---
abstract: '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 $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}$ of discriminant $\Delta$ and some element $\lambda \in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, 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 $(f(\lambda))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)(x - \lambda'')$ for distinct elements $\lambda, \lambda'' \in K$. We then show that the hypothesis in the former statement holds for almost all $\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.'
author:
- Jeffrey Yelton
bibliography:
- 'bibfile.bib'
title: 'Boundedness results for $2$-adic Galois images associated to hyperelliptic Jacobians'
---

Introduction {#S1}
============

Let $K$ be a number field with absolute Galois group $G_{K}$, and let $C$ be a hyperelliptic curve defined over $K$; i.e. $C$ is a smooth projective curve defined by an equation of the form $y^{2} = f(x)$ for some squarefree polynomial $f$ of degree $d \geq 3$. (Note that in the case of $d = 3$, $C$ is an elliptic curve.) It is well known that the genus of $C$ is given by $g = \lfloor (d + 1) / 2 \rfloor$. We denote the Jacobian variety of $C$ by $J$; it is an abelian variety of dimension $g$. For each prime $\ell$, we let $T_{\ell}(J)$ denote the $\ell$-adic Tate module of $J$, which is a free ${\mathbb{Z}}_{\ell}$-module of rank $2g$. We write $\rho_{\ell} : G_{K} \to {\mathrm{Aut}}(T_{\ell}(J))$ for the natural $\ell$-adic Galois action on this Tate module. The Tate module $T_{\ell}(J)$ is endowed with the Weil pairing defined with respect to the canonical principal polarization on $J$, which we write as $e_{\ell} : T_{\ell}(J) \times T_{\ell}(J) \to {\mathbb{Z}}_{\ell}$; it is a ${\mathbb{Z}}_{\ell}$-bilinear skew-symmetric pairing. Let ${\mathrm{Sp}}(T_{\ell}(J))$ denote the group of symplectic automorphisms of $T_{\ell}(J)$ with respect to the pairing $e_{\ell}$, and let $${\mathrm{GSp}}(T_{\ell}(J)) := \{\sigma \in \mathrm{Aut}_{{\mathbb{Z}}_{\ell}}(T_{\ell}(J))\ |\ e_{\ell}(P^{\sigma}, Q^{\sigma}) = e_{\ell}(P, Q)^{\chi_{\ell}(\sigma)}\ \forall P, Q \in T_{2}(J)\}$$ denote the group of symplectic similitudes, where $\displaystyle \chi_{\ell} : G_{K} \to {\mathbb{Z}}_{\ell}^{\times}$ is the $\ell$-adic cyclotomic character.

It is well known that the image $G_{\ell}$ of $\rho_{\ell}$ is always a closed subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ and that in fact there is some hyperelliptic Jacobian $J$ of a given dimension $g$ such that the inclusion $G_{\ell} \subseteq {\mathrm{GSp}}(T_{\ell}(J))$ has finite index (or equivalently, that $G_{\ell}$ is an open subgroup of the $\ell$-adic Lie group ${\mathrm{GSp}}(T_{\ell}(J))$); see for instance [@yelton2015images Theorem 1.1]. Note that the subgroup $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J)) \subset G_{\ell}$ coincides with the image of the Galois subgroup which fixes the extension $K(\mu_{\ell}) / K$ obtained by adjoining all $\ell$-power roots of unity to $K$. Since $K$ is a number field, the extension $K(\mu_{\ell}) / K$ is infinite; it follows that $G_{\ell} \not\subset {\mathrm{Sp}}(T_{\ell}(J))$ and that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ if and only if $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J))$ has finite index in ${\mathrm{Sp}}(T_{\ell}(J))$.

There have been many results stating that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ under various hypotheses for the polynomial defining the hyperelliptic curve. For instance, Y. Zarhin has proven this for large enough genus in the case of hyperelliptic curves defined by equations of the form $y^{2} = f(x)$ or $y^{2} = f(x)(x - \lambda)$ with $\lambda \in K$, where the Galois group of $f$ is the full symmetric or alternating group ([@zarhin2002very Theorem 2.5] and [@zarhin2010families Theorem 8.3]; see also [@zarhin2013two Theorem 1.3] for a variant of this where the curve is defined using two parameters). A. Cadoret and A. Tamagawa have also proven ([@cadoret2012uniform Theorems 1.1 and 5.1]) that for any family of hyperelliptic Jacobians over a smooth, geometrically connected, separated curve over $K$, this openness condition will be satisfied for the $\ell$-adic Galois action associated to all but finitely many fibers, and that in fact the indices of the $\ell$-adic Galois images corresponding to these fibers are uniformly bounded. However, there have been very few results which give explicit bounds for the index of $G_{\ell}$ in ${\mathrm{GSp}}(T_{\ell}(J))$ in such cases.

Our aim in this paper is to give some similar results on the openness of the $2$-adic Galois images in the group of symplectic similitudes associated to Jacobians of hyperelliptic curves whose defining polynomials satisfy certain hypotheses, and to provide formulas giving explicit bounds for the indices of the $2$-adic Galois images in these cases. (Unfortunately, our method currently cannot tell us anything about the $\ell$-adic Galois images for odd primes $\ell$. However, we are hopeful that it can be strengthened to show the openness of the $\ell$-adic Galois images as well under the same or similar hypotheses as is implied by the Mumford-Tate conjecture, and to show that the $\ell$-adic Galois images contain the full symplectic group for almost all $\ell$.)

