In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says
"Hecke [proved] a beautiful theorem on the different of k, namely that the class of the absolute different in the ideal class group is a square. This theorem – an analogue of the fact that the Euler characteristic of a Riemann surface is even – is the crowning moment (coronidis loco) in both Hecke's book and Andre Weil's Basic Number Theory."
About the same matter, J.V. Armitage says (in his review of the 1981 translation of Hecke's book):
"That beautiful theorem deservedly occupies the 'coronidis loco' in Weil's Basic number theory and was the starting point for the work on parity problems in algebraic number theory and algebraic geometry, which has borne such rich fruit in the past fifteen years."
What is a reference for learning about the parity
problems that Armitage alludes to?
It can be impossible to verbalize the reasons for aesthetic preferences, but
Why might Weil, Patterson and Armitage
have been so favorably impressed by
the theorem that the ideal class of
the different of a number field is a
square in the ideal class group?
Weil makes no comment on why he chose to end Basic Number Theory with the above theorem. It should be borne in mind that Weil's book covers the class number formula and all of class field theory, so that the standard against which the above theorem is being measured in the above quotes is high!
Best Answer
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.