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The distribution of group structures on elliptic curves over finite prime fields. (English) Zbl 1130.11054
Let \(E\) be an elliptic curve defined over a finite field \(\mathbb F_p\) with \(p\) elements. The group \(E(\mathbb F_p)\) of \(\mathbb F_p\)-rational points on \(E\) is isomorphic to \(\mathbb Z/m \times \mathbb Z/n\) for integers \(m, n\) with \(m \mid n\), and its order differs from \(p+1\) by at most \(2\sqrt{p}\) by a theorem of Hasse. The author now considers questions of the following type: given a group \(H = \mathbb Z/\ell^\alpha \times \mathbb Z/\ell^\beta\) with \(0 \leq \alpha \leq \beta\), what is the probability that the \(\ell\)-primary part of \(E(\mathbb F_p)\) is isomorphic to \(H\)? Using results of E. W. Howe [Compos. Math. 85, No. 2, 229–247 (1993; Zbl 0793.14023)], it is shown that this probability exists, and its value is computed with an error term \(O(x^{-1/2})\), the involved constants depending on \(\ell\), \(\alpha\), and \(\beta\). Next it is shown that this probability agrees with the Haar measure of a certain subset of \(\text{GL}(2,\mathbb Z_\ell)\).
These results are then applied to compute the probabilities for certain events, such as the divisibility of the group order by \(n\), or the cyclicity of \(E(\mathbb F_p)\).

11N45 Asymptotic results on counting functions for algebraic and topological structures
11G20 Curves over finite and local fields
Zbl 0793.14023
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