We state our main results below. In these statements as well as in the rest of the paper, we use the following notation. For any integer $N \geq 1$, we denote the level-$N$ congruence subgroup of ${\mathrm{Sp}}(T_{2}(J))$ by $\Gamma(N) := \{\sigma \in {\mathrm{Sp}}(T_{2}(J)) \ | \ \sigma \equiv 1 \ (\mathrm{mod} \ N)\}$. We denote the ring of integers of a number field $K$ by $\mathcal{O}_{K}$. Finally, we write $v_{2} : {\mathbb{Q}}^{\times} \to {\mathbb{Z}}$ for the (normalized) $2$-adic valuation on ${\mathbb{Q}}$.

\[thm main1\]

Let $K$ be a number field, and let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 2$ with discriminant $\Delta$. Let $J$ be the Jacobian of the hyperelliptic curve with defining equation $y^{2} = f(x)(x - \lambda)$ for some $\lambda \in \mathcal{O}_{K}$, and define the $2$-adic Galois image $G_{2}$ as above. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$. If in addition $d = 3$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + 1})$.

\[thm main2\]

Assume the same notation as in the statement of Theorem \[thm main1\], except that the defining equation of the hyperelliptic curve is $y^{2} = f(x)(x - \lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in \mathcal{O}_{K}$. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(\lambda - \lambda')$ or $(2\Delta)$ and a prime $\mathfrak{p}'$ which divides $(\lambda - \lambda')$ but not $(f(\lambda))$ or $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$ if $v_{2}(m') \leq v_{2}(m)$ or $d = 2g$ and $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + v_{2}(m') + 2})$ otherwise, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$ and $m' \geq 1$ is the greatest integer such that $\mathfrak{p}'^{m'} \mid (\lambda - \lambda')$. If in addition $d = 2$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{\max \{v_{2}(m), v_{2}(m')\} + 1})$.

\[rmk almost all fibers\]

For a fixed irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$ with discriminant $\Delta$, it is not hard to show using Faltings’ Theorem that the hypotheses in Theorems \[thm main1\] is satisfied for almost all $\lambda \in \mathcal{O}_{K}$, as we will see in §\[S5\] (Remark \[rmk not PID\] below). It follows immediately that there are also infinitely many choices of $(\lambda, \lambda') \in K \times K$ satisfying the hypotheses in Theorem \[thm main2\] for this polynomial $f$.

\[rmk end\]

It is known that there is a finite algebraic extension $K'$ of $K$ over which every endomorphism of an abelian variety over a field $K$ is defined (see [@silverberg1992fields Theorem 2.4]), so that each endomorphism commutes with the action of ${\mathrm{Gal}}(\bar{K} / K')$ on torsion points. Note that the only endomorphisms in ${\mathrm{End}}(T_{\ell}(J))$ which commute with everything in an open subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ are scalars. It therefore follows from the above theorems that the endomorphism ring of any hyperelliptic Jacobian $J$ satisfying the hypotheses of Theorem \[thm main1\] or of Theorem \[thm main2\] coincides with ${\mathbb{Z}}$ and that such a $J$ is absolutely simple.

The key ingredient used in proving the above theorems is a method of describing Galois actions on $\ell$-adic Tate modules of hyperelliptic Jacobians defined over strictly Henselian local fields of residue characteristic $p \neq 2, \ell$ by looking at the valuations of the differences between the roots of the defining polynomial, which is derived from results shown in joint work with H. Hasson ([@hasson2017prime]). This way of looking at $\ell$-adic Galois actions associated to hyperelliptic Jacobians over local fields is very similar to the “method of clusters" used by S. Anni and V. Dokchitser in [@anni2017constructing]. Our approach seems quite powerful and should lead to many similar boundedness results in a number of situations where one can compute valuations of the differences between the roots of the defining polynomial with respect to various primes of the ground field. Unfortunately, in practice, these valuations (or even the roots themselves) may be difficult to calculate, and so our main focus here is on obtaining results such as the ones stated above where the hypotheses are very easy to verify.

The rest of this paper is organized as follows. In §\[S2\], we use the main results of [@hasson2017prime], which show that over a strictly Henselian local field of characteristic $p \neq 2$, for primes $\ell \neq p$, the $\ell$-adic Galois action factors through the tame quotient of the absolute Galois group and can be described in terms of Dehn twists with respect to certain loops on a complex hyperelliptic curve. In particular cases such as when exactly two roots of the defining polynomial coalesce in the reduction over the residue field, we will show (Proposition \[prop Galois action local\]) that such a Dehn twist induces a transvection in the symplectic group. We will later put local data together to show that over a number field $K$, the $2$-adic Galois image contains certain powers of several sufficiently “independent" transvections. In §\[S3\], we will demonstrate using elementary matrix algebra that the group generated by these powers of transvections contains a certain congruence subgroup. In §\[S4\], we will use what we have shown in §\[S2\] and §\[S3\] to prove Theorems \[thm main1\] and \[thm main2\] as well as to prove an auxiliary result that applies to a more general situation (Theorem \[thm several primes\]). Finally, in §\[S5\], we will assume that $K$ has class number $1$ and show using Theorem \[thm main1\] that for a given $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 3$, the $2$-adic Galois image associated to the hyperelliptic curves defined by $y^{2} = f(x)(x - \lambda)$ for all but finitely many $\lambda \in \mathcal{O}_{K}$ contains a principal congruence subgroup which depends only on $d$ (Theorem \[thm uniform bounds\]). In fact, for $d \geq 4$ even, in Theorem \[thm uniform bounds\](c) we will provide a uniform bound for indices of the $2$-adic Galois images associated to almost all fibers of such a one-parameter family over the $K$-line, as is guaranteed by [@cadoret2012uniform Theorems 1.1 and 5.1].

The author is grateful to a MathOverflow user whose comment on question 264281 helped to inspire the arguments for the results in §\[S5\].

