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$$p$$-adic height pairings on abelian varieties with semistable ordinary reduction. (English) Zbl 1168.11314
Let $$A_F$$ be an abelian variety defined over a number field $$F$$. Among the various approaches to the study of special values of the $$L$$-function of $$A_F$$, which play a fundamental role in the celebrated conjectures of Birch and Swinnerton-Dyer, there is the particular one that aims at describing their $$p$$-adic properties with respect to any fixed prime number $$p$$. In this context, the search for suitably defined “$$p$$-adic regulators” of the group of $$\mathbb Q$$-rational points of $$A_F$$, that is, for natural $$p$$-adic height pairings on abelian varieties over number fields in general, appears as a crucial strategic task. In the recent two decades, several such $$p$$-adic height pairings have been established. The first construction was proposed by P. Schneider [Invent. Math. 69, 401–409 (1982; Zbl 0509.14048)], and a second, slightly simpler one was given by B. Mazur and J. Tate [Arithmetic and geometry, Papers dedicated to I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 195–237 (1983; Zbl 0574.14036)] just one year later. Both concepts were based on biextensions of abelian varieties, and the latter was used in the formulation of the $$p$$-adic analogue of the Birch and Swinnerton-Dyer conjecture by Mazur-Tate. Since then it is known that if the abelian variety $$A_F$$ has good ordinary reduction at all the primes of $$F$$ above a fixed prime $$p$$, then these two $$p$$-adic height pairings actually coincide.
However, they are really different in general, as A. Werner [Doc. Math., J. DMV 3, 301–319 (1998; Zbl 1019.11017)] has demonstrated in 1998 by using advanced methods of rigid analytic uniformization theory. On the other hand, further constructions of $$p$$-adic height pairings for abelian varieties over number fields were provided by the work of R. Coleman and B. Gross [Algebraic number theory – in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 73–81 (1989; Zbl 0758.14009)], Y. I. Zarkhin [Sémin. Théor. Nombres, Paris/Fr. 1987–88, Prog. Math. 81, 317–341 (1990; Zbl 0707.14040), and J. Nekovář Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)] in the early 1990s, this time by using local splittings of the Hodge filtration of the first de Rham cohomology group of the abelian variety $$A_F$$.
In order to apply these more recent concepts of $$p$$-adic height pairings effectively to the construction of new $$p$$-adic regulators, it appears to be utmost important to relate them to the fundamental original approach by Mazur-Tate. As early as in 1991, R. Coleman [Invent. Math. 103, No. 3, 631–650 (1991; Zbl 0763.14009)] had proved their overall coincidence in the case of abelian varieties with good ordinary reduction at all the primes of $$F$$ above $$p$$, but the general case represents a still open problem.
In the paper under review, the authors provide another step forward in this direction, and that by investigating the case of an abelian variety $$A_F$$ with semi-stable ordinary reduction at all the primes of $$F$$ above a fixed prime $$p$$.
In fact, their main result (Theorem 3.6.) guarantees that, for such an abelian variety $$A_F$$, the $$p$$-adic height pairing constructed via “unit-root splitting” (in the sense of Coleman-Gross and others, as mentioned above) and the one introduced by Mazur-Tate are indeed equal. The method of proof is based on the decomposition of height pairings into local contributions with respect to the finite places of $$F$$ over the prime $$p$$. After base change of $$A_F$$ to the completion of $$F$$ at a fixed place, the authors show then that the problem leads to the study of the abelian quotient $$B$$ of a semi-abelian rigid analytic variety. Finally, a refined, technically utmost challenging analysis is carried out to prove that earlier results of the second author [Doc. Math., J. DMV 3, 301–319 (1998; Zbl 1019.11017)], of the first author [Isr. J. Math. 120, Part B, 429–447 (2000; Zbl 1045.14503)], and of B. le Stum [C. R. Acad. Sci., Paris, Sér. I 303, 989–992 (1986; Zbl 0615.14012)] on the structure of $$B$$ can be applied in such a way that the main result becomes a consequence of Coleman’s theorem [Invent. Math. 103, No. 3, 631–650 (1991; Zbl 0763.14009)] from 1991.
Along the way, the authors also show that the Hodge filtration of the first de Rham cohomology group of a semi-abelian variety is determined by its invariant differentials. This result, presented in an appendix to the current paper, answers a question raised by B. le Stum in 1985.

##### MSC:
 11G50 Heights 11G10 Abelian varieties of dimension $$> 1$$ 14F40 de Rham cohomology and algebraic geometry 14K15 Arithmetic ground fields for abelian varieties 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G22 Rigid analytic geometry
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