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$$p$$-adic heights on curves. (English) Zbl 0758.14009
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).
[For the entire collection see Zbl 0721.00006.]
$$k$$ denotes a non-archimedean local field of characteristic zero and $$\chi: k^*\to\mathbb Q_ p$$ denotes a continuous character. Let $$J$$ be the Jacobian variety of a curve $$X$$ over $$k$$ (having a $$k$$-rational point). The aim of the paper is to construct a $$p$$-adic height pairing on $$J$$.
In the case that the residue characteristic of $$k$$ is different from $$p$$, arithmetic intersection theory is used to produce a unique pairing $$\langle a,b\rangle$$, with values in $$\mathbb Q_ p$$, defined on relatively prime divisors $$a$$ and $$b$$ on $$X$$ (defined over $$k)$$ and satisfying: continuous, symmetric, bi-additive and $$\langle(f),b\rangle=\chi(f(b))$$ for $$f\in k(X)^*$$. In the case $$k\supset\mathbb Q_ p$$, a rigid analytic analysis of differentials of the third kind and the de Rham cohomology is made to arrive at a definition of the pairing. The pairing which is constructed depends on a suitable choice of a direct sum decomposition $$H^ 1_{DR}(X/k)=H^ 0(X,\Omega_ X)\oplus W$$. In case $$X$$ has a good ordinary reduction one can take the unit root space as a choice for $$W$$. With this choice the pairing coincides with the canonical $$p$$-adic height pairings constructed by P. Schneider [Invent. Math. 69, 401–409 (1982; Zbl 0509.14048 and 79, 329–374 (1985; Zbl 0571.14021)] and by B. Mazur and J. Tate in Arithmetic and geometry, Pap. dedic. Shafarevich, Vol. I. Arithmetic, Prog. Math. 35, 195–237 (1983; Zbl 0574.14036)]. A proof of the last statement is given in the sequel of this paper [R. F. Coleman, “The universal vectorial bi-extension and $$p$$-adic heights”, Invent. Math. 103, No. 3, 631–650 (1991; Zbl 0763.14009)].

##### MSC:
 14G20 Local ground fields in algebraic geometry 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14H25 Arithmetic ground fields for curves