Hyperelliptic Jacobians over local fields and tame Galois actions {#S2}
=================================================================

We retain all notation introduced in the previous section. In this section, we write $\widehat{{\mathbb{Z}}}$ for the profinite completion of ${\mathbb{Z}}$ and use the symbol $\widehat{\pi}_{1}$ to denote the profinite completion of the fundamental group of a topological space. For any profinite group $G$, we write $G^{(p')}$ for its maximal prime-to-$p$ quotient. For any profinite group $G$, let $G^{(p')}$ denote its prime-to-$p$ quotient. Note that since $G^{(p')}$ is a characteristic quotient of $G$, any action on $G$ induces an action on $G^{(p')}$.

Now we choose a prime $\mathfrak{p}$ of $K$ of residue characteristic $p \neq 2$. Fix a strict Henselization of the localization of $K$ at the prime $\mathfrak{p}$ and denote it by $\mathcal{R}_{\mathfrak{p}}$ and its fraction field by $\mathcal{K}_{\mathfrak{p}}$; this comes with an embedding $\mathcal{K}_{\mathfrak{p}} \hookrightarrow \bar{K}$. Let $\pi \in K$ be a uniformizer of the discrete valuation ring $\mathcal{R}_{\mathfrak{p}}$. We fix a compatible system of $N$th roots of unity $\zeta_{N} \in \bar{K}$ for $N = 1, 2, 3, ...$; that is, we require that $\zeta_{N'N}^{N'} = \zeta_{N}$ for any integers $N, N' \geq 1$. Note that since $R$ is strictly Henselian, $\zeta_{N} \in R \subset K$ for any $N$ not divisible by $p$. Let $G_{K, \mathfrak{p}}$ denote the absolute Galois group of $\mathcal{K}_{\mathfrak{p}}$, and let $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ denote its tame quotient. It follows from a special case of Abhyankar’s Lemma that the maximal tamely ramified extension $\mathcal{K}_{\mathfrak{p}}^{{\mathrm{tame}}}$ is given by $\mathcal{K}_{\mathfrak{p}}(\{\pi^{1/N}\}_{(N, p) = 1})$, where $\pi^{1/N}$ denotes an $N$th root of $\pi$, and that $G_{K, \mathfrak{p}}^{{\mathrm{tame}}} \cong \widehat{{\mathbb{Z}}}^{(p')}$ is topologically generated by the automorphism which acts on $K^{{\mathrm{tame}}}$ by fixing $K$ and sending each $\pi^{1/N}$ to $\zeta_{N}\pi^{1/N}$.

We fix, once and for all, an embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ where $\zeta_{N}$ is sent to $e^{2 \pi \sqrt{-1} / N}$ for $N \geq 1$, so that we have an inclusion $\mathcal{K}_{\mathfrak{p}} \subset {\mathbb{C}}$. Let $d \geq 2$ be an integer and choose distinct integral elements $\alpha_{1}, ... , \alpha_{d} \in K$. Choose polynomials $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{d} \in {\mathbb{C}}[x]$ satisfying $\tilde{\alpha}_{i}(\pi) = \alpha_{i}$ for $1 \leq i \leq d$ and such that the $x$-adic valuation of $\tilde{\alpha}_{i}$ and $\tilde{\alpha}_{j}$ and the $\pi$-adic valuation of $\alpha_{i}$ and $\alpha_{j}$ are equal (such polynomials exist as is shown in the discussion in [@hasson2017prime §3.3]). Let $\varepsilon > 0$ be a real number small enough that $\tilde{\alpha}_{i}(z) \neq \tilde{\alpha}_{j}(z)$ for all $i \neq j$ and for all $z \in B_{\varepsilon}^{*} := \{z \in {\mathbb{C}}\ | \ |z| < \varepsilon\} \smallsetminus \{0\}$. We define a family $\mathcal{F} \to B_{\varepsilon}^{*}$ of $d$-times-punctured Riemann spheres by letting $$\mathcal{F} = {\mathbb{P}}_{{\mathbb{C}}}^{1} \times B_{\varepsilon}^{*} \smallsetminus \bigcup_{i = 1}^{d} \{(\tilde{\alpha}_{i}(z), z) \ | \ z \in B_{\varepsilon}^{*}\}.$$ Choose a basepoint $z_{0} \in B_{\varepsilon}^{*}$. The fundamental group $\pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts by monodromy on the fundamental group $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ of the fiber over $z_{0}$ with basepoint $\infty$. We write $\rho_{{\mathrm{top}}} : \pi_{1}(B_{\varepsilon}^{*}, z_{0}) \to {\mathrm{Aut}}(\pi_{1}(\mathcal{F}_{z_{0}}, \infty))$ for this action. The action $\rho_{{\mathrm{top}}}$ extends uniquely to a continuous action of the profinite completion $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ on the profinite completion $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ (see the discussion in [@hasson2017prime §1.1]), which we also denote by $\rho_{{\mathrm{top}}}$. We note that $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ is isomorphic to $\widehat{{\mathbb{Z}}}$ and is topologically generated by the element $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ represented by the loop given by $t \mapsto e^{2 \pi \sqrt{-1} t}z_{0}$ for $t \in [0, 1]$.

Meanwhile, the absolute Galois group $G_{K, \mathfrak{p}}$ acts naturally on the étale fundamental group $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)$ via the $K_{\mathfrak{p}}$-point lying under the geometric point $\infty : {\mathrm{Spec}}({\mathbb{C}}) \to {\mathbb{P}}_{K_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$. After identifying $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ with $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ via Riemann’s Existence Theorem and the inclusion of algebraically closed fields $\bar{\mathcal{K}}_{\mathfrak{p}} \subset {\mathbb{C}}$, we write $\rho_{{\mathrm{alg}}} : G_{K, \mathfrak{p}} \to {\mathrm{Aut}}(\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')})$ for this action. We denote the actions on prime-to-$p$ quotients of étale fundamental groups induced by $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ by $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ respectively.

