Variation of the Faltings height in a \(\overline{\mathbb{Q}}\)-isogeny class of elliptic curves. (Variation de la hauteur de Faltings dans une classe de \(\overline{\mathbb{Q}}\)-isogénie de courbe elliptique.) (French) Zbl 0952.11018
This paper proves some bounds on the variation of the Faltings height of a non-CM elliptic curve within a \(\overline{\mathbb Q}\)-isogeny class. The principal result in this direction is the following. Let \(E\) be a semi-stable elliptic curve over a number field \(K\) without complex multiplication. Let \(n\) be a positive integer, and let \(\prod_{i=1}^r p_i^{\alpha_i}\) be its prime factorization. Let \(P\) be a torsion point of \(E\) of order exactly \(n\), and let \(\pi\:E\to E'\) be the \(K(P)\)-isogeny whose kernel is generated by \(P\). Then
\[
h(E')=h(E)+\tfrac 12\log n-\sum_i ((p_i^{\alpha_i}-1)/((p_i^2-1)p_i^{\alpha_i-1}))\log p_i + O(1),
\]
where \(O(1)\) depends only on \(E\). Moreover, if \(\text{Gal}(\overline K/K)\) acts transitively on the set of points of order \(n\) of \(E\), then the \(O(1)\) can be omitted.
The paper also proves a similar result for the norm of the conductor.
The paper also proves a similar result for the norm of the conductor.
Reviewer: P.Vojta (Berkeley)
MSC:
| 11G50 | Heights |
| 11G05 | Elliptic curves over global fields |
| 11R32 | Galois theory |
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
References:
| [1] | S. J. Arakelov, Intersection theory of divisors on an arithmetic surface , Math. USSR-Izv. 8 (1974), 1167-1180. · Zbl 0355.14002 · doi:10.1070/IM1974v008n06ABEH002141 |
| [2] | Gerd Faltings, Calculus on arithmetic surfaces , Ann. of Math. (2) 119 (1984), no. 2, 387-424. JSTOR: · Zbl 0559.14005 · doi:10.2307/2007043 |
| [3] | M. Flexor and J. Oesterlé, Sur les points de torsion des courbes elliptiques , Astérisque (1990), no. 183, 25-36, dans Séminaire sur les Pinceaux de Courbes Elliptiques, (Paris, 1988), Soc. Math. France, Paris. · Zbl 0737.14004 |
| [4] | Paul Hriljac, Splitting fields of principal homogeneous spaces , Number theory (New York, 1984-1985), Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 214-229. · Zbl 0623.14013 |
| [5] | Serge Lang, Introduction to Arakelov theory , Springer-Verlag, New York, 1988. · Zbl 0667.14001 |
| [6] | B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld) , Invent. Math. 44 (1978), no. 2, 129-162. · Zbl 0386.14009 · doi:10.1007/BF01390348 |
| [7] | Michel Raynaud, Hauteurs et isogénies , Astérisque (1985), no. 127, 199-234, dans Seminar on Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/84), Soc. Math. France, Paris. · Zbl 1182.14049 |
| [8] | Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Invent. Math. 15 (1972), no. 4, 259-331. · Zbl 0235.14012 · doi:10.1007/BF01405086 |
| [9] | Joseph H. Silverman, Heights and elliptic curves , Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 253-265. · Zbl 0603.14020 |
| [10] | Joseph H. Silverman, Hecke points on modular curves , Duke Math. J. 60 (1990), no. 2, 401-423. · Zbl 0744.14014 · doi:10.1215/S0012-7094-90-06016-8 |
| [11] | Lucien Szpiro, Degrés, intersections, hauteurs , Astérisque (1985), no. 127, 11-28, dans Seminar on Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/84), Soc. Math. France, Paris. · Zbl 1182.11029 |
| [12] | L. Szpiro, Sur les propriétés numériques du dualisant relatif d’une surface arithmétique , The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 229-246. · Zbl 0759.14018 |
| [13] | Emmanuel Ullmo, Points entiers, points de torsion et amplitude arithmétique , Amer. J. Math. 117 (1995), no. 4, 1039-1055. JSTOR: · Zbl 0863.14016 · doi:10.2307/2374958 |
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