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I learnt Hasse and Artin reciprocity laws when I was learning class field theory. Recently, I was looking for some facts about simple algebras written in Weil’s famous text Basic Number Theory. And I saw in his book (pp. 255-256, Theorem 2 and its Corollary) a comment on relations of Hasse and Artin’s reciprocity laws. So let me formalize my question in full detail below.

In Weil’s book, pp.255, the theorem 2 is just the well-known Hasse’s reciprocity. Weil’s notations are somewhat old-fashioned.In its modern form, it’s the exact sequence $$0\rightarrow Br(K)\rightarrow \oplus_v Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z} \rightarrow 0.$$ Here $K$ is a global field and the direct sum runs through all of its places. The two maps have their well-known definitions (base change and summation of Hasse invariants; Weil used multiplicative notations.)

And on pp.256, the corollary has the following modern paraphrase:

Given a character $\chi$ of the absolute Galois group of $Gal(K_s/K)$ (which amounts to the same thing of a character of the abelian Galois group), with $\chi_v$ its associated characters of local Galois groups (obtained from the restriction of Galois elements, local to global subfield). From the theory of cyclic algebras, if we denote by $L/K$ the corresponding cyclic extension associated to $\chi$, then central simple $K$-algebras which split over $L$ are similar to cyclic algebras $(\chi,a),(a\in K^*)$ (Weil’s notation of cyclic algebras). We can define $$ \prod_v(\chi_v,*):\mathbb{A}_K ^*\rightarrow \oplus_v Br(K_v),$$ and a $\textbf{key point}$ here is: from the general case of the Hasse reciprocity above, we can see that in this situation, $K^*\subset \mathbb{A}_K ^*$ is killed by the above map $ \prod_v(\chi_v,*)$, because it factors through $(\chi,*):K^*\rightarrow Br(K)$. And this is exactly the content of the corollary on Weil’s book.

Of course, this is the core of the proof of global class field theory. In this way, for each $\chi$, we obtain a homomorphism $$C_K:= \mathbb{A}_K ^*/K^* \rightarrow \mathbb{Q}/\mathbb{Z} \subset \mathbb{C}^*$$ (identification:$\mathbb{Q}/\mathbb{Z}$ as all the roots of unity). Then we get the pairing of $X(K)$(the character group of $Gal(K^{ab}/K)$ ) and $C_K$, and finally we use Pontryagin duality to get a map from $C_K$ to $Gal(K^{ab}/K)$, which establishes the global CFT.

What made me puzzled is Weil’s comment: enter image description here

He commented that this observation is nothing new but the famous Artin reciprocity. The usual form of Artin reciprocity is given by the Artin map and Frobenius elements. It has its adelic form, but I didn’t see the relations between these two reciprocity laws.

I knew from Professor Roquette’s nice book enter image description here

enter image description here

That they’re INDEED equivalent. But I didn’t found any literature dealing with this. Could any expert give me some help? Thanks a lot in advance!

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See Theorem 7.5 and the bottom paragraph on page 22 here. Note every central simple algebra over a number field is a cyclic algebra (a very nontrivial result), which is why understanding the Brauer group of a number field $K$ can be achieved by working with (finite) cyclic extensions of $K$.

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  • $\begingroup$ Oh, thank you very much, professor Conrad! I should read it among your note series earlier!!! $\endgroup$
    – youknowwho
    Jul 11, 2022 at 6:29

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