Abstract
Let K be a global field. The aim of this paper is to study the basic properties of the global non-Abelian norm-residue symbol \(\mathsf{NR}_{K}^{\underline{\varphi }^{\mathrm{Weil}} }\) of K, which is defined following the Chevalley-Miyake philosophy of idèles by “glueing” the local non-Abelian norm-residue symbol \(\{\bullet,K_{\nu }\}_{\varphi _{\nu }}\) of K ν in the sense of Koch, for each \(\nu \in\mathbb{h}_{K}\).
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Notes
- 1.
The author would like to thank Pierre Cartier, A. D. Raza Choudary, and Michel Waldschmidt for inviting him to deliver a talk in the 6th World Conference on 21st Century Mathematics 2013, which took place in the Abdus Salam School of Mathematical Sciences, Lahore, on March 6–9, 2013. He would also like to thank the Abdus Salam School of Mathematical Sciences for arranging his stay and for the hospitality he received in Lahore on March 4–11, 2013, which he enjoyed very much. Finally, the author thanks the referee for his or her suggestions, which improved the presentation of this work a lot.
- 2.
By Čebotarev density theorem, \(\mathrm{Spl}: L/K\mapsto \mathrm{Spl}(L/K)\) is an injective and order-reversing mapping from finite Galois extensions L of the global field K into the power set of \(\mathbb{h}_{K}\cup\mathbb{a}_{K}\). The image of the map “Spl” for finite Abelian extensions L of K has a description in terms of the Abelian global reciprocity map (•, L∕K) relative to the extension L∕K.
- 3.
Initially, Koch started this theory for metabelian extensions of local fields (look at [9]) using explicit computations with formal Lubin-Tate groups, unlike the more general approach of Fesenko, Laubie, Serbest, and others, which uses APF-extensions and the fields of norm construction of Fontaine and Wintenberger.
- 4.
If {G i } i ∈ I is a collection of topological groups and
is the free product of this collection together with the canonical embeddings 
, for each i
o
∈ I, then the universal mapping property of free products states that, if for each i
o
∈ I, \(\phi _{i_{o}}: G_{i_{o}} \rightarrow H\) is a continuous homomorphism, then there exists a unique continuous homomorphism 
, such that \(\phi \circ \iota _{i_{o}} =\phi _{i_{o}}\), for every i
o
∈ I. - 5.
Which is unique if K is a function field and unique up to composition with an inner automorphism of W K defined by an element of the connected component W K o of W K if K is a number field.
- 6.
In fact, if \(\mathsf{s}_{K} = \mathsf{a}_{K}\), then \(\mathcal{J}_{K}^{\underline{\varphi }_{K}^{c}} = \mathsf{s}_{K}^{-1}\left (K^{\times }\right )\) by Theorem 2.
is the free product of this collection together with the canonical embeddings 
, for each i
o
∈ I, then the universal mapping property of free products states that, if for each i
o
∈ I, 
, such that References
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Dedicated to my teacher Goro Shimura.
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İkeda, K.İ. (2015). Basic Properties of the Non-Abelian Global Reciprocity Map. In: Cartier, P., Choudary, A., Waldschmidt, M. (eds) Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics, vol 98. Springer, Basel. https://doi.org/10.1007/978-3-0348-0859-0_5
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