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Let E be an elliptic curve over the rationals. There are two different ways in which the set of rational points on E can be said to be divisible by a prime p. We will call one of these types of divisibility local and the other global. Global divisibility will imply local divisibility; however, the converse is not guaranteed. In this project we focus on the cases where p=5 and p=7 to determine the probability that E has global divisibility by p, given that E has local divisibility by p.
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Kenney, Meagan, "The Conditional Probability That an Elliptic Curve Has a Rational Subgroup of Order 5 or 7" (2019). Senior Projects Spring 2019. 202.