For the statement of Theorem \[thm comparison punctured projective line\](a) below, we require the terminology of Dehn twists. Let $\gamma : [0, 1] \to M$ be a simple loop on any complex manifold $M$; we will often identify $\gamma$ with its image in $M$. We define the *Dehn twist* on $M$ with respect to the loop $\gamma$. It is an element of the mapping class group of $M$ represented by a self-homeomorphism of $M$ which can be visualized in terms of a small tubular neighborhood of $\gamma \subset M$, in the following way: the Dehn twist keeps the outer edge of the tubular neighborhood fixed while twisting the inner edge one full rotation counterclockwise and acts as the identity everywhere else on $M$. Since this Dehn twist depends only on the homology class $[\gamma] \in H_{1}(M, {\mathbb{Z}})$ of any loop $\gamma$, we will denote it by $D_{[\gamma]}$. (See [@farb2011primer Chapter 3] for more details.)

The following theorem is a compilation of all the necessary results describing and comparing $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ that are proven in [@hasson2017prime] (Theorems 1.2 and 2.3 and Remark 3.10 of that paper).

\[thm comparison punctured projective line\]

In the above situation, we have the following.

a\) Let $\mathcal{I}$ be the set of all pairs $(I, n)$ where $I \subseteq \{1, ... , d\}$ is a subset and $n \geq 1$ is an integer such that $x^{n} \mid \tilde{\alpha}_{i} - \tilde{\alpha}_{j} \in {\mathbb{C}}[x]$ for all $i, j \in I$ and such that $I$ is maximal among intervals with this property. If $\varepsilon$ is small enough, there exist pairwise nonintersecting loops $\gamma_{I, n} : [0, 1] \to \mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ for each $(I, n) \in \mathcal{I}$ such that $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ in the same way that the product $\prod_{(I, d) \in \mathcal{I}} D_{[\gamma_{I, n}]}$ of Dehn twists on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ does. These loops $\gamma_{I, n}$ each have the property of separating the subset $\{\tilde{\alpha}_{i}(z_{0})\}_{i \in I}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$, and two such loops $\gamma_{I, n}$ and $\gamma_{I', n'}$ are homologous if and only if $I = I'$.

b\) The actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ factor through $\pi_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')}$ and $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ respectively.

c\) We have isomorphisms $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')} \stackrel{\sim}{\to} G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ inducing an isomorphism of the actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$. Moreover, we can choose the isomorphism $\phi$ so that it takes any element represented by a loop on $\mathcal{F}_{z_{0}}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\tilde{\alpha}_{i}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d} \cup \{\infty\}$.

In other words, the prime-to-$p$ monodromy action $\rho_{{\mathrm{top}}}^{(p')}$ can be described in terms of Dehn twists with respect to loops surrounding certain subsets of the removed points which coalesce at a certain rate as one approaches the center of $B_{\varepsilon}^{*}$; moreover, this is isomorphic to the algebraic action $\rho_{{\mathrm{alg}}}^{(p')}$ via an isomorphism of prime-to-$p$ étale fundamental groups which takes the image of a loop wrapping around a given $a_{i}(z_{0})$ to the image of a loop wrapping around $\alpha_{i}$.

We now want to relate this to the action of $G_{K, \mathfrak{p}}$ on the prime-to-$p$ étale fundamental group of a smooth hyperelliptic curve over $K_{\mathfrak{p}}$. Let $\mathfrak{C}$ be a smooth, projective hyperelliptic curve over ${\mathbb{C}}$ of degree $d$ and genus $g$ defined by an equation of the form $y^{2} = \prod_{i = 1}^{d} (x - z_{i})$ for distinct roots $z_{i} \in \mathcal{K}_{\mathfrak{p}}$; if $d$ is odd (resp. even), then $d = 2g + 1$ (resp. $d = 2g + 2$). The hyperelliptic curve $\mathfrak{C}$ comes with a surjective degree-$2$ morphism $\mathfrak{C} \to {\mathbb{P}}_{{\mathbb{C}}}^{1}$ defined by projecting onto the $x$-coordinate. It is well known that this projection ramifies at $\infty$ if and only if $d = 2g + 1$; in this case, we write $z_{2g + 2} = \infty$. Then the projection is ramified at exactly the $(2g + 2)$-element set of $z_{i}$’s. Write $\mathfrak{B} \subset \mathfrak{C}({\mathbb{C}})$ for the set of inverse images of these ramification points in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ ($\mathfrak{B}$ is the set of *branch points* of $\mathfrak{C}$). Clearly the restriction to $\mathfrak{C} \smallsetminus \mathfrak{B}$ of the above projection map yields a finite degree-$2$ étale morphism $\mathfrak{C} \smallsetminus \mathfrak{B} \to {\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$. Choose a basepoint $P$ of the topological space ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{d}\}$ and a basepoint $Q$ of the topological space $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ such that $Q$ lies in the inverse image of $P$. Then after making identifications via Riemann’s Existence Theorem, we get an inclusion and surjections $$\label{eq maps of fundamental groups} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P) \rhd \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}), Q) \twoheadrightarrow H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$$ induced by the maps ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\} \leftarrow \mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B} \hookrightarrow \mathfrak{C}({\mathbb{C}})$ and by identifying the first singular homology group of $\mathfrak{C}({\mathbb{C}})$ with the abelianization of its fundamental group. The inclusion in (\[eq maps of fundamental groups\]) is an inclusion of a characteristic subgroup, so any automorphism of $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ induces an automorphism of $\widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, P)$.

