A new proof is given for the explicit formulae for the non-archimedean canonical height on an elliptic curve. This arises as a direct calculation of the Haar integral in the elliptic Jensen formula.
In complex analysis, Jensen's formula is the following statement where denotes any complex number. This formula is fundamental to the development of Mahler's measure of a polynomial. For a full discussion of this subject, and a proof of Jensen's formula, see . It is known (see –) that the global canonical height of a rational point on an elliptic curve defined over is analogous to Mahler's measure. In  and , we gave a new approach to the canonical height where each local height arises as an integral of the kind in Jensen's formula.
Let denote any local field containing , with denoting the absolute value on . Let denote an elliptic curve defined over and let denote a -rational point. Write for the coordinates of with respect to a minimal defining equation. Let denote the local canonical height of . In , we pointed out the formula where is any compact group containing and denotes the Haar measure on , normalised to give measure 1 to itself. The proof of (1) is trivial: just integrate the local parallelogram law. In particular, (1) holds with , the topological closure of the group generated by . If is torsion then the group is finite with the discrete topology.
The point of view in this paper is to assume (1) and use this, with , to give a new proof of the explicit formulae for the local canonical heights. This is a different point of view to that in , where the explicit formulae are shown to be the unique functions which satisfy the parallelogram law. What is gained is a new interpretation for the exotic formulae for the local canonical heights. Presumably, one could take (1) as the definition of the local canonical height and work back to the parallelogram law, but this is not pursued here.
The explicit formula in the archimedean case was worked out in  so it is sufficient to look at the non-archimedean case. Let denote a prime and let denote a finite extension of , the -adic rational field. Write for the unique extension of the -adic absolute value to , so that . Let denote the valuation ring of and let denote the residue field. The curve and points upon it can be reduced to give a curve . The reduced curve might be singular. If the reduced curve is singular, the reduction of might or might not be singular.
Theorem 1 is the elliptic analogue of Jensen's formula and it is true for any compact group which contains by (1). Theorem 1 gives an alternative derivation of the explicit formula for the local canonical height of (see ) in the good reduction case. Note that in , the height is normalised to make it isomorphism invariant.
Proof. Let denote the subgroup of such that for all we have Then is topologically cyclic, generated by say, where . The measure of itself is . For any , consider the integral over the coset , written The integral in (3) is written in the classical notation to signify as the variable of integration.
Suppose firstly that . Then and (2) gives By the translation invariance of the measure, If then the cosets are distinct. For , consider Expand the brackets in (6) using the binomial theorem, use (2) and extract the dominant term to give Therefore the total contribution from the cosets is Using (2) and (4), and remembering to give measure to , (8) collapses to For cosets with not mod , so each coset contributes . There are of these cosets in total so In the case where , the identity coset gives the formula in (5). For the non-identity coset, note that when , . Then a new dominant term emerges in (6) giving Clearly the contributions from the two cosets cancel each other.
Next suppose that . Deal firstly with the case that . The integrals , for not mod all vanish. This uses the non-singular reduction hypothesis. The reduced curve is a group and if and only if reduces to the point at infinity. As in (8), the total contribution from the cosets with mod is But the term in (11) must vanish because we cannot have , otherwise . In the case when , for the identity coset, the formula in (5) remains valid. For the non-identity coset, we note that and this causes a new dominant term to emerge in (6) giving (10) as above. Once again the two contributions cancel and the proof of Theorem 1 is complete.
I am going to compute the local height only in the case when is point of split multiplicative singular reduction on . It is always possible to assume the reduction is of this type, by passing to a finite extension of . Use the Tate curve together with the -parametrisation. All the definitions needed come from Chapter V of . The Tate curve has the form The points on the projective curve are isomorphic to the group where has . The explicit formula for the and -coordinates of a non-identity point are given in terms of the parameter as follows: Formula (13) makes it obvious that and . Similarly for formula (14) and the -variable. If corresponds to the point , take , a compact group. Assume is chosen to lie in a fundamental domain, which means that , where and denote rationals.
Theorem 2 gives an alternative derivation of the explicit formula for the local canonical height of in the case of split multiplicative reduction. This formula agrees with the one in Chapter VI of  but note that in , heights are normalised to make them isomorphism invariant.
Proof. Assume firstly that . If then Theorem 1 applies so assume and show the integral in (15) vanishes. Write for the subgroup of consisting of all with . Then is topologically cyclic, generated by say. Consider the integral over the coset , written where in (16), the classical notation is chosen once again to point to the variable . Assuming firstly that , and referring to the explicit formula for the -coordinate in (14), the only non-zero integrals come from the cosets with . Obviously, When , take note that and use the addition formula for the Tate curve, Therefore From (14), when . Also, from (12), for any , Just as in (6), expand the brackets in (18) using the binomial theorem, use (12) and (19) then extract the dominant term, to obtain Sum the contribution from the three cosets with , and use (19), to give
These calculations assumed . In the case when , (17) remains valid. For the non-identity coset, the cancelling in (18) works out differently. From the addition law, Therefore, if , it follows that . It is this fact which causes two extra terms in (18) to cancel and leaves Clearly now the contributions from the two cosets cancel each other.
Finally, deal with the case where . To ease the computation, assume , where . To ease the computation further, assume . In general, one would take smallest with and . Suppose firstly that . Let . Using the same notation as before, the contributions from the cosets with are all equal to , remembering that the measure of each coset is . There are of these cosets giving a total contribution of For the cosets , take account that . Taking the dominant term in (18), Including now the contribution from the identity coset gives where, in (25), (19) has been used. Combining (23) and (25) gives as required. If then it is easy to check that the integrals over the two cosets combine to give as they should.