The $u$-invariant of the function fields  of $p$-adic curves

Raman Parimala; V. Suresh

doi:http://doi.org/10.4007/annals.2010.172.1391

Abstract

The $u$-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the $u$-invariant of function fields of $p$-adic curves is $8$. In this paper, we answer this question in the affirmative for function fields of nondyadic $p$-adic curves.

Show/hide bibliography for this article

[A] $Go to document$ K. Jón. Arason, "Cohomologische invarianten quadratischer Formen," J. Algebra, vol. 36, iss. 3, pp. 448-491, 1975.

Show bibtex

@article {A, MRKEY = {0389761}, AUTHOR = {Arason, J{ó}n Kr.}, TITLE = {Cohomologische invarianten quadratischer {F}ormen}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {36}, YEAR = {1975}, NUMBER = {3}, PAGES = {448--491}, ISSN = {0021-8693}, MRCLASS = {10C05 (18F25)}, MRNUMBER = {52 \#10592}, MRREVIEWER = {Horst Pfeuffer}, DOI = {10.1016/0021-8693(75)90145-3}, ZBLNUMBER = {0314.12104}, }
[AEJ] $Go to document$ K. Jón. Arason, R. Elman, and B. Jacob, "Fields of cohomological $2$-dimension three," Math. Ann., vol. 274, iss. 4, pp. 649-657, 1986.

Show bibtex

@article {AEJ, MRKEY = {848510}, AUTHOR = {Arason, J{ó}n Kr. and Elman, Richard and Jacob, Bill}, TITLE = {Fields of cohomological {$2$}-dimension three}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {274}, YEAR = {1986}, NUMBER = {4}, PAGES = {649--657}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {12G10 (11E81 12G05)}, MRNUMBER = {87m:12006}, MRREVIEWER = {Ron Brown}, DOI = {10.1007/BF01458600}, ZBLNUMBER = {0576.12025}, }
[C] J. -L. Colliot-Thélène, "Birational invariants, purity and the Gersten conjecture," in $K$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Providence, RI: Amer. Math. Soc., 1995, pp. 1-64.

Show bibtex

@incollection {C, MRKEY = {1327280}, AUTHOR = {Colliot-Th{é}l{è}ne, J.-L.}, TITLE = {Birational invariants, purity and the {G}ersten conjecture}, BOOKTITLE = {{$K$}-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras}, VENUE={{S}anta {B}arbara, {CA}, 1992}, SERIES = {Proc. Sympos. Pure Math.}, NUMBER = {58}, PAGES = {1--64}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1995}, MRCLASS = {14F20 (14C25)}, MRNUMBER = {96c:14016}, MRREVIEWER = {Luca Barbieri Viale}, ZBLNUMBER = {0834.14009}, }
[K] $Go to document$ K. Kato, "A Hasse principle for two-dimensional global fields," J. Reine Angew. Math., vol. 366, pp. 142-183, 1986.

Show bibtex

@article {K, MRKEY = {833016}, AUTHOR = {Kato, Kazuya}, TITLE = {A {H}asse principle for two-dimensional global fields}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {366}, YEAR = {1986}, PAGES = {142--183}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11G35 (11E81 11R70)}, MRNUMBER = {88b:11036}, MRREVIEWER = {J. Browkin}, DOI = {10.1515/crll.1986.366.142}, ZBLNUMBER = {0576.12012}, }
[L] T. Y. Lam, Introduction to Quadratic Forms over Fields, Providence, RI: Amer. Math. Soc., 2005, vol. 67.

Show bibtex

@book {L, MRKEY = {2104929}, AUTHOR = {Lam, T. Y.}, TITLE = {Introduction to Quadratic Forms over Fields}, SERIES = {Grad. Stud. Math.}, VOLUME = {67}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2005}, PAGES = {xxii+550}, ISBN = {0-8218-1095-2}, MRCLASS = {11Exx}, MRNUMBER = {2005h:11075}, MRREVIEWER = {K. Szymiczek}, ZBLNUMBER = {1068.11023}, }
[Li1] J. Lipman, "Introduction to resolution of singularities," in Algebraic Geometry, Providence, R.I.: Amer. Math. Soc., 1975, vol. 29, pp. 187-230.

