\(p\)-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)].

\(k\) 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)].

Reviewer: M. van der Put (Groningen)

##### MSC:

14G20 | Local ground fields in algebraic geometry |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14H25 | Arithmetic ground fields for curves |