Comparison of \(p\)-adic theta functions (Q2756087)

For a rational prime \(p\), denote by \(K\) a finite extension of \({\mathbb Q}_p\) and by \(m\) the maximal ideal of the ring of integers \({\mathfrak O}_K\). NEWLINENEWLINENEWLINELet \(A\) be an elliptic curve over \(K\), with identity element \(e\), and, for any positive integer \(n\), denote by \(n\iota:A\to A\) the multiplication by \(n\) of the points of \(A\). Let \(D\) be a \(K\)-rational divisor on \(A\) with degree \(0\), disjoint from \(e\) modulo \(m\), and for any subgroup \(G\) of points of \(A\), denote by \(I=I_G\) the ideal in the group-ring \({\mathbb Z}[G]\) of the zero-cycles of degree \(0\) and denote by \([D,{\mathfrak a}]\) the pairing of \(D\) with a cycle \({\mathfrak a}\in I^2\). NEWLINENEWLINENEWLINEFor a suitable subgroup \(G\) of \(A(K)\) and \({\mathfrak a}\in I_G\), Néron proved that the limit \(\vartheta_D^\nu({\mathfrak a}) = \lim_{r\to\infty}[D, p^r{\mathfrak a}-(p^r\iota){\mathfrak a}]^{1/p^r}\) exists and defined a function, \(a\mapsto \vartheta_D^\nu((a)-(e))\), analytic on \(G\), with values in \(1+m\). This is the Néron theta function associated to \(D\) [cf. \textit{A. Néron}, Sémin. Delange-Pisot-Poitou, Prog. Math. 22, 149-174 (1982; Zbl 0492.14035)]. NEWLINENEWLINENEWLINEGiven a uniformization \(1\to q^{\mathbb Z}\to K^\times \buildrel{\pi}\over{\to} A\to 0\) of \(A\), denote by \(\theta_D\) a theta function (in the sense of Tate) associated to \(D\), normalized by putting \(\theta_D(e)=1\). If \(G\) is the subgroup of definition of the Néron theta function and \({\mathfrak a}=(\pi(a))-(e)\in I_G\), the author states that \([D, p^r{\mathfrak a}-(p^r\iota){\mathfrak a}]\theta_D(a^{p^r})=\theta_D(a)^{p^r}\) and that, for \(|a|<1\), the limit \(\lim_{r\to\infty}\theta_D(a^{p^r})^{1/p^r}\) exists in \(1+m\). As a consequence the author states that, in a suitable neighborhood of the origin of \(A\), the quotient of the two theta functions is analytic and its values are units.