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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 $$E(\mathbb F_p)$$.

##### MSC:
 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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