Universal norms on abelian varieties over global function fields. (English) Zbl 1082.11038
Summary: We examine the Mazur-Tate canonical height pairing [B. Mazur and J. Tate, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 195–237 (1983; Zbl 0574.14036)] defined between an abelian variety over a global field and its dual. We show in the case of global function fields that certain of these pairings are annihilated by universal norms coming from Carlitz cyclotomic extensions. Furthermore, for elliptic curves we find conditions for the triviality of these universal norms.
MSC:
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11G50 | Heights |
| 14K15 | Arithmetic ground fields for abelian varieties |
| 14G25 | Global ground fields in algebraic geometry |
Citations:
Zbl 0574.14036References:
| [1] | Barsotti, I., Theta functions in positive characteristic, Astérisque, 63, 5-16 (1979) · Zbl 0423.14026 |
| [2] | Bloch, S., A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math., 58, 65-76 (1980) · Zbl 0444.14015 |
| [3] | Breen, L., Fonctions thêta et théorème du cube. Fonctions thêta et théorème du cube, Lecture Notes in Mathematics, 980 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0558.14029 |
| [4] | Candilera, M.; Cristante, V., Bi-extensions associated to divisors on abelian varieties and theta functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10, 437-491 (1983) · Zbl 0576.14043 |
| [5] | Cristante, V., Theta functions and Barsotti-Tate groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7, 125-181 (1980) · Zbl 0438.14027 |
| [6] | Cristante, V., \(p\)-adic theta series with integral coefficients, Astérisque, 119-120, 169-182 (1984) · Zbl 0559.14030 |
| [7] | Goss, D., Basic Structures of Function Field Arithmetic (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0874.11004 |
| [8] | Grothendieck, A., Groupes de monodromie en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (7 I). Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (7 I), Lecture Notes in Mathematics, 288 (1972), Springer-Verlag: Springer-Verlag Berlin · Zbl 0237.00013 |
| [9] | Lubin, J.; Rosen, M. I., The norm map for ordinary abelian varieties, J. Algebra, 52, 236-240 (1978) · Zbl 0417.14035 |
| [10] | Mazur, B.; Tate, J., Canonical height pairings via biextensions, (Artin, M.; Tate, J., Arithmetic and Geometry: Papers Dedicated to I. R. Shafarevich on the Occasion of His Sixtieth Birthday (1983), Birkhäuser: Birkhäuser Boston), 195-237 · Zbl 0574.14036 |
| [11] | Mazur, B.; Tate, J., The \(p\)-adic sigma function, Duke Math. J., 62, 663-688 (1991) · Zbl 0735.14020 |
| [12] | Milne, J. S., Jacobian varieties, (Cornell, G.; Silverman, J. H., Arithmetic Geometry (1986), Springer-Verlag: Springer-Verlag New York), 167-212 · Zbl 0604.14018 |
| [13] | Mumford, D., On the equations defining abelian varieties I, Invent. Math., 1, 287-354 (1966) · Zbl 0219.14024 |
| [14] | Mumford, D., Bi-extensions of formal groups, Algebraic Geometry. Algebraic Geometry, (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) (1969), Oxford Univ. Press: Oxford Univ. Press London, p. 307-322 · Zbl 0216.33101 |
| [15] | Néron, A., Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. (2), 82, 249-331 (1965) · Zbl 0163.15205 |
| [16] | Norman, P., \(p\)-adic theta functions, Amer. J. Math., 107, 617-661 (1985) · Zbl 0587.14028 |
| [17] | Norman, P., Explicit \(p\)-adic theta functions, Invent. Math., 83, 41-57 (1986) · Zbl 0602.14045 |
| [18] | Papanikolas, M. A., Canonical heights on elliptic curves in characteristic \(p\), Compositio Math., 122, 299-313 (2000) · Zbl 1159.11312 |
| [19] | Schneider, P., \(p\)-adic height pairings I, Invent. Math., 69, 401-409 (1982) · Zbl 0509.14048 |
| [20] | Turner, S., Nonarchimedean theta functions and the Néron symbol, J. Number Theory, 21, 1-16 (1985) · Zbl 0587.14029 |
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.
