Searcharxiv⌕ Search

arXiv subjects

Jeffrey Yelton

Publications and source records attributed to Jeffrey Yelton.

16 recordsLinked to original sources
arXiv2026

Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case

Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$ which we call the \emph{relatively stable model} $\Yrst$ of $Y$, and we discuss its properties, focusing on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve viewed as a degree-$2$ cover of the projective line $X := \proj_K^1$. Over residue characteristic different from $2$, it follows from known results that the toric rank (i.e.\ the number of loops in the graph of components) of the special fiber of $\Yrst$ can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial $f$. We instead consider the ``wild" case of residue characteristic $2$ and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of $f$ gives rise to a loop in the graph of components of the special fiber of $\Yrst$ if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the $2$-rank of the special fiber of $\Yrst$.

math.AG↗
arXiv2025

A cluster criterion for potential degeneracy of superelliptic curves

Let $K$ be a field with a discrete valuation; let $p$ be a prime; and let $C$ be the curve defined by an equation of the form $y^p = f(x)$. We show that the curve $C$ has a model over an algebraic extension of $K$ whose special fiber consists of genus-$0$ components and has at worst nodal singularities if and only if the cluster data of the roots of $f$ satisfies a certain criterion, and when these hold, we show explicitly how to build the minimal regular model of $C$. We develop an interpretation of cluster data in terms of a convex hull in the Berkovich projective line and express the above results directly in terms of this convex hull.

math.AG↗
arXiv2025

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

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 $C$.

math.NT↗
arXiv2024

Branch points of split degenerate superelliptic curves I: construction of Schottky groups

Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $Γ\lhd Γ_0 < \mathrm{PGL}_2(K)$ is a Schottky group normally contained in a larger group which is generated by order-$p$ elements each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a certain subset of the projective line $\mathbb{P}_K^1$ by the action of $Γ$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of fixed points is mapped modulo $Γ$ to the set of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. We produce an algorithm for determining whether an input even-cardinality subset $S \subset K \cup \{\infty\}$ consists of fixed points of generators of such a group $Γ_0$ and which, in the case of a positive answer, modifies $S$ into a subset $S^{\mathrm{min}} \subset K \cup \{\infty\}$ with particularly nice properties. Our results do not involve any restrictions on the prime $p$ or on the residue characteristic of $K$ and allow these to be the same.

math.NT↗
arXiv2024

Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put

Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $Γ\lhd Γ_0 < \mathrm{PGL}_2(K)$ is a Schottky group normally contained in a larger group which is generated by order-$p$ elements each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a certain subset of the projective line $\mathbb{P}_K^1$ by the action of $Γ$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of fixed points is mapped bijectively modulo $Γ$ to the set $\mathcal{B}$ of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. A conjecture of Gerritzen and van der Put, in the case that $C$ is hyperelliptic and $K$ has residue characteristic $\neq 2$, compares the cluster data of $S$ with that of $\mathcal{B}$. We show that this conjecture requires a slight modification in order to hold and then prove a much stronger version of the modified conjecture that holds for any $p$ and any residue characteristic.

math.NT↗
arXiv2023

Clusters and semistable models of hyperelliptic curves in the wild case

Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$, which we call the relatively stable model $\mathcal{Y}^{\mathrm{rst}}$ of $Y$, and we discuss its properties. We focus on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve, viewed as a degree-$2$ cover of the projective line $X := \mathbb{P}_K^1$, and demonstrate a practical way to compute the relatively stable model. In the case of residue characteristic $p \neq 2$, the components of the special fiber $(\mathcal{Y}^{\mathrm{rst}})_s$ correspond precisely to the non-singleton clusters of roots of the defining polynomial $f$, i.e. the subsets of roots of $f$ which are closer to each other than to the other roots of $f$ with respect to the induced discrete valuation on the splitting field; this relationship, however, is far less straightforward in the $p=2$ case, which is our main focus (the techniques we introduce nevertheless also allow us to recover the simpler, already-known results in the $p\neq 2$ case). We show that, when $p = 2$, for each cluster containing an even number of roots of $f$, there are $0$, $1$, or $2$ components of $(\mathcal{Y}^{\mathrm{rst}})_s$ corresponding to it, and we determine a direct method of finding and describing them. We also define a polynomial $F(T) \in K[T]$ whose roots allow us to find the components of $(\mathcal{Y}^{\mathrm{rst}})_s$ which are not connected to even-cardinality clusters.

math.NT↗
arXiv2021

Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobians

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - λ)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of discriminant $Δ$ and some element $λ\in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(λ))$ but not $(2Δ)$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(λ))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x - λ)(x - λ')$ for distinct elements $λ, λ' \in K$. We then show that the hypothesis in the former statement holds for almost all $λ\in \mathcal{O}_{K}$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

math.NT↗
arXiv2021

Lifting images of standard representations of symmetric groups

We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 2}$ under the standard representation. We show that when $g \geq 2$, the only closed subgroup $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 2}$ is its full inverse image in $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$, while all subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 1}$ are open and contain the level-$8$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$. As an immediate application, we are able to strengthen a result of Zarhin on $2$-adic Galois representations associated to hyperelliptic curves. We also prove an elementary corollary concerning even-degree polynomials with full Galois group.

