5 problems

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…

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 $f=\mathfrak{a}0\sum a_nq^n$ be a newform with rational Fourier coefficients, and let $m\geq 2$ be an integer. Define $\delta_m(f)$ as the relative density of the primes…

Maximum VC-dimension conjecture for power subgroups. One has