On Selmer groups of cyclic twist families of elliptic curves over global function fields [NT seminar]

Let $K = \mathbb{F}_q(t)$ be the global function field of characteristic coprime to 2 and 3. Let $E$ be a non-isotrivial elliptic curve over $K$. Fix a prime number $l$ that is coprime to the characteristic of $K$, and that the primitive $l$-th roots of unity is contained in the constant field of $K$. Let $L$ be a cyclic $\mathbb{Z}/l\mathbb{Z}$ geometric Galois extension over $K$. We will explore two approaches - a probabilistic approach and a geometric approach - to compute a lower bound on the probability that the rank of $E(L)$ is equal to the rank of $E(K)$. If time allows, we will also explore new geometric insights that can be obtained from comparing the two approaches.