We write $\mathfrak{J}$ for the Jacobian of $\mathfrak{C}$. There is a well-known identification of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ with $H_{1}(\mathfrak{J}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ and in turn with $T_{\ell}(\mathfrak{J})$ for any prime $\ell$. Moreover, the intersection pairing on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ defined above carries over to the canonical Riemann form on the complex abelian variety $\mathfrak{J}({\mathbb{C}})$ and in turn to the Weil pairing $e_{\ell}$ on $T_{\ell}(\mathfrak{J})$ (see the results in [@lang2012introduction §IV.4 and §VIII.1] and in [@mumford1974abelian §24]). Given an element $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, we also write $c$ for the element $c \otimes 1 \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell} = T_{\ell}(\mathfrak{J})$. It is not difficult to show that the action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(\mathfrak{J})$ induced by $\rho_{{\mathrm{alg}}}^{(p')}$ via these identifications is the natural $\ell$-adic Galois action $\rho_{\ell}$: see, for instance, step 5 of the proof of [@yelton2015images Proposition 2.2].

\[dfn symplectic basis\]

Given any complex hyperelliptic curve $\mathfrak{C}$ as above, we define the following objects.

a\) For any $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, the *transvection* with respect to $c$, denoted $T_{c} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$, is the automorphism given by $v \mapsto v + \langle v, c \rangle c$. As above, we may identify $c$ with its image in $T_{\ell}(\mathfrak{J})$, and then $T_{c}$ is identified with the automorphism of $T_{\ell}(\mathfrak{J})$ given by $v \mapsto e_{\ell}(v, c) c$.

b\) A *sympectic basis* of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is an ordered basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ satisfying the following properties:

   (i) each $a_{i}'$ (resp. each $b_{i}'$) is represented by a loop on $\mathfrak{C}({\mathbb{C}})$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates $\{z_{2i - 1}, z_{2i}\}$ (resp. $\{z_{2i}, ... , z_{2g + 1}\}$) from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$; and

   (ii) the (skew-symmetric) intersection pairing $\langle \cdot, \cdot \rangle$ on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is determined by $\langle a_{i}', b_{i}' \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}', a_{j}' \rangle = \langle b_{i}', b_{j}' \rangle = \langle a_{i}', b_{j}' \rangle = 0$ for $1 \leq i < j \leq g$. (See, for instance, the first figure in [@arnold1968remark], or [@farb2011primer Figure 6.1].)

We note that a transvection in ${\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$ respects the intersection pairing $\langle \cdot, \cdot \rangle$; similarly, a transvection in ${\mathrm{Aut}}(T_{\ell}(\mathfrak{J}))$ respects the Weil pairing $e_{\ell}$ and thus lies in ${\mathrm{Sp}}(T_{\ell}(\mathfrak{J}))$.

From now on, given a complex hyperelliptic curve $\mathfrak{C}$, we fix a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. In order to prove Proposition \[prop Galois action local\] below, we require a couple of lemmas, the first of which is purely topological.

\[lemma lifts to Dehn twists\]

Assume all of the above notation.

a\) Let $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ be the image of a simple closed loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}, P\}$ which separates the set of $\alpha_{i}$’s into two even-cardinality subsets. Then the inverse image of $\gamma$ under the ramified degree-$2$ covering map $\mathfrak{C}({\mathbb{C}}) \to {\mathbb{P}}_{{\mathbb{C}}}$ consists of two connected components, each of which are simple closed loops whose homology classes $\pm c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ differ by sign. As a particular case, if $\gamma_{i} \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ is the image of a loop which separates $\{z_{i}, z_{2g + 1}\}$ from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$ for some $i$, then the homology classes of these simple closed loops on $\mathfrak{C}({\mathbb{C}})$ lying above $\gamma_{i}$ are given by $\pm c_{i}'$ for some element $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ which is equivalent modulo $2$ to $$\label {eq c_i} \begin{cases} a_{(i + 1)/2}' + ... + a_{g}' + b_{(i + 1)/2}' & i \ \mathrm{odd}\\ a_{i/2 + 1}' + ... + a_{g}' + b_{i/2 + 1}' & i \ \mathrm{even} \end{cases}$$

b\) With $\gamma$ and $c$ as above, the Dehn twist $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the inclusion and quotient maps in (\[eq maps of fundamental groups\]), which is given by $T_{c}^{2} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$.

The essential ideas of this argument are contained in the proof of [@mumford1984tata Lemma 8.12]. The maximal abelian exponent-$2$ quotient of $\pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ is isomorphic to $({\mathbb{Z}}/ 2{\mathbb{Z}})^{2g + 1}$ and is identified with the group of partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into two subsets (where the addition law is given by symmetric differences), by sending the homology class of any loop $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ to the subsets of $z_{i}$’s lying in each connected component of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \gamma$. We have a homomorphism of homology groups (composed with reduction modulo $2$) $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \to H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ coming from the inclusion of fundamental groups in (\[eq maps of fundamental groups\]). By [@arnold1968remark Lemma 1], this homomorphism factors through $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$. It is clear from this that we have an inclusion of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$ as the subgroup of $H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ which is generated by the partitions $\{z_{2i - 1}, z_{2i}\} \cup \{z_{1}, ... , z_{2i - 2}, z_{2i + 1}, ... , z_{2g + 2}\}$ and $\{z_{2i}, ... , z_{2g + 1}\} \cup \{z_{1}, ... , z_{2i - 1}, z_{2g + 2}\}$ for $1 \leq i \leq g$. It follows from an easy combinatorial argument that this subgroup consists of the partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into even-cardinality subsets. Thus, any loop $\gamma \in \pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ whose image modulo $2$ is such a partition lifts to a loop $c \in \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$; the only other choice of loop on $\mathfrak{C}({\mathbb{C}})$ whose image is $\gamma$ must be based at $\iota(Q)$ and equal to the composition of the path $\gamma$ with $\iota$, where $\iota : \mathfrak{C}({\mathbb{C}}) \to \mathfrak{C}({\mathbb{C}})$ is the only nontrivial deck transformation. Since $\iota$ acts on the homology group by sign change (see [@farb2011primer §7.4]), this loop must be $-c$. If the image of $\gamma$ modulo $2$ is the partition of $\{z_{i}, z_{2g + 1}\}$ and its complement, then the desired statement in (a) follows by the straightforward verification that this partition is equal to the symmetric sum of the partitions corresponding to the basis elements $a_{i}'$ and $b_{i}'$ appearing in the formulas given in (\[eq c\_i\]). Thus, (a) is proved.

