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Iwasawa-theory of abelian varieties at primes of non-ordinary reduction. (English) Zbl 0847.14026
The well known Birch and Swinnerton-Dyer conjectures for abelian varieties can be reformulated for \(p\)-adic \(L\)-functions. There are two approaches to a definition of \(p\)-adic \(L\)-function of an abelian variety \(A\), the analytical one and the algebraic one. The last one working with the Selmer groups of infinite cyclotomic extensions is more directly related with the Birch-Swinnerton-Dyer formulas for value at \(s = 1\). The algebraic definition was given by B. Mazur [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] and the formulas for \(L(1)\) were established by P. Schneider [Invent. Math. 71, 251-293 (1983; Zbl 0511.14010)]. The severe restriction used in these papers was that the abelian variety must have an ordinary reduction at the primes \(p\). This restriction was overcome by B. Perrin-Riou in Invent. Math. 99, No. 2, 247-292 (1990; Zbl 0715.11030) for supersingular elliptic curves.
The author gives here an extension to the case of general abelian varieties \(A\). It is defined some module \(L_0\) over Iwasawa algebra of \(A\) and the \(L\)-function is a characteristic function related to \(L_0\). The part of the Birch-Swinnerton-Dyer formulas is checked.

14K05 Algebraic theory of abelian varieties
11R23 Iwasawa theory
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G25 Varieties over finite and local fields
11G10 Abelian varieties of dimension \(> 1\)
Full Text: DOI EuDML
[1] [CS]Cornell, G. andSilverman, J.H.: Arithmetic geometry. Springer 1986
[2] [De]Demazure, M.: Lectures on p-divisible groups. Lecture Notes in Math.302. Springer 1972. · Zbl 0247.14010
[3] [Fo]Fontaine, J.-M.: Groupes p-divisibles sur les corps locaux. Asterisque47–48 (1977)
[4] [Gr]Greenberg, R.: Iwasawa theory forp-adic representations. Adv. Stud. Pure Math.17 (1989)
[5] [Ka]Kato, K.: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functiones viaB dR. In: Lecture Notes in Math.1553, pp. 50–163. Springer 1994
[6] [Katz]Katz, N.: Slope Filtration of F-Crystals. Asterisque63, 113–164 (1979) · Zbl 0426.14007
[7] [Man]Manin, J.: Cyclotomic fields and modular curves. Russ. Math. Surv.26, 7–78 (1971) · Zbl 0266.14012
[8] [Maz]Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Inv. Math.18, 183–266 (1972) · Zbl 0245.14015
[9] [MTT]Mazur, B., Tate, J. andTeitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inv. Math.84, 1–48 (1986) · Zbl 0699.14028
[10] [Mi1]Milne, J.S.: Étale Cohomology. Princeton University Press 1980 · Zbl 0433.14012
[11] [Mi2]Milne, J.S.: Arithmetic Duality Theorems. Academic Press 1986
[12] [PR1]Perrin-Riou, B.: Théorie d’Iwasawa p- adique locale et globale. Inv. Math.99, 247–292 (1990) · Zbl 0715.11030
[13] [PR2]Perrin-Riou, B.: Fonctions L p-adiques d’une courbe elliptique et points rationnels. Ann. Inst. Fourier Grenoble43, 945–995 (1993) · Zbl 0840.11024
[14] [PR3]Perrin-Riou, B.: Théorie d’Iwasawa des representations p-adique sur un corps local. Inv. Math.115, 81–149 (1994) · Zbl 0838.11071
[15] [Sch1]Schneider, P.: Iwasawa L-Functions of Varieties over Algebraic Number Fields. Inv. Math.71, 251–293 (1983) · Zbl 0511.14010
[16] [Sch2]Schneider, P.: p-adic height pairings. II. Inv. Math.79, 329–374 (1985) · Zbl 0571.14021
[17] [Sch3]Schneider, P.: Arithmetic of formal groups and applications I: Universal norm subgroups. Inv. Math.87, 587–602 (1987) · Zbl 0608.14034
[18] [CL]Serre, J.-P.: Corps locaux. Hermann, Paris 1962
[19] [Ta1]Tate, J.: WC-Groups over p-adic Fields. Sem. Bourbaki156, 1–13 (1957)
[20] [Ta2]Tate, J.: p-divisible Groups. In: Proc. Conf. Local Fields, Driebergen 1966, pp. 158–183. Springer 1967
[21] [Vi]Visik, M.M.: Non-archimedean measures connected with Dirichlet series. Math. USSR Sbornik Vol.28 (1976)
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