On the distance of an algebraic point from the origin in Abelian varieties (Q5935871)

Let \(A\) be a complex Abelian variety equipped with an invertible ample and symmetric fibre \(F\). Let \(p\in A\). We denote by \(r_F(p)\) the distance of \(p\) from the origin 0 of \(A\). If \(A\) is defined over a number field, then we denote by \(\widehat{h}_F\) the canonical height of Néron-Tate with respect to \(F\). Finally, we put \(h(A)= \max\{1, h_{\text{Fal}}(A)\}\), where \(h_{\text{Fal}}(A)\) is the semistable Faltings height of \(A\). NEWLINENEWLINENEWLINELet \(k\) be a number field of degree \(d\), \(A\) an Abelian variety of dimension \(g\) defined over \(k\) with an invertible ample and symmetric fibre \(F\) defined over \(k\) and \(C\) be a positive real number. In this paper, the author proves that if \(K\) is a number field of degree \(D\) with \(K\supseteq k\) and \(q\) a point of \(A(K)\) of infinite order modulo every propery Abelian subvariety of \(A\) satisfying NEWLINE\[NEWLINE\widehat{h}_F(q)\leq \min\{CD^{-1/g}, dh(A)+\log D\},NEWLINE\]NEWLINE then NEWLINE\[NEWLINE\log r_F(q)\geq -c_1C^{g/(1+2g)} (h(A)+\log D)^{(1+g)/(1+2g)} D^{2g/(1+2g)},NEWLINE\]NEWLINE where \(c_1\) is a positive real number depending on \(g\), \(d\) and \(\dim H^0(A,F)\). NEWLINENEWLINENEWLINEThis result extends some results of \textit{M. Mignotte} and \textit{M. Waldschmidt} [J. Number Theory 47, 43-62 (1994; Zbl 0801.11033)] to Abelian varieties over a number field.