Split degenerate superelliptic curves and $\ell$-adic images of inertia
Jeffrey Yelton
TL;DR
This work analyzes the action of inertia on the $oldsymbol{ extit{l}}$-adic Tate module of the Jacobian $J$ of a split degenerate superelliptic curve $C: y^p=f(x)$ over a discretely valued field. Using Mumford’s non-archimedean uniformization, it expresses the inertia action as a product of transvections associated to a carefully constructed $oldsymbol{ extit{l}}$-adic submodule and a canonical principal polarization, with exponents determined solely by the cluster data of the roots of $f$. A detailed framework is developed: (i) lift to Berkovich space, (ii) relate clusters to monodromy via the Bruhat–Tits tree, (iii) derive explicit valuations of a period matrix, and (iv) connect these valuations to a monodromy pairing that matches Grothendieck’s and the canonical polarization. The main theorem yields an explicit formula for $ ho_oldsymbol{ extit{l}}(oldsymbol{ extsigma})$ as a product of commuting transvections $t_{oldsymbol{w}_{oldsymbol{ rak s}}}^{m_{oldsymbol{ rak s}}}$ over $oldsymbol{ rak s} olinebreak i oldsymbol{ rak{C}}_0$, with explicit $m_{oldsymbol{ rak s}}$ given by depths of clusters, offering a generalization of previous inertia-transvection descriptions to general $p$ and arbitrary residue characteristic. In the special $oldsymbol{ extell}=p$ case, the paper also identifies the $oldsymbol{1-oldsymbol{ extzeta}_p}$-torsion subgroup generators via divisors $(oldsymbol{ extalpha}_i,0)-(oldsymbol{ extbeta}_i,0)$ and their Abel–Jacobi images, providing concrete torsion-structure information. Overall, the results give a precise, cluster-driven description of inertia action on $oldsymbol{ extl}$-adic Jacobian data for split degenerate superelliptic curves, with potential implications for global Galois representations and reduction-type questions.
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 $C$.