Linear forms in elliptic logarithms

LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE <I>p</I>-ADIC CASE

“…Then Lemma 7.1 assures that, for any L × L minor determinant of M, we have the estimate (13). Under the condition (20), the assumption of …”

“…[15,[46][47][48]). The dependence on the height of algebraic coefficients of the linear forms in n elliptic logarithms is refined as the best possible one by David and Hirata-Kohno in [13].…”

An effective result of André-Oort type

“…Proposition 1 (Theorem 1.6 of [7] for k = 1 and E = e). There exists an effective absolute constant c 1 > 0 with the following property: Assume that u / ∈ h(C) for any connected algebraic subgroup…”

“…Then we apply an elliptic transcendence measure originally due to Masser [19] and deduce an upper bound on |∆ 1 | and |∆ 2 | of the sort stated in Theorem 2. Instead of the original result in [19] we use here a more explicit transcendence measure given by David and Hirata-Kohno [7]. If x σ is close to (∞, ∞) ∈ Y cusp , then we construct a linear form Λ in logarithms of algebraic numbers with algebraic coefficients from the fact that near ∞ the j-function can be well approximated by the first two terms of its q-expansion e −2πiτ +744+e 2πiτ P (e 2πiτ ), A lack of sufficiently good bounds for linear forms in elliptic logarithms leads to the exponent (8 + ε) in Theorem 2.…”

“…If x σ is close to (∞, ∞) ∈ Y cusp , then we construct a linear form Λ in logarithms of algebraic numbers with algebraic coefficients from the fact that near ∞ the j-function can be well approximated by the first two terms of its q-expansion e −2πiτ +744+e 2πiτ P (e 2πiτ ), A lack of sufficiently good bounds for linear forms in elliptic logarithms leads to the exponent (8 + ε) in Theorem 2. An elliptic analogue of the LangWaldschmidt conjecture (Conjecture 1.7 of [7]) would give (2 + ε) instead. In addition, it is noteworthy that we invoke Baker's Theorem for linear forms with (essentially) purely imaginary coefficients.…”