Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Witt equivalence of function fields over global fields


Authors: Paweł Gładki and Murray Marshall
Journal: Trans. Amer. Math. Soc. 369 (2017), 7861-7881
MSC (2010): Primary 11E81, 12J20; Secondary 11E04, 11E12
DOI: https://doi.org/10.1090/tran/6898
Published electronically: April 11, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Witt equivalent fields can be understood to be fields having the same symmetric bilinear form theory. Witt equivalence of finite fields, local fields and global fields is well understood. Witt equivalence of function fields of curves defined over archimedean local fields is also well understood. In the present paper, Witt equivalence of general function fields over global fields is studied. It is proved that for any two such fields $ K,L$, any Witt equivalence $ K \sim L$ induces a cannonical bijection $ v \leftrightarrow w$ between Abhyankar valuations $ v$ on $ K$ having residue field not finite of characteristic $ 2$ and Abhyankar valuations $ w$ on $ L$ having residue field not finite of characteristic $ 2$. The main tool used in the proof is a method for constructing valuations due to Arason, Elman and Jacob [J. Algebra 110 (1987), 449-467]. The method of proof does not extend to non-Abhyankar valuations. The result is applied to study Witt equivalence of function fields over number fields. It is proved, for example, that if $ k$, $ \ell $ are number fields and $ k(x_1,\dots ,x_n) \sim \ell (x_1,\dots ,x_n)$, $ n\ge 1$.


References [Enhancements On Off] (What's this?)

