×

On Tate duality for Jacobian varieties. (English) Zbl 1049.11129

Let \(k\) be a \(p\)-adic number field and let \(A(k)\) be an abelian variety over \(k\) with dual \(A^t(k)\). The author studies Tate’s pairing \(A(k)\times H^1(k,A^t(k))\to \mathbb Q/\mathbb Z\) in the case that \(A=J\) is the Jacobian of a curve. He gives a description of the annihilator of \(U^nj(k)\) with respect to the Tate pairing, where \(\{U^nJ(k) | n=0,1,\dots\}\) is a filtration of \(J(k)\) given by the Néron model of \(J\).

MSC:

11S25 Galois cohomology
14H40 Jacobians, Prym varieties
14G20 Local ground fields in algebraic geometry
11G10 Abelian varieties of dimension \(> 1\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abhyankar, S., Resolution of singularities for arithmetical surfaces, (Schilling, O., Arithmetical Algebraic Geometry (1965), Harper, Row: Harper, Row New York), 111-152
[2] Artin, M., Neron models, (Cornell, G.; Silverman, J., Arithmetic Geometry (1986), Springer: Springer Berlin), 213-230 · Zbl 0603.14028
[3] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, 21, Springer, Berlin, 1990.; S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, 21, Springer, Berlin, 1990.
[4] Coates, J.; Greenberg, R., Kummer theory for abelian varieties over local fields, Invent. Math., 124, 129-174 (1996) · Zbl 0858.11032
[5] H. Hironaka, Desingularization of excellent surfaces, Lectures at Advanced Science Seminar in Algebraic Geometry, Bowdoin College, Summer 1967, noted by Bruce Bennett.; H. Hironaka, Desingularization of excellent surfaces, Lectures at Advanced Science Seminar in Algebraic Geometry, Bowdoin College, Summer 1967, noted by Bruce Bennett.
[6] Kato, K., Swan conductors for characters of degree one in the imperfect residue field case, Contemp. Math., 83, 101-131 (1989)
[7] Lichtenbaum, S., Duality theorems for curves over \(p\)-adic fields, Invent. Math., 7, 120-136 (1969) · Zbl 0186.26402
[8] McCallum, W., Duality theorems for Néron models, Duke Math. J., 53, 1093-1124 (1986) · Zbl 0623.14023
[9] McCallum, W., Tate duality and wild ramification, Math. Ann., 288, 553-558 (1990) · Zbl 0767.11057
[10] Milne, J. S., Etale cohomology, Princeton Mathematical Series, Vol. 33 (1980), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0433.14012
[11] Saito, S., Arithmetic on two dimensional local rings, Invent. Math., 85, 379-414 (1986) · Zbl 0609.13003
[12] Serre, J. P., Corps Locaux (1962), Hermann: Hermann Paris · Zbl 0137.02601
[13] Tate, J., WC-groups over \(p\)-adic fields, Sem. Bourbaki, 156, 13p (1957)
[14] Yamazaki, T., Reduced norm map of division algebras over complete discrete valuation fields of certain type, Comput. Math., 112, 127-145 (1998) · Zbl 0990.11072
[15] Yamazaki, T., On Swan conductors for Brauer groups of curves over local fields, Proc. Amer. Math. Soc., 127, 1269-1274 (1999) · Zbl 0921.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.