Degrees of the elliptic Teichmüller lift
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 \[ a_1= a^2_0 b^2_0+ {b^4_0\over a_0},\quad b_1= 4a^6_0 b_0+ a^3_0 b^3_0+ b^5_0, \] \[ \begin{multlined} a_2= 2a^{25}_0+ a^{22}_0 b^2_0+ a^{19}_0 b^4_0+ 3a^{16}_0 b^6_0+ 2a^{13}_0 b^8_0+\\ a^7_0 b^{12}_0+ 4a_0 b^{16}_0+ {3b^{18}_0\over a^2_0}+ {4b^{20}_0\over a^5_0}+ {4b^{22}_0\over a^8_0}+ {4b^{24}_0\over a^{11}_0},\end{multlined} \] \[ b_2= a^{36}_0 b_0+ 4a^{33}_0 b^3_0+ 3a^{27}_0 b^7_0+ 4a^{21}_0 b^{11}_0+ 4a^{15}_0 b^{15}_0+ a^{12}_0 b^{17}_0+ 3a^6_0 b^{21}_0+ b^{25}_0. \]
scientific article; zbMATH DE number 1815497