings of integers in number fields and algebraic curves.

If you want to think more arithmetically, it is a kind of reciprocity law. It expresses in some way a condition on the ramification of an arbitrary extension of number fields: however the ramification occurs, overall it must be such that the different ramified primes balance out in some way in order to have $\prod_{\wp} \wp^{e_{\wp}}$ be trivial in the class group mod $2$ (where $\wp^{e_{\wp}}$ is the local different at a prime $\wp$). (And to go back to the analogy: this is supposed to be in analogy with the fact that if $\omega$ is any meromorphic differential on a curve, then the sum of the orders of all the divisors and poles of $\omega$ is even.) Note that Hecke proved his theorem as an application of quadratic reciprocity in an arbitrary number field.

The parity problems Armitage alludes to includes Hecke's theorem, as well as to other (and related) parity problems brought fourth by Froehlich in his theory of Galois modules. For a couple of references, see Chapter 11 of "Reciprocity Laws", e.g. at link text

franz lemmermeyer