New York Journal of Mathematics New York J. Math. 5 (1999) 115{10. Explicit Local eights raham Everest Abstract. 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 aar integral in the elliptic Jensen formula. Contents 1. The Elliptic Jensen Formula 115. Singular Reduction 117 References 10 1. The Elliptic Jensen Formula In complex analysis, Jensen's formula is the following statement 1 0 log je it ; aj dt = log maxf1 jajg where a 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 [1]{[3]) that the global canonical height of a rational point on an elliptic curve dened over Q is analogous to Mahler's measure. In [1] and [], we gave a new approach to the canonical height where each local height arisesasanintegral of the kind in Jensen's formula. Let K denote any local eld containing Q, withj:j denoting the absolute value on K. Let E denote an elliptic curve dened over K and let Q E(K) denote a K-rational point. Write Q =(x Q y Q ) for the coordinates of Q with respect to a minimal dening equation. Let (Q) denote the local canonical height ofq. In [1], we pointed out the formula (1) log jx ; x Q jd =(Q) where is any compact group containing Q and denotes the aar measure on, normalised to give measure 1 to itself. The proof of (1) is trivial: just Received July 19, 1999. Mathematics Subject Classication. 11C08. Key words and phrases. Elliptic Curve, Canonical eight, Jensen's Formula. My thanks go to the referee for helpful comments. 115 c1999 State University of New York ISSN 1076-9803/99
116 raham Everest integrate the local parallelogram law. In particular, (1) holds with = < Q >, the topological closure of the group generated by Q. If Q is torsion then the group is nite with the discrete topology. The point of view in this paper is to assume (1) and use this, with = <Q>, to give a new proof of the explicit formulae for the local canonical heights. This is a dierent point of view to that in [3], 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 denition of the local canonical height andwork 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 sucient to look at the non-archimedean case. Let p denote a prime and let K denote a nite extension of Q p, the p-adic rational eld. Write j:j for the unique extension of the p-adic absolute value to K, sothatjpj =1=p. Let O K denote the valuation ring of K and let F denote the residue eld. The curve and points upon it can be reduced to give acurve E(F ). The reduced curve might be singular. If the reduced curve is singular, the reduction of Q might ormight not be singular. Theorem 1. Suppose Q is a point of non-singular reduction and = <Q>. Then log jx ; x Q j d = log maxf1 jx Q jg: Theorem 1 is the elliptic analogue of Jensen's formula and it is true for any compact group which contains Q by (1).Theorem 1 gives an alternative derivation of the explicit formula for the local canonical height of Q (see [3]) in the good reduction case. Note that in [3], the height is normalised to make it isomorphism invariant. I am going to give aproof of Theorem 1 assuming p 6= 3. This assumption allows me to use the usual Weierstrass equation, () y = x 3 + ax + b a b O K : Also, I assume Q is non-torsion: it makes little dierence. Proof. Let denote the subgroup of such that for all P we have jx P j > maxf1 jx Q jg: Then is topologically cyclic, generated by mq say, where 1 < m N. The measure of itself is 1=m. For any R, consider the integral over the coset R +, written (3) I R = log jx P +R ; x Q j dp: The integral in (3) is written in the classical notation to signify P as the variable of integration. Suppose rstly that jx Q j > 1. Then jy Q j > 1and()gives (4) jy Q j = jx Q j 3 also jy P j = jx P j 3 for P : By the translation invariance of the measure, (5) I R = log jx P j dp for R O mod :
Explicit Local eights 117 If <mthen the cosets Q + are distinct. For P, consider (6) jx P Q ; x Q j = yp x P 1 y Q y P 1 ; x ; Q x P ; x P ; x Q : Expand the brackets in (6) using the binomial theorem, use () and extract the dominant termtogive (7) jx P Q ; x Q j = y P y Q x P = y Q x P : y P Therefore the total contribution from the cosets O Q + is (8) log y Qx P dp + y P log jx P j dp = log y Q x3 P y P dp: Using () and (4), and remembering to give measure 1=m to, (8) collapses to (9) 3 m log jx Qj: For cosets with R not O Q mod, jx P +R ; x Q j = jx Q j so each coset contributes 1 m log jx Qj. There are m ; 3 of these cosets in total so log jx ; x Q j d = m ; 3 m log jx Qj + 3 m log jx Qj =logjx Q j: In the case where m =, the identity coset gives the formula in (5). For the nonidentity coset, note that when m =, jy Q j < jx Q j. Then a new dominant term emerges in (6) giving (10) jx P +Q ; x Q j = j1=x P j that is I Q = ; log jx P j dp: Clearly the contributions from the two cosets cancel each other. Next suppose that jx Q j1. Deal rstly with the case that m>. The integrals I R, for R not O Q mod all vanish. This uses the non-singular reduction hypothesis. The reduced curve E(F ) is a group and jx P +R ; x Q j < 1ifandonly if R Q reduces to the point at innity. As in (8), the total contribution from the cosets with R O Q mod is (11) m log jy Qj: Butthetermin(11)must vanish because we cannot have jy Q j < 1, otherwise m =. In the case when m =,fortheidentity coset, the formula in (5) remains valid. For the non-identity coset, we note that jy Q j < 1 and this causes a new dominant term to emerge in (6) giving (10) asabove. Once again the two contributions cancel and the proof of Theorem 1 is complete.. Singular Reduction I am going to compute the local height only in the case when Q is point of split multiplicative singular reduction on E. It is always possible to assume the reduction is of this type, by passing to a nite extension of K. Use the Tate curve together
