4 problems

Let $L(\chi_n,s)$ be the $L$-function of a primitive character of order $p^n$, let $\ell(n)$ be its degree, let $\ell_\alpha(n)$ count the slopes equal to a fixed rational number…

Let $(C_n)_{n\geq 0}$ be a $\mathbb{Z}_p$-tower of curves from algebraic geometry. Let $g_n$ be the genus of $C_n$, let $d_\alpha(n)$ be the slope-counting quantity defined in the…

Let $P(C_n,s)$ be the numerator of the zeta function of the curve $C_n$. Let $\mathbb{Q}_n$ and $\mathbb{Q}_{p,n}$ be its splitting fields over $\mathbb{Q}$ and $\mathbb{Q}_p$, res…

Let $(C_n)_{n\geq 0}$ be a $\mathbb{Z}_p$-tower of curves over a finite field, with genus $g_n=g(C_n)$. Assume the tower comes from algebraic geometry. Genus stability conjecture.…