6 problems

Let $E/\mathbb Q$ be an elliptic curve, let $p$ be a prime, and let $G\subseteq \operatorname{GL}(2,\mathbb Z/p\mathbb Z)$ be the image of the mod-$p$ Galois representation…

The no-large-prime coincidence conjecture. If $p\geq 5$, then there are no non-CM elliptic curves $\mathcal{E}$ over $\mathbb{Q}$ with a $p$-coincidence. This is a more focused for…

Let $\mathcal{E}$ be a non-CM elliptic curve over $\mathbb{Q}$. For a prime $p$, call an equality … a $p$-coincidence. The 2-and-3 coincidence conjecture. For non-CM elliptic curve…

Let $\mathcal{E}$ be a non-CM elliptic curve over $\mathbb{Q}$, let $j$ be a positive integer, and let $\mathcal{C}_{\mathcal{E},j}$ denote the constant considered in the paper. A…

For an elliptic curve $E$ defined over $\mathbb{Q}$, an $(m,n)$-coincidence means an equality of division fields $\mathbb{Q}(E[m])=\mathbb{Q}(E[n])$ with $m\neq n$. The known coinc…

Let $\mathcal{E}$ be the elliptic curve under consideration, let $p$ be a prime, and let $\zeta_p$ denote a primitive $p$th root of unity. Suppose that … for some…