\(p\)-adic height pairings on abelian varieties with semistable ordinary reduction.

*(English)*Zbl 1168.11314Let \(A_F\) 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.

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.

Reviewer: Werner Kleinert (Berlin)

##### 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 |

##### Keywords:

arithmetic algebraic geometry; heights; \(L\)-functions; \(p\)-adic regulators; abelian varieties; rigid analytic uniformization; de Rham cohomology; Hodge filtration##### References:

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