We study probabilistic behaviors of elliptic curves with torsion points. First, we compute the probability for elliptic curves over the rationals with a non-trivial torsion subgroup $G$ whose size $\leq 4$ to satisfy a certain local condition.
We have a good interpretation of the probabilities we obtain, and for multiplicative reduction case, we have a heuristic to explain the probability. Furthermore, for $G=\mathbb{Z}/ 2\mathbb{Z} $ or $ \mathbb{Z} /2 \mathbb{Z} \times \mathbb{Z} /2 \mathbb{Z} $, we give an explicit upper bound of the $n$-th moment of analytic ranks of elliptic curves with a torsion subgroup $G$ for every positive integer $n$, and show that the probability for elliptic curves with a torsion group $G$ with a high analytic rank is small under GRH for elliptic $L$-function. From the results we have obtained, we conjecture that the condition of having the analytic rank $0$ or $1$ is independent of the condition of having the torsion subgroup $G= \mathbb{Z} /2 \mathbb{Z}$ or $ \mathbb{Z} /2 \mathbb{Z} \times \mathbb{Z} /2 \mathbb{Z}$.
(Send me(Bo-Hae Im) an email to get the Zoom link, if you would like to join this seminar.)
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