Every self-homeomorphism of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$ fixing $P$ lifts uniquely to a self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ under which the image of a loop wrapping around a single point in $\mathfrak{B}$ is also a loop wrapping around a single point in $\mathfrak{B}$. Since the kernel of the quotient map $\pi_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$ is generated by squares of homotopy classes of such loops, it follows that $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the maps in (\[eq maps of fundamental groups\]). It is clear that the unique self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ lifting a representative of $D_{[\gamma]}$ must represent the composition of Dehn twists on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$ with respect to the two lifts of $\gamma$ on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$, which are $\pm c$. Therefore, the induced automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) = H_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \{Q\}, {\mathbb{Z}})$ is determined by the product of $D_{c}$ and $D_{-c}$. Since Dehn twists do not depend on the orientation of loops, this product is $D_{c}^{2}$. Now (b) follows from the well known fact (see [@farb2011primer Proposition 6.3]) that the Dehn twist $D_{c}$ acts on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ as the transvection $T_{c}$ since $\mathfrak{C}({\mathbb{C}})$ is a compact smooth manifold.

\[lemma branch point infty\]

Let $\alpha_{1}, ... , \alpha_{d}$ be distinct elements in $\mathcal{R}_{\mathfrak{p}}$. There exist elements $\alpha_{1}', ... , \alpha_{d + 1}' \in \mathcal{R}_{\mathfrak{p}}$ satisfying the following:

\(i) The elements $\alpha_{i}' - \alpha_{j}'$ and $\alpha_{i} - \alpha_{j}$ have equal valuation for $1 \leq i < j \leq d$, and $\alpha_{d + 1}' - \alpha_{i}' \in \mathcal{R}_{\mathfrak{p}}^{\times}$ for $1 \leq i \leq d$.

\(ii) Let $\alpha_{d + 1} = \infty \in {\mathbb{P}}_{\mathcal{K}_{\mathfrak{p}}}^{1}$. There is a $\mathcal{K}_{\mathfrak{p}}$-isomorphism $$\psi : {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\} \stackrel{\sim}{\to} {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$$ such that the induced isomorphism $\psi_{*} : \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}, \psi^{-1}(\infty)) \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}, \infty)$ yields an isomorphism of $\rho_{{\mathrm{alg}}}$ with the analogously defined representation $\rho_{{\mathrm{alg}}}'$. The isomorphism $\psi_{*}$ takes any element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d + 1}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}'\}$ from its complement in $\{\alpha_{j}'\}_{j = 1}^{d + 1}$.

Choose $\beta \in \mathcal{R}_{\mathfrak{p}}^{\times}$ satisfying $\beta \not\equiv \alpha_{j}$ (mod $\pi$) for $1 \leq j \leq d$ (this is always possible because the residue field $\mathcal{R}_{\mathfrak{p}} / (\pi)$ is infinite). Let $\alpha_{j}' = \alpha _{j}\beta / (\beta - \alpha_{j}) \in \mathcal{R}_{\mathfrak{p}}$ for $1 \leq j \leq d$ and $\alpha_{d + 1}' = \beta \in \mathcal{R}_{\mathfrak{p}}$, and let $\psi$ be the $\mathcal{K}_{\mathfrak{p}}$-morphism given by $x \mapsto x\beta / (\beta - x)$. Then property (i) follows from straightforward computation. Since $\psi$ is defined over $\mathcal{K}_{\mathfrak{p}}$, the isomorphism $\psi_{*}$ is equivariant with respect to the action of $G_{K, \mathfrak{p}} = {\mathrm{Gal}}(\bar{\mathcal{K}}_{\mathfrak{p}} / \mathcal{K}_{\mathfrak{p}})$. Moreover, after base change to ${\mathbb{C}}$, $\psi$ is a homeomorphism of punctured Riemann spheres, and the property given in (ii) follows.

We are finally ready to state and prove the main result of this section, which is essentially a more concrete version of a particular case of Grothendieck’s criterion for semistable reduction ([@grothendieck1972modeles Proposition 3.5(iv)]). For the statement of the below proposition, we fix a symplectic basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$; the image $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\} \subset T_{\ell}(J)$ forms a symplectic basis of $T_{\ell}(J)$ with respect to the Weil pairing.

\[prop Galois action local\]

Let $C$ be a hyperelliptic curve of genus $g$ over $\mathcal{K}_{\mathfrak{p}}$ given by an equation of the form $y^{2} = h(x)$ for some squarefree polynomial $h$ of degree $d = 2g + 1$ or $d = 2g + 2$ with distinct roots $\alpha_{1}, ... , \alpha_{d} \in \mathcal{R}_{\mathfrak{p}}$, and let $J$ be its Jacobian. Suppose that exactly $2$ of the roots, $\alpha_{i}$ and $\alpha_{j}$, are equivalent modulo $\pi$, and let $m \geq 1$ be the maximal integer such that $\pi^{m} \mid (\alpha_{i} - \alpha_{j})$. Then for any prime $\ell \neq 2, p$, the image of the natural action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(J)$ is topologically generated by the element $T_{c}^{2m} \in {\mathrm{Sp}}(T_{\ell}(J))$ for some $c \in T_{\ell}(J)$ determined by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ whose image separates $\{\alpha_{i}, \alpha_{j}\}$ from the rest of the roots. As a particular case, if $i \leq 2g$ and $j = 2g + 1$, then this $\ell$-adic Galois image is topologically generated by $T_{c_{i}}^{2m}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ equivalent modulo $2$ to $a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2}$ (resp. $a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1}$) if $i$ is odd (resp. even).

