Linear forms in elliptic logarithms

Abstract

New lower bounds for linear forms in n (≥ 2) elliptic logarithms in the CM case are established. The estimate is better than all previous estimates with respect to some of the parameters that appear. It may be interesting to notice that the product log A 1 … log A n in the lower bound (see the Corollary of Theorem 1) is of exactly the same form as in the lower bounds for linear forms in logarithms of algebraic numbers (see A. Baker [ in “Transcendence Theory: Advances and Applications”. (A. Baker and D. W. Masser, Eds.), pp. 1–27, Academic Press, New York, 1977] ) and this is the first time such a parallelism has been achieved. To obtain the above lower bounds a zero estimate on the group variety G a n × E ( C n × E) is established (with E being an elliptic curve with CM), which is sharper than that derived from the general results in D. W. Masser and G. Wüstholz ( Inventiones Math. 63 (1981), 81–95).