118 raham Everest with the q-parametrisation. All the denitions needed come from Chapter V of [3]. The Tate curve has the form (1) y + xy = x 3 + ax + b a b O K : The points on the projectivecurve are isomorphic to the group K =q where q K has jqj < 1. The explicit formula for the x and y-coordinates of a non-identity point are given in terms of the parameter u K as follows: (13) X q n u X x = x u = (1 ; q n n u) ; nq n (1 ; q n ) n1 (14) X q n u X y = y u = (1 ; q n n u) + nq n 3 (1 ; q n ) : n1 Formula (13) makes it obvious that x u = x uq and x u = x u ;1. Similarly for formula (14) and the y-variable. If Q corresponds to the point u K, take = <u>,a compact group. Assume u is chosen to lie in a fundamental domain, which means that jqj = p ;k < juj = p ;r 1, where r and k denote rationals. Theorem. Suppose Q is a point of split multiplicative reduction corresponding to u K with jqj = p ;k < juj = p ;r 1 and = <u>. Then (15) log jx ; x u j d = ( ;logj1 ; uj if juj =1, ; r k ; ( r k ) log jqj if juj < 1. Theorem gives an alternative derivation of the explicit formula for the local canonical height of Q in the case of split multiplicative reduction. This formula agrees with the one in Chapter VI of [3] but note that in [3], heights are normalised to make them isomorphism invariant. Proof. Assume rstly that juj =1. If ju;1j < 1 then Theorem 1 applies so assume ju ; 1j = 1 and show the integral in (15) vanishes. Write for the subgroup of consisting of all v with jv ; 1j < 1. Then is topologically cyclic, generated by u m say. Consider the integral over the coset w, written (16) I w = log jx wv ; x u j dv where in (16), the classical notation is chosen once again to point to the variable v. Assuming rstly that m >, and referring to the explicit formula for the x-coordinate in (14), the only non-zero integrals come from the cosets with w =1 u 1. Obviously, (17) I w = log jx v jdv when w =1: When w = u 1, take note that x u = x u ;1 and use the addition formula for the Tate curve, yv ; y u yv ; y u x vu 1 = + ; x v ; x u : x v ; x u x v ; x u
Therefore (18) jx vu 1 ; x u j = yv Explicit Local eights 119 x v 1 ; yu =y v 1 ; x u =x v + y v x v 1 ; yu =y v 1 ; x u =x v ; x v ; x u : From (14), jx u j = jy u j = 1 when juj = ju ; 1j =1. Also, from (1), for any v, (19) jy v j = jx v j 3 : Just as in (6), expand the brackets in (18) using the binomial theorem, use (1) and (19) then extract the dominant term, to obtain (0) jx vu 1 ; x u j = jy v =x v j that is I u1 = log jy v =x v j dv: Sum the contribution from the three cosets with w =1 u 1, and use (19), to give (1) log jy v =x v j dv + log jx v j dv = log jyv =x3 vj dv =0: These calculations assumed m>. In the case when m =, (17) remains valid. For the non-identity coset, the cancelling in (18) works out dierently. From the addition law, 3x x u = u + a 3x + u + a ; x u : y u + x u y u + x u Therefore, if jx u j > 1, it follows that jy u + x u j < 1. It is this fact which causes two extra terms in (18) to cancel and leaves () jx vu ; x u j = j1=x v j that is I u = ; log jx v j dv: Clearly now the contributions from the two cosets cancel each other. Finally, deal with the case where juj < 1. To ease the computation, assume k = mr, where m N. To ease the computation further, assume ju m =q ; 1j < 1. In general, one would take m N smallest with qju m O K and ju m =q ; 1j < 1. Suppose rstly that m>. Let = <u m >. Using the same notation as before, the contributions from the cosets w with w 6= 1 u 1 are all equal to 1 m log jx uj, remembering that the measure of each coset is 1=m. There are m;3 of these cosets giving a total contribution of (m ; 3) (3) m log jx uj: For the cosets u 1,take account that jx u 1 j < 1. Taking the dominant termin (18), (4) I u 1 = log jx u y v =x v j dv: Including now the contribution from the identity coset gives (5) I 1 + I u + I u ;1 = log jx u y v =x3 v j dv = m log jx uj where, in (5), (19) has been used. Combining (3) and (5) gives 1 1 1 ; log jx u j = 1 m m ; log jqj m
10 raham Everest as required. If m =thenitiseasytocheck that the integrals over the two cosets combine to give 1 log jqj as they should. 4 References [1] raham Everest and Brid Ni Fhlathuin, The elliptc Mahler measure, Math. Proc. Camb. Phil. Soc. 10 (1996), 13{5, MR 97e:11064, bl 865.11068. [] raham Everest and Thomas Ward, eights of Polynomials and Entropy in Algebraic Dynamics, Springer-verlag, Berlin, 1999. [3] Joseph. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer-verlag, New York, 1994, MR 96b:11074, bl 911.14015. School of Mathematics, University of East Anglia, Norwich, Norfolk, NR4 7TJ, England g.everest@uea.ac.uk http://www.mth.uea.ac.uk/~h090/ This paper is available via http://nyjm.albany.edu:8000/j/1999/5-9.html.