Show bibtex

@incollection {Li1, MRKEY = {0389901}, AUTHOR = {Lipman, Joseph}, TITLE = {Introduction to resolution of singularities}, BOOKTITLE = {Algebraic Geometry}, VENUE={{H}umboldt {S}tate {U}niv., {A}rcata, {C}alif., 1974}, SERIES={{P}roc. {S}ympos. {P}ure {M}ath.}, VOLUME={29}, PAGES = {187--230}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, R.I.}, YEAR = {1975}, MRCLASS = {14E15 (14J15)}, MRNUMBER = {52 \#10730}, MRREVIEWER = {Jean Giraud}, ZBLNUMBER = {0306.14007}, }
[Li2] J. Lipman, "Desingularization of two-dimensional schemes," Ann. Math., vol. 107, iss. 1, pp. 151-207, 1978.

Show bibtex

@article {Li2, MRKEY = {0491722}, AUTHOR = {Lipman, Joseph}, TITLE = {Desingularization of two-dimensional schemes}, JOURNAL = {Ann. Math.}, VOLUME = {107}, YEAR = {1978}, NUMBER = {1}, PAGES = {151--207}, MRCLASS = {14J10}, MRNUMBER = {58 \#10924}, MRREVIEWER = {A. H. Wallace}, ZBLNUMBER = {0349.14004}, }
[M] A. S. Merkur$’$ev, "On the norm residue symbol of degree $2$," Dokl. Akad. Nauk SSSR, vol. 261, iss. 3, pp. 542-547, 1981.

Show bibtex

@article {M, MRKEY = {638926}, AUTHOR = {Merkur$'$ev, A. S.}, TITLE = {On the norm residue symbol of degree {$2$}}, JOURNAL = {Dokl. Akad. Nauk SSSR}, FJOURNAL = {Doklady Akademii Nauk SSSR}, VOLUME = {261}, YEAR = {1981}, NUMBER = {3}, PAGES = {542--547}, ISSN = {0002-3264}, MRCLASS = {12A60 (13A20)}, MRNUMBER = {83h:12015}, MRREVIEWER = {Thomas Zink}, ZBLNUMBER={0496.16020}, }
[Mi] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980.

Show bibtex

@book {Mi, MRKEY = {559531}, AUTHOR = {Milne, James S.}, TITLE = {\'{E}tale Cohomology}, SERIES = {Princeton Mathematical Series}, NUMBER = {33}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1980}, PAGES = {xiii+323}, ISBN = {0-691-08238-3}, MRCLASS = {14-02 (14F20 18F99)}, MRNUMBER = {81j:14002}, MRREVIEWER = {G. Horrocks}, ZBLNUMBER = {0433.14012}, }
[PS] $Go to document$ R. Parimala and V. Suresh, "Isotropy of quadratic forms over function fields of $p$-adic curves," Inst. Hautes Études Sci. Publ. Math., iss. 88, pp. 129-150 (1999), 1998.

Show bibtex

@article {PS, MRKEY = {1733328}, AUTHOR = {Parimala, R. and Suresh, V.}, TITLE = {Isotropy of quadratic forms over function fields of {$p$}-adic curves}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, NUMBER = {88}, YEAR = {1998}, PAGES = {129--150 (1999)}, ISSN = {0073-8301}, CODEN = {PMIHA6}, MRCLASS = {11E04 (11E08 11E81 12G05)}, MRNUMBER = {2000m:11034}, MRREVIEWER = {Detlev W. Hoffmann}, URL = {http://www.numdam.org/item?id=PMIHES_1998__88__129_0}, ZBLNUMBER = {0972.11020}, }
[S1] D. J. Saltman, "Division algebras over $p$-adic curves," J. Ramanujan Math. Soc., vol. 12, iss. 1, pp. 25-47, 1997.