  • [1] Jón Kr. Arason, Richard Elman, and Bill Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), no. 2, 449-467. MR 910395, https://doi.org/10.1016/0021-8693(87)90057-3
  • [2] Jón Kristinn Arason and Albrecht Pfister, Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math. 12 (1971), 173-176 (German). MR 0294251
  • [3] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • [4] Ricardo Baeza and Remo Moresi, On the Witt-equivalence of fields of characteristic $ 2$, J. Algebra 92 (1985), no. 2, 446-453. MR 778461, https://doi.org/10.1016/0021-8693(85)90133-4
  • [5] Jenna P. Carpenter, Finiteness theorems for forms over global fields, Math. Z. 209 (1992), no. 1, 153-166. MR 1143220, https://doi.org/10.1007/BF02570827
  • [6] P. E. Conner and Jürgen Hurrelbrink, The $ 4$-rank of $ K_2(O)$, Canad. J. Math. 41 (1989), no. 5, 932-960. MR 1015590, https://doi.org/10.4153/CJM-1989-043-0
  • [7] P. E. Conner, R. Perlis, and K. Szymiczek, Wild sets and $ 2$-ranks of class groups, Acta Arith. 79 (1997), no. 1, 83-91. MR 1438119
  • [8] Alfred Czogała, On reciprocity equivalence of quadratic number fields, Acta Arith. 58 (1991), no. 1, 27-46. MR 1111088
  • [9] M. A. Dickmann and F. Miraglia, Special groups: Boolean-theoretic methods in the theory of quadratic forms, Mem. Amer. Math. Soc. 145 (2000), no. 689, xvi+247. With appendixes A and B by Dickmann and A. Petrovich. MR 1677935, https://doi.org/10.1090/memo/0689
  • [10] Ido Efrat, Valuations, orderings, and Milnor $ K$-theory, Mathematical Surveys and Monographs, vol. 124, American Mathematical Society, Providence, RI, 2006. MR 2215492
  • [11] H. Eisenbeis, G. Frey, and B. Ommerborn, Computation of the $ 2$-rank of pure cubic fields, Math. Comp. 32 (1978), no. 142, 559-569. MR 0480416
  • [12] Antonio J. Engler and Alexander Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496
  • [13] Nicolas Grenier-Boley and Detlev W. Hoffmann, Isomorphism criteria for Witt rings of real fields, Forum Math. 25 (2013), no. 1, 1-18. With an appendix by Claus Scheiderer. MR 3010846, https://doi.org/10.1515/form.2011.097
  • [14] D. K. Harrison, Witt rings, University of Kentucky Notes, Lexington, Kentucky (1970).
  • [15] Bill Jacob, On the structure of Pythagorean fields, J. Algebra 68 (1981), no. 2, 247-267. MR 608534, https://doi.org/10.1016/0021-8693(81)90263-5
  • [16] Bill Jacob, Quadratic forms over dyadic valued fields. I. The graded Witt ring, Pacific J. Math. 126 (1987), no. 1, 21-79. MR 868606
  • [17] Bill Jacob, Quadratic forms over dyadic valued fields. II. Relative rigidity and Galois cohomology, J. Algebra 148 (1992), no. 1, 162-202. MR 1161571, https://doi.org/10.1016/0021-8693(92)90242-E
  • [18] Stanislav Jakubec, František Marko, and Kazimierz Szymiczek, Parity of class numbers and Witt equivalence of quartic fields, Math. Comp. 64 (1995), no. 212, 1711-1715. MR 1308455, https://doi.org/10.2307/2153380
  • [19] Jerrold L. Kleinstein and Alex Rosenberg, Succinct and representational Witt rings, Pacific J. Math. 86 (1980), no. 1, 99-137. MR 586872
  • [20] Manfred Knebusch, Alex Rosenberg, and Roger Ware, Structure of Witt rings and quotients of Abelian group rings, Amer. J. Math. 94 (1972), 119-155. MR 0296103
  • [21] Przemysław Koprowski, Local-global principle for Witt equivalence of function fields over global fields, Colloq. Math. 91 (2002), no. 2, 293-302. MR 1898636, https://doi.org/10.4064/cm91-2-8
  • [22] Przemyslaw Koprowski, Witt equivalence of algebraic function fields over real closed fields, Math. Z. 242 (2002), no. 2, 323-345. MR 1980626, https://doi.org/10.1007/s002090100336
  • [23] Marc Krasner, Approximation des corps valués complets de caractéristique $ p\not =0$ par ceux de caractéristique 0, Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957, pp. 129-206 (French). MR 0106218
  • [24] Marc Krasner, A class of hyperrings and hyperfields, Internat. J. Math. Math. Sci. 6 (1983), no. 2, 307-311. MR 701303, https://doi.org/10.1155/S0161171283000265
  • [25] Franz-Viktor Kuhlmann, Places of algebraic function fields in arbitrary characteristic, Adv. Math. 188 (2004), no. 2, 399-424. MR 2087232, https://doi.org/10.1016/j.aim.2003.07.021
  • [26] M. Kula, L. Szczepanik, and K. Szymiczek, Quadratic form schemes and quaternionic schemes, Fund. Math. 130 (1988), no. 3, 181-190. MR 970903
  • [27] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929
  • [28] Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
  • [29] Murray Marshall, Abstract Witt rings, Queen's Papers in Pure and Applied Mathematics, vol. 57, Queen's University, Kingston, Ont., 1980. MR 674651
  • [30] M. Marshall, The elementary type conjecture in quadratic form theory, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 275-293. MR 2060204, https://doi.org/10.1090/conm/344/06224
  • [31] M. Marshall, Real reduced multirings and multifields, J. Pure Appl. Algebra 205 (2006), no. 2, 452-468. MR 2203627, https://doi.org/10.1016/j.jpaa.2005.07.011
  • [32] Ch. G. Massouros, Methods of constructing hyperfields, Internat. J. Math. Math. Sci. 8 (1985), no. 4, 725-728. MR 821630, https://doi.org/10.1155/S0161171285000813
  • [33] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. MR 0506372
  • [34] Ján Mináč and Michel Spira, Witt rings and Galois groups, Ann. of Math. (2) 144 (1996), no. 1, 35-60. MR 1405942, https://doi.org/10.2307/2118582
  • [35] Jean Mittas, Sur une classe d'hypergroupes commutatifs, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A485-A488 (French). MR 0260651
  • [36] R. Perlis, K. Szymiczek, P. E. Conner, and R. Litherland, Matching Witts with global fields, Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991) Contemp. Math., vol. 155, Amer. Math. Soc., Providence, RI, 1994, pp. 365-387. MR 1260721, https://doi.org/10.1090/conm/155/01393
  • [37] Paulo Ribenboim, Théorie des valuations, Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l'Université de Montréal, Montreal, Que., 1968 (French). MR 0249425
  • [38] Ursula Schneiders, Estimating the $ 2$-rank of cubic fields by Selmer groups of elliptic curves, J. Number Theory 62 (1997), no. 2, 375-396. MR 1432782, https://doi.org/10.1006/jnth.1997.2062
  • [39] Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137-1152. MR 0352049
  • [40] T. A. Springer, Quadratic forms over fields with a discrete valuation. I. Equivalence classes of definite forms, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 352-362. MR 0070664
  • [41] T. A. Springer, Quadratic forms over fields with a discrete valuation. II. Norms, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 238-246. MR 0076794
  • [42] Kazimierz Szymiczek, Matching Witts locally and globally, Math. Slovaca 41 (1991), no. 3, 315-330. MR 1126669
  • [43] Kazimierz Szymiczek, Witt equivalence of global fields, Comm. Algebra 19 (1991), no. 4, 1125-1149. MR 1102331, https://doi.org/10.1080/00927879108824194
  • [44] Kazimierz Szymiczek, Witt equivalence of global fields. II. Relative quadratic extensions, Trans. Amer. Math. Soc. 343 (1994), no. 1, 277-303. MR 1176087, https://doi.org/10.2307/2154533
  • [45] Kazimierz Szymiczek, $ 2$-ranks of class groups of Witt equivalent number fields, Ann. Math. Sil. 12 (1998), 53-64. Number theory (Cieszyn, 1998). MR 1673064
  • [46] Roger Ware, Valuation rings and rigid elements in fields, Canad. J. Math. 33 (1981), no. 6, 1338-1355. MR 645230, https://doi.org/10.4153/CJM-1981-103-0
  • [47] Lawrence C. Washington, Class numbers of the simplest cubic fields, Math. Comp. 48 (1987), no. 177, 371-384. MR 866122, https://doi.org/10.2307/2007897
  • [48] Ernst Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44 (German). MR 1581519, https://doi.org/10.1515/crll.1937.176.31

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11E81, 12J20, 11E04, 11E12

Retrieve articles in all journals with MSC (2010): 11E81, 12J20, 11E04, 11E12


Additional Information

Paweł Gładki
Affiliation: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland – and – Department of Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: pawel.gladki@us.edu.pl

Murray Marshall
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, Canada

DOI: https://doi.org/10.1090/tran/6898
Keywords: Symmetric bilinear forms, quadratic forms, Witt equivalence of fields, function fields, global fields, valuations, Abhyankar valuations
Received by editor(s): April 21, 2015
Received by editor(s) in revised form: November 28, 2015
Published electronically: April 11, 2017
Additional Notes: The research of the second author was supported in part by NSERC of Canada.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society