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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!

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What does "Coronidis Loco" translate to? I looked it up and only got the sources you were just talking about. –  Harry Gindi Jan 20 '10 at 0:26
    
I'm not actually sure, from Patterson's article I assume that it means "crowning moment." –  Jonah Sinick Jan 20 '10 at 0:32
    
What language is that? –  Harry Gindi Jan 20 '10 at 1:01
    
It's Latin. latin-dictionary.net/info/word/14361.html. A Google search shows the use of that phrase in quite a few web pages with Latin passages, as well as uses in English as a final achievement. –  Douglas Zare Jan 20 '10 at 1:33
    
When I did a google search, I got a bunch of references to Weil and this situation. –  Harry Gindi Jan 20 '10 at 1:38
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2 Answers

up vote 15 down vote accepted

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.

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Perhaps the word "square" in the parenthetical remark "(Recall that the discriminant is the square of the different.)" sould be changed to "norm". –  Chandan Singh Dalawat Jan 20 '10 at 4:38
    
Thanks; it's now fixed! –  Emerton Jan 20 '10 at 6:23
    
@Emerton. This was very helpful. I have added the tag quadratic-reciprocity, accordingly. Have I understood correctly that the theta divisor makes sense only in the case of complex algebraic geometry? –  Anweshi Jan 20 '10 at 17:16
    
Unless I am misunderstanding you, theta divisors are a general concept in the geometry of curves and their Jacobians which can be studied for curves over any field. (Of course, there are questions of rationality when the field is not algebraically close.) –  Emerton Jan 20 '10 at 17:41
    
@Emerton. I had encountered the theta divisor in the book of Lange and Birkenhanke on complex abelian varieties and had imagined it to be an analytic construction. Going through your answer, it seems that it can indeed be made algebraic. Thanks for the explanation. –  Anweshi Jan 20 '10 at 18:03
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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

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Ah! I presume you are the author of the book. I take this opportunity to ask an aside question. Perhaps, you will be kind enough to provide a reference for learning the proof of Kronecker-Weber as proved by Kronecker and Weber? –  Anweshi Jan 24 '10 at 12:12
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I'm not sure whether this belongs here, but: Kronecker did not prove his result; he sketched the proof for cyclotomic fields unramified at 2. Weber's proofs contained errors. For a modern exposition, see Neumann, Olaf Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber". J. Reine Angew. Math. 323, 105-126 (1981). franz –  Franz Lemmermeyer Jan 24 '10 at 14:01
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