| 词条 | Arithmetic topology |
| 释义 |
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds. AnalogiesThe following are some of the analogies used by mathematicians between number fields and 3-manifolds:[1]
Expanding on the last two examples, there is an analogy between knots and prime numbers in which one considers "links" between primes. The triple of primes {{nowrap|(13, 61, 937)}} are "linked" modulo 2 (the Rédei symbol is −1) but are "pairwise unlinked" modulo 2 (the Legendre symbols are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"[2] or "mod 2 Borromean primes".[3] HistoryIn the 1960s topological interpretations of class field theory were given by John Tate[4] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier[5] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots[6] which was further explored by Barry Mazur.[7][8] In the 1990s Reznikov[9] and Kapranov[10] began studying these analogies, coining the term arithmetic topology for this area of study. See also
Notes1. ^Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844. 2. ^{{Citation |last=Vogel |first=Denis |date=13 February 2004 |title=Massey products in the Galois cohomology of number fields |url=http://www.ub.uni-heidelberg.de/archiv/4418 |id={{URN|nbn|de:bsz:16-opus-44188}}}} 3. ^{{Citation |last=Morishita |first=Masanori |date=22 April 2009 |title=Analogies between Knots and Primes, 3-Manifolds and Number Rings |arxiv=0904.3399|bibcode=2009arXiv0904.3399M }} 4. ^J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295). 5. ^M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole {{webarchive |url=https://web.archive.org/web/20110526230017/http://www.jmilne.org/math/Documents/WoodsHole3.pdf |date=May 26, 2011 }}, 1964. 6. ^Who dreamed up the primes=knots analogy? {{webarchive |url=https://web.archive.org/web/20110718061649/http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html |date=July 18, 2011 }}, neverendingbooks, lieven le bruyn's blog, may 16, 2011, 7. ^Remarks on the Alexander Polynomial, Barry Mazur, c.1964 8. ^B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552. 9. ^A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399. 10. ^M. Kapranov, [https://books.google.com/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119 Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151. Further reading
External links
3 : Number theory|3-manifolds|Knot theory |
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