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.
-
[A]
K. Jón. Arason, "Cohomologische invarianten quadratischer Formen," J. Algebra, vol. 36, iss. 3, pp. 448-491, 1975.
@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]
K. Jón. Arason, R. Elman, and B. Jacob, "Fields of cohomological $2$-dimension three," Math. Ann., vol. 274, iss. 4, pp. 649-657, 1986.
@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.
@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]
K. Kato, "A Hasse principle for two-dimensional global fields," J. Reine Angew. Math., vol. 366, pp. 142-183, 1986.
@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.
@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.
@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.
@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.
@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.
@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]
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.
@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.
@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.
@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]
D. J. Saltman, "Cyclic algebras over $p$-adic curves," J. Algebra, vol. 314, iss. 2, pp. 817-843, 2007.
@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.
@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.
@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},
}