8 problems

Let $E$ be an elliptic curve over $\mathbf Q$, let $m\geq 1$, and let $\mathcal C$ be a set of primes contained in the set of primes congruent to $1$ modulo $m$. Write…

Let $E_t^d$ be the CM elliptic curve and let $w_d$ denote the order of the relevant unit group. For a prime $p$, write $\left(\frac{a_p(E_t^d)}{p}\right)_m$ for the $m$th power res…

Fix an imaginary quadratic field $K$ of class number one and an elliptic curve $E$ with complex multiplication by $\mathcal O_K$, and let $w$ be the order of $\mathcal O_K^\times$.…

Let $f$ be a rational newform of weight $k\geq 2$ with complex multiplication by $\mathbf{Q}(\sqrt{-d})$, where $d>3$, and let $m$ be relatively prime to $2(k-1)$. CM power-residue…

Let $\rho$ be a motivic Galois representation with open image as above, let $m\geq 2$, and let $\mathcal{P}$ be a Cebatorev set, meaning that, up to finitely many primes, it is spe…

Let $\rho:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GL}_n\mathbf{Q}_\ell$ be a motivic Galois representation, and let $\delta_m(\rho)$ be the density rat…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with complex multiplication by an imaginary quadratic field $K$, and let $w_K$ denote the number of roots of unity in $K$. For each p…

Maximum VC-dimension conjecture for power subgroups. One has