The \(p\)-adic height pairings of Coleman-Gross and of Nekovář.

*(English)*Zbl 1153.11316
Kisilevsky, Hershy (ed.) et al., Number theory. Papers from the 7th conference of the Canadian Number Theory Association, University of Montreal, Montreal, QC, Canada, May 19–25, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3331-6/pbk). CRM Proceedings & Lecture Notes 36, 13-25 (2004).

Introduction: In [Adv. Stud. Math. 7, 73–81 (1989; Zbl 0758.14009)], R. Coleman and B. Gross proposed a definition of a \(p\)-adic height pairing on curves over number fields with good reduction at primes above \(p\). The pairing was defined as a sum of local terms and the most interesting terms are the ones corresponding to primes above \(p\) where the definition depends on Coleman’s theory of \(p\)-adic integration. Later, J. Nekovář constructed in [Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)] a general \(p\)-adic height pairing for Galois representations which are cristalline at primes above \(p\), and satisfying certain additional technical assumptions.

Wintenberger raised the question of the equality of the two height pairings for curves. In other words: does the Nekovář height pairing applied to \(H_{\text{ét}}^1\) of a curve recover the Coleman-Gross pairing. The purpose of this small note is to answer this in the affirmative. More precisely, let \(F\) be a number field and let \(X\) be a smooth and proper curve over \(F\), with good reduction at all places \(v\) above a fixed prime \(p\). To define either the height of Coleman and Gross or the height of Nekovář the following choices must be made.

With these choices the Coleman-Gross height pairing is a pairing

\[ h_{\text{CG}}: \text{Div}_0(X)\times \text{Div}_0(X)\to\mathbb Q_p, \]

where \(\text{Div}_0(X)\) denotes the group of zero divisors on \(X\) (defined over \(F\)), while the Nekovář height pairing is a pairing

\[ h_{\text{Nek}}: H_f^1(F,V)\times H_f^1(F,V)\to\mathbb Q_p, \]

where \(V=V_p(J_X)\cong H_{\text{ét}}^1(X,\mathbb Q_p(1))\) is the Tate-module of the jacobian of \(X\) and \(H_f\) is finite cohomology in the sense of Bloch and Kato. The relation between the two pairings is provided by the étale Abel-Jacobi map

\[ \alpha_X: \text{Div}_0(X)\to H_f^1(F,V). \]

Our main result is the following.

Theorem 1.1. With the same choices of \((\ell,W_v)\) we have

\[ h_{\text{Nek}}(\alpha_X(y),\alpha_X(z))= h_{\text{CG}}(y,z). \]

The proof goes as follows. We first recall the construction of the Coleman-Gross height in Section 2. This is expressed in terms of local heights. Likewise, the Nekovář height can be expressed as a sum of local heights, although it also has a global description which we will not need here. The decomposition into local heights depends on a choice of a “mixed extension”. We recall this in Section 3 as well as the particular mixed extension we will use. With this mixed extension, it is known that the local heights at places not dividing \(p\) are equal to the local heights of Coleman-Gross. This leaves the comparison of the local heights above \(p\). This is done in the rest of the paper.

For the entire collection see [Zbl 1051.11006].

Wintenberger raised the question of the equality of the two height pairings for curves. In other words: does the Nekovář height pairing applied to \(H_{\text{ét}}^1\) of a curve recover the Coleman-Gross pairing. The purpose of this small note is to answer this in the affirmative. More precisely, let \(F\) be a number field and let \(X\) be a smooth and proper curve over \(F\), with good reduction at all places \(v\) above a fixed prime \(p\). To define either the height of Coleman and Gross or the height of Nekovář the following choices must be made.

- –
- A “global log” – a continuous idele class character \(\ell:\mathbb A_F^\times/F^\times\to\mathbb Q_p\).
- –
- for each \(v|p\) a choice of a subspace \(W_v\in H_{dR}^1(X\otimes F_v/F_v)\) complementary to the space of holomorphic forms.

With these choices the Coleman-Gross height pairing is a pairing

\[ h_{\text{CG}}: \text{Div}_0(X)\times \text{Div}_0(X)\to\mathbb Q_p, \]

where \(\text{Div}_0(X)\) denotes the group of zero divisors on \(X\) (defined over \(F\)), while the Nekovář height pairing is a pairing

\[ h_{\text{Nek}}: H_f^1(F,V)\times H_f^1(F,V)\to\mathbb Q_p, \]

where \(V=V_p(J_X)\cong H_{\text{ét}}^1(X,\mathbb Q_p(1))\) is the Tate-module of the jacobian of \(X\) and \(H_f\) is finite cohomology in the sense of Bloch and Kato. The relation between the two pairings is provided by the étale Abel-Jacobi map

\[ \alpha_X: \text{Div}_0(X)\to H_f^1(F,V). \]

Our main result is the following.

Theorem 1.1. With the same choices of \((\ell,W_v)\) we have

\[ h_{\text{Nek}}(\alpha_X(y),\alpha_X(z))= h_{\text{CG}}(y,z). \]

The proof goes as follows. We first recall the construction of the Coleman-Gross height in Section 2. This is expressed in terms of local heights. Likewise, the Nekovář height can be expressed as a sum of local heights, although it also has a global description which we will not need here. The decomposition into local heights depends on a choice of a “mixed extension”. We recall this in Section 3 as well as the particular mixed extension we will use. With this mixed extension, it is known that the local heights at places not dividing \(p\) are equal to the local heights of Coleman-Gross. This leaves the comparison of the local heights above \(p\). This is done in the rest of the paper.

For the entire collection see [Zbl 1051.11006].