On a theorem of Mestre and Schoof. (English) Zbl 1223.11072

For an elliptic curve \(E\) defined over the finite field \({\mathbb F}_q\) with \(q\) elements let \( \lambda(E)\) be the exponent of the group \(E({\mathbb F}_q) \). The authors prove that if \(q\not\in\{3,4,5,7,9,11,16,17,23,25,29,49\} \) then there exists a unique integer \(t\) with \(\mid t\mid\leq 2\sqrt{q}\) such that \(\lambda(E)\mid q+1-t \) and \(\lambda(E')\mid q+1+t \), where \(E' \) is the quadratic twist of \(E\).


11G07 Elliptic curves over local fields
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[1] René Schoof, Counting points on elliptic curves over finite fields. Journal de Théorie des Nombres de Bordeaux 7 (1995), 219-254. · Zbl 0852.11073
[2] Andrew V. Sutherland, Order computations in generic groups. PhD thesis, M.I.T., 2007, available at .
[3] Lawrence C. Washington, Elliptic curves: Number theory and cryptography, 2nd ed. CRC Press, 2008. · Zbl 1200.11043
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