math.NT↗
arXiv2021

Divisibility of torsion subgroups of abelian surfaces over number fields

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A(\mathbb{F}_p) \equiv 0 \pmod{\ell^2}$ for a set of primes $\mathfrak{p} \subset \mathcal{O}_K$ of density 1. When $\ell=2$ Serre has shown that there does not necessarily exist a $K$-isogenous $A'$ such that $\#A'(K)_{\mathrm{tors}} \equiv 0 \pmod{4}$. We extend those results to all odd $\ell$ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod-$\ell^2$ representation.

math.NT↗
arXiv2020

Semistable models of elliptic curves over residue characteristic 2

Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - λ)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only on the valuation of $λ$. We provide several examples along with an easy corollary concerning $2$-torsion.

math.NT↗
arXiv2020

Prime-to-$p$ étale fundamental groups of punctured projective lines over strictly Henselian fields

Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p \geq 0$ with algebraic closure $\bar{K}$, and let $α_{1}, ..., α_{d} \in \mathbb{P}_{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ étale fundamental group of $\mathbb{P}_K^1\smallsetminus\{α_1,...,α_d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a_{1}, ..., a_{d}$ be distinct power series in $\mathbb{C}[[x]]$ with the same intersection behavior as the $α_i$'s, converging on an open disk centered at $0$, and choose a point $z_{0} \neq 0$ lying in this open disk. We compare the natural action of $\mathrm{Gal}(K)$ on the prime-to-$p$ étale fundamental group of $\mathbb{P}_{\bar{K}} \smallsetminus \{α_{1}, ..., α_{d}\}$ to the topological action of looping $z_0$ around the origin on the fundamental group of $\mathbb{P}_{\mathbb{C}}^1 \smallsetminus \{a_1(z_0),...,a_d(z_0)\}$. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-$p$ $G$-Galois cover of $\mathbb{P}_{\bar K}^1 \smallsetminus \{α_1,...,α_d\}$ satisfies that its field of moduli (as a $G$-Galois cover) has degree over $K$ dividing the exponent of $G / Z(G)$.

math.AG↗
arXiv2019

An abelian subfield of the dyadic division field of a hyperelliptic Jacobian

Given a field $k$ of characteristic different from $2$ and an integer $d \geq 3$, let $J$ be the Jacobian of the "generic" hyperelliptic curve given by $y^2 = \prod_{i = 1}^d (x - α_i)$, where the $α_i$'s are transcendental and independent over $k$; it is defined over the transcendental extension $K / k$ generated by the symmetric functions of the $α_i$'s. We investigate certain subfields of the field $K_{\infty}$ obtained by adjoining all points of $2$-power order of $J(\bar{K})$. In particular, we explicitly describe the maximal abelian subextension of $K_{\infty} / K(J[2])$ and show that it is contained in $K(J[8])$ (resp. $K(J[16])$) if $g \geq 2$ (resp. if $g = 1$). On the way we obtain an explicit description of the abelian subextension $K(J[4])$, and we describe the action of a particular automorphism in $\mathrm{Gal}(K_{\infty} / K)$ on these subfields.

math.NT↗
arXiv2017

A note on 8-division fields of elliptic curves

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form $y^{2} = f(x)$ with degree $4$. We obtain generators over $K$ of the $8$-division field $K(E[8])$ of $E$ given as formulas in terms of the roots of the polynomial $f$, and we explicitly describe the action of a particular automorphism in $\mathrm{Gal}(K(E[8]) / K)$.

math.NT↗
arXiv2014

Dyadic torsion of 2-dimensional hyperelliptic Jacobians

Let $k$ be a field of characteristic $0$, and let $α_{1}$, $α_{2}$, ..., $α_{5}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions of the $α_{i}$'s. Let $J$ be the Jacobian of the hyperelliptic curve defined over $K$ which is given by the equation $y^{2} = \prod_{i = 1}^{5} (x - α_{i})$. We define a tower of field extensions $K = K_{0}' \subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the generators of each $K_{n}'$ over $K_{n - 1}'$, and let $K_{\infty}' = \bigcup_{n = 0}^{\infty} K_{n}'$. We show that $K_{\infty}'(μ_{2})$ is the subextension of the field $K(J[2^{\infty}]) := \bigcup_{n = 0}^{\infty} K(E[2^{n}])$ corresponding to a central order-$2$ Galois subgroup of $\mathrm{Gal}(K(J[2^{\infty}]) / K(μ_{2}))$, and a generator of $K(J[2^{\infty}])$ over $K_{\infty}'(μ_{2})$ is given.

math.NT↗
arXiv2014

Images of 2-adic representations associated to hyperelliptic Jacobians

Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(α_{1}, α_{2}, ... , α_{2g + 1})$, where the $α_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - α_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $Γ(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generators for the algebraic extension $K(J[4]) / K$ generated by coordinates of the $4$-torsion points of $J$.

math.NT↗