Show bibtex

@article {S1, MRKEY = {1462850}, AUTHOR = {Saltman, David J.}, TITLE = {Division algebras over {$p$}-adic curves}, JOURNAL = {J. Ramanujan Math. Soc.}, FJOURNAL = {Journal of the Ramanujan Mathematical Society}, VOLUME = {12}, YEAR = {1997}, NUMBER = {1}, PAGES = {25--47}, ISSN = {0970-1249}, MRCLASS = {16H05 (12E15 13A20 16K20)}, MRNUMBER = {98d:16032}, MRREVIEWER = {Timothy J. Ford}, ZBLNUMBER = {0902.16021}, }
[S2] D. J. Saltman, "Correction to: “Division algebras over $p$-adic curves” [J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47; MR1462850 (98d:16032)]," J. Ramanujan Math. Soc., vol. 13, iss. 2, pp. 125-129, 1998.

Show bibtex

@article {S2, MRKEY = {1666362}, AUTHOR = {Saltman, David J.}, TITLE = {Correction to: ``{D}ivision algebras over {$p$}-adic curves'' [{J}. {R}amanujan {M}ath. {S}oc. {\bf 12} (1997), no. 1, 25--47; {MR}1462850 (98d:16032)]}, JOURNAL = {J. Ramanujan Math. Soc.}, FJOURNAL = {Journal of the Ramanujan Mathematical Society}, VOLUME = {13}, YEAR = {1998}, NUMBER = {2}, PAGES = {125--129}, ISSN = {0970-1249}, MRCLASS = {16H05 (12E15 13A20 16K20)}, MRNUMBER = {99k:16036}, MRREVIEWER = {Timothy J. Ford}, ZBLNUMBER={0920.16008}, }
[S3] $Go to document$ D. J. Saltman, "Cyclic algebras over $p$-adic curves," J. Algebra, vol. 314, iss. 2, pp. 817-843, 2007.

Show bibtex

@article {S3, MRKEY = {2344586}, AUTHOR = {Saltman, David J.}, TITLE = {Cyclic algebras over {$p$}-adic curves}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {314}, YEAR = {2007}, NUMBER = {2}, PAGES = {817--843}, ISSN = {0021-8693}, CODEN = {JALGA4}, MRCLASS = {16K20 (11S15 16K50)}, MRNUMBER = {2008i:16018}, MRREVIEWER = {Jan van Geel}, DOI = {10.1016/j.jalgebra.2007.03.003}, ZBLNUMBER = {1129.16014}, }
[Sc] W. Scharlau, Quadratic and Hermitian Forms, New York: Springer-Verlag, 1985, vol. 270.

Show bibtex

@book {Sc, MRKEY = {770063}, AUTHOR = {Scharlau, Winfried}, TITLE = {Quadratic and {H}ermitian Forms}, SERIES = {Grundl. Math. Wissen.}, VOLUME = {270}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1985}, PAGES = {x+421}, ISBN = {3-540-13724-6}, MRCLASS = {11Exx (11-02 12D15 15A63 16A16 16A28)}, MRNUMBER = {86k:11022}, MRREVIEWER = {R. Ware}, ZBLNUMBER = {0584.10010}, }
[Sh] I. R. Shafarevich, Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes, Bombay: Tata Institute of Fundamental Research, 1966, vol. 37.

Show bibtex

@book {Sh, MRKEY = {0217068}, AUTHOR = {Shafarevich, I. R.}, TITLE = {Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes}, SERIES = {Tata Inst. Fund. Res. Lect. Math. Phys.}, VOLUME={37}, PUBLISHER = {Tata Institute of Fundamental Research}, ADDRESS = {Bombay}, YEAR = {1966}, PAGES = {iv+175}, MRCLASS = {14.15}, MRNUMBER = {36 \#163}, MRREVIEWER = {Joseph Lipman}, ZBLNUMBER = {0164.51704}, }

The $u$-invariant of the function fields of $p$-adic curves

Abstract

Authors