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We study the ramification properties of the extensions $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ under the hypothesis that $m$ is odd and if $p\mid m$ than either $p\nmid v_p\left(a\right)$ or $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(a\right)$ and $v_p\left(m\right)$ are the exponents with which $p$ divides $a$ and $m$). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the $p$-adique valuation of the discriminant of the studied global extensions with $m=p^r$.

In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.