We first assume that $d = 2g + 2$. Let $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{2g + 2} \in {\mathbb{C}}[x]$ be the elements constructed from the $\alpha_{i}$’s as above, and define the family $\mathcal{F} \to B_{\varepsilon}^{*}$ as above. It is clear from the hypothesis on the roots $\alpha_{j}$ that the set $\mathcal{I}$ in the statement of Theorem \[thm comparison punctured projective line\] consists of only the elements $(\{i, 2g + 1\}, n)$ for $1 \leq n \leq m$. Theorem \[thm comparison punctured projective line\](a) then implies that a topological generator of $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ via the monodromy action $\rho_{{\mathrm{top}}}$ as $D_{[\gamma_{i}]}^{m}$, where $\gamma_{i}$ is a loop on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the subset $\{\tilde{\alpha}_{i}(z_{0}), \tilde{\alpha}_{2g + 1}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{2g + 2}$. Let $\mathfrak{C}$ be the complex hyperelliptic curve of degree $d = 2g + 2$ ramified over ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ at the points $\tilde{\alpha}_{1}(z_{0}), ... , \tilde{\alpha}_{d}(z_{0}) \in {\mathbb{C}}$, which has a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. By Lemma \[lemma lifts to Dehn twists\], the automorphism of $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ determined by $D_{[\gamma_{i}]}^{m}$ induces the automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}$ given by $T_{c_{i}}^{2m} : v \mapsto v + 2m \langle v, c_{i}' \rangle c_{i}'$, where $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is equivalent modulo $2$ to the formula in (\[eq c\_i\]). Now parts (b) and (c) of Theorem \[thm comparison punctured projective line\] say that there is an isomorphism $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ making the action $\rho_{{\mathrm{top}}}$ isomorphic to the Galois action $\rho_{{\mathrm{alg}}}$. Since $p \neq 2$ and $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ and $C({\mathbb{C}}) \smallsetminus \{(\alpha_{i}, 0)\}_{i = 1}^{2g + 2}$ are the only degree-$2$ covers of $\mathcal{F}_{z_{0}}$ and ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}\}$ respectively, we see that the isomorphism $\phi$ induces an isomorphism $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')} \stackrel{\sim}{\to} H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which we also denote by $\phi$. It is clear from the property of $\phi$ given in Theorem \[thm comparison punctured projective line\](c) and from our characterization of homology groups with coefficients in ${\mathbb{Z}}/ 2{\mathbb{Z}}$ in the proof of Lemma \[lemma lifts to Dehn twists\] that $\phi(a_{j}') \equiv a_{j}$ and $\phi(b_{j}') \equiv b_{j}$ (mod $2$) for $1 \leq j \leq g$.

In the case that $d = 2g + 1$, we get the same results by applying Lemma \[lemma branch point infty\], which allows us to replace $\alpha_{1}, ... , \alpha_{2g + 1}, \alpha_{2g + 2} := \infty$ with elements $\alpha_{1}', ... , \alpha_{2g + 2}' \in \mathcal{R}_{\mathfrak{p}}$ whose differences have the same valuations with respect to $\pi$.

Putting this all together, we see that the tame Galois action on $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by $\rho_{{\mathrm{alg}}}$ sends a generator of $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ to the automorphism of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ given by $v \mapsto v + 2m \langle v, c_{i} \rangle_{\phi} c_{i}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), \infty) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which is equivalent modulo $2$ to the formula given in the statement. Here $\langle \cdot, \cdot \rangle_{\phi}$ is the skew-symmetric pairing on $H_{1}(C({\mathbb{Z}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by the interection pairing on $H_{1}(\mathfrak{C}, {\mathbb{Z}})$ via $\phi$; note that $\langle \cdot, \cdot \rangle_{\phi}$ is normalized so that $\langle H_{1}(C({\mathbb{C}}), {\mathbb{Z}}), H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \rangle_{\phi} = {\mathbb{Z}}$. As above, we identify the maximal pro-$\ell$ quotient of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ with $T_{\ell}(J)$ and see that the natural $\ell$-adic Galois action $\rho_{\ell}$ factors through $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and takes a generator to the automorphism of $T_{\ell}(J)$ given by $v \mapsto v + 2m\langle v, c_{i} \rangle_{\phi} c_{i}$. But this automorphism must lie in ${\mathrm{Sp}}(T_{\ell}(J))$ by the Galois equivarience of the Weil pairing $e_{\ell}$ and the fact that $\mathcal{K}_{\mathfrak{p}}$ contains all $\ell$-power roots of unity. It is now an easy exercise to verify that this implies that $\langle v, c_{i} \rangle_{\phi} = \pm e_{\ell}(v, c_{i})$ for all $v \in T_{\ell}(J)$, and so the image of $\rho_{\ell}$ is generated by $T_{c_{i}}^{2m} : v \mapsto v + e_{\ell}(v, c_{i}) c_{i}$, as desired.

In order to prove Theorems \[thm main1\] and \[thm main2\], we will put some local data together to show using Proposition \[prop Galois action local\] that the $\ell$-adic Galois image contains $\{T_{c_{1}}^{2m}, ... , T_{c_{2g}}^{2m}\}$ for homology classes $c_{i}$ as above and a certain integer $m$. Therefore, it is of interest to investigate the subgroup that this set generates. Unfortunately, since the $c_{i}$’s are only known modulo $2$, not much can be deduced except if $\ell = 2$. In this case, the subgroup of ${\mathrm{Sp}}(T_{2}(J))$ generated by the above set can be determined, which is the goal of the next section.

