Split degenerate superelliptic curves and $\ell$-adic images of inertia

Abstract

Let $K$ be a field with a discrete valuation, and let $p$ and $\ell$ be (possibly equal) primes which are not necessarily different from the residue characteristic. Given a superelliptic curve $C:y^p=f(x)$ which has split degenerate reduction over $K$, with Jacobian denoted by $J/K$, we describe the action of an element of the inertia group $I_K$ on the $\ell$-adic Tate module $T_{\ell }(J)$ as a product of powers of certain transvections with respect to the $\ell$-adic Weil pairing and the canonical principal polarization of $J$. The powers to which the transvections are taken are given by a formula depending entirely on the cluster data of the roots of the defining polynomial $f$. This result is demonstrated using Mumford's non-archimedean uniformization of the curve $C$.

32 pages, 7 sections, 0 figures. Main revisions from V1 include correcting a significant notational typo in the statement of Theorem 1.3, rearranging some material in Section 3, simplifying the definitions of types of vertices (now Definition 3.7) and changing later language accordingly, revising proofs of Corollary 3.10 and Lemma 6.3, slightly revising introduction