Abstract.
Let E be an elliptic curve defined over a number field K, without complex multiplication, S a finite subset of E(K) and l a rational prime being ‘‘good modulus’’ for E/K. The main result of the paper asserts that if |S|≤l
2+l and for almost all prime ideals P of 
K
S contains an element R satisfying R mod P = lQ with Q∈E(
K
/P) then S contains an element R which satisfies R=lQ with some Q∈E(K). It improves the result of S.Wong [6], where the above statement is proved under the stronger assumption |S|≤l
2. Moreover we show that the bound |S|≤l
2+l is optimal.
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Skałba, M. Power residue problem on elliptic curves. manuscripta math. 114, 37–43 (2004). https://doi.org/10.1007/s00229-004-0444-2
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DOI: https://doi.org/10.1007/s00229-004-0444-2