Degrees of the elliptic Teichmüller lift
type: Item
instance of: scholarly article
title: Degrees of the elliptic Teichmüller lift
zbMATH Open document ID: 1043.11053
author: Luís R. A. Finotti
published in: Journal of Number Theory
publication date: 2002-10-16T00:00:00Z
review text: Let \(k\) be a perfect field of characteristic \(p> 0\) and \(E/k\) be an ordinary elliptic curve defined over \(k\). This curve can be given by a Weierstrass equation over \(k\): \[ E/k: y^2+ a_0 xy+ b_0 y= x^3+ c_0 x^2+ d_0 x+ e_0. \] It has a canonical lifting to an elliptic curve \(\widehat E/W(k)\) over the ring of Witt vectors \(W(k)\): \[ \widehat E: Y^2+ aXY+ bY= X^3+ cX^2+ dX+ e, \] \(a= (a_0,a_1,\dots)\), \(b= (b_0,b_1,\dots)\dots, e= (e_0,e_1,\dots)\) being elements of \(W(k)\). Let \(\overline k\) be an algebraic closure of \(k\). We have an injective group homomorphism \[ \tau: E(\overline k)\to \widehat E(W(\overline k)), \] given by a section of the reduction map, called the elliptic Teichmüller lift of \(E\): \[ (x_0, y_0)\mapsto ((x_0, x_1,x_2,\dots), (y_0,y_1, y_2,\dots)). \] The coordinate functions \(x_n\) and \(y_n\) are regular except at the origin \(O\) of \(E\). The author obtains bounds for the order of poles of \(x_n\) and \(y_n\) at \(O\). Let \(v\) be the valuation on the function field of \(E\) given by the order of vanishing of functions at \(O\). For all \(n\geq 0\), he proves the inequalities \[ v(x_n)\geq -((n+ 2)p^n- np^{n-1})\quad\text{and}\quad v(y_n)\geq -((n+ 3)p^n- np^{n-1}). \] He also proves that the bound for \(x_n\) is not the exact order if and only if \(p\) divides \(n+ 1\), and that the bound for \(y_n\) is not the exact order if and only if \(p\) divides \((n+ 1)(n+ 2)/2\). Besides, the author gives an algorithm to compute the canonical lifting \(\widehat E\) and the elliptic Teichmüller lift explicitly up to the third coordinate in case \(p\neq 2,3\). For instance, suppose \(p= 5\) and the Weierstrass equation of \(E/k\) is given by \[ y^2= x^3+ a_0 x+ b_0. \] Since the curve is ordinary, we have \(a_0\neq 0\). Concerning the canonical lifting, he obtains \[ a_1= a^2_0 b^2_0+ {b^4_0\over a_0},\quad b_1= 4a^6_0 b_0+ a^3_0 b^3_0+ b^5_0, \] \[ \begin{multlined} a_2= 2a^{25}_0+ a^{22}_0 b^2_0+ a^{19}_0 b^4_0+ 3a^{16}_0 b^6…
reviewed by: Alain Kraus
Mathematics Subject Classification ID: 11G07
zbMATH DE Number: 1815497
zbMATH Keywords: elliptic curves
zbMATH Keywords: canonical lightings
zbMATH Keywords: elliptic Teichmüller lift
MaRDI profile type: Publication
full work available at URL: jnth.2001.2756
OpenAlex ID: W2023207137
cites work: Geometry of \(p\)-jets
cites work: Frobenius and the Hodge filtration (estimates)
cites work: Q3856855
cites work: Euclidean weights of codes from elliptic curves over rings
DOI: 10.1006/JNTH.2001.2756
description: scientific article; zbMATH DE number 1815497