Subgroups generated by powers of transvections {#S3}
==============================================

For this section, we let $M$ be a free ${\mathbb{Z}}$-module of rank $2g$, equipped with a nondegenerate skew-symmetric ${\mathbb{Z}}$-bilinear pairing $\langle \cdot, \cdot \rangle : M \times M \to {\mathbb{Z}}$. For any ring $A$, this pairing induces in an obvious way a nondegenerate skew-symmetric $A$-bilinear pairing on the free $A$-module $M \otimes A$ which we also denote by $\langle \cdot, \cdot \rangle$. We write ${\mathrm{Sp}}(M \otimes A)$ for the symplectic group of $A$-automorphisms of $M \otimes A$ which respect this pairing; for any integer $N \geq 1$, we write $\Gamma(N)$ for the level-$N$ congruence subgroup of ${\mathrm{Sp}}(M \otimes A)$ as defined in §\[S1\]. For any element $c \in M \otimes A$, we write $T_{c} \in {\mathrm{Sp}}(M \otimes A)$ for the transvection with respect to $c$ and the pairing $\langle \cdot, \cdot \rangle$, as in Definition \[dfn symplectic basis\](a). We fix a basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $M$ and assume that the pairing is determined by $\langle a_{i}, b_{i} \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}, a_{j} \rangle = \langle b_{i}, b_{j} \rangle = \langle a_{i}, b_{j} \rangle = 0$ for $1 \leq i < j \leq g$. We also write $a_{i}, b_{i} \in M \otimes A$ for the elements $a_{i} \otimes 1$ and $b_{i} \otimes 1$ respectively for $1 \leq i \leq g$. Our main goal is to prove the following purely algebraic result.

\[prop open subgroup\]

Let $c_{1}, ... , c_{2g} \in M$ be elements such that $c_{i}$ is equivalent modulo $2$ to $$\begin{cases} a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2} & i \ \mathrm{odd}\\ a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1} & i \ \mathrm{even} \end{cases}$$ for $1 \leq i \leq 2g$. Then given integers $n, n' \geq 1$, the subgroup $G \subset \Gamma(2^{n})$ generated by the elements $T_{c_{1}}^{2^{n}}, ... , T_{c_{2g - 1}}^{2^{n}}, T_{c_{2g}}^{2^{n'}} \in \Gamma(2^{n})$ contains $\Gamma(2^{2n})$ (resp. $\Gamma(2^{n + n'})$) if $n' \leq n$ (resp. if $n' > n$). In particular, as a topological subgroup of ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, $G$ is open, and its associated Lie algebra $\mathfrak{g}$ coincides with the $2$-adic symplectic Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$.

Note that the image of any transvection $T \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$ under the logarithm map is $T - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and that $\mathfrak{g}$ is generated as a Lie algebra by the logarithms of the transvections $T_{c_{i}}$. Therefore, in order to prove the second statement of this proposition, it suffices to show that $\{T_{c_{s}} - 1\}_{1 \leq s \leq 2g}$ generates the full Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$. However, in order to get both statements, we will prove something slightly stronger which is given by the following lemma.

\[lemma transvection commutator basis\]

Let $t_{i} = T_{c_{i}} - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and let $\bar{t}_{i}$ denote the reduction of $t_{i}$ modulo $2$ for $1 \leq i \leq 2g$. Then an ${\mathbb{F}}_{2}$-basis for $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ is given by the set $\{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{1 \leq i < j \leq 2g}$.

Let $\bar{c}_{i}$ denote the reductions modulo $2$ of the basis element $c_{i}$ for $1 \leq i \leq 2g$. We know from the characterization of the $c_{i}$’s modulo $2$ given in the statement that $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ is an ordered basis of $M \otimes {\mathbb{F}}_{2}$ with $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1 \in {\mathbb{F}}_{2}$ for $1 \leq i < j \leq 2g$. With respect to this basis, we may view $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ as the Lie algebra $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$. Now it is well known that $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ is a $(2g^{2} + g)$-dimensional vector space over ${\mathbb{F}}_{2}$. Since there are $2g^{2} + g$ elements listed in the set of $\bar{t}_{s}$’s and their commutators, it suffices to prove that this set is linearly independent over ${\mathbb{F}}_{2}$, using only the fact that $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1$ for $1 \leq i < j \leq 2g$. In order to do this, we first note that for any elements $a, b \in M \otimes {\mathbb{F}}_{2}$, the commutator of the logarithms of the transvections with respect to $a$ and to $b$ is the ${\mathbb{F}}_{2}$-linear operator in $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $v \mapsto \langle v, a \rangle \langle a, b \rangle b + \langle v, b \rangle \langle b, a \rangle a$ for all $v \in M \otimes {\mathbb{F}}_{2}$. Therefore, for $1 \leq i < j \leq 2g$, the linear operator $[\bar{t}_{i}, \bar{t}_{j}] = [\bar{t}_{j}, \bar{t}_{i}]$ is given by $$\label{eq commutator} v \mapsto \langle v, \bar{c}_{i} \rangle \bar{c}_{j} + \langle v, \bar{c}_{j} \rangle \bar{c}_{i}.$$ We compute using this formula that the upper left-hand $2 \times 2$ submatrices of $\bar{t}_{1}$, $\bar{t}_{2}$, and $[\bar{t}_{1}, \bar{t}_{2}]$ respectively are $$\label{eq base case} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},$$ and therefore these elements of $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ are linearly independent over ${\mathbb{F}}_{2}$. This in particular proves the statement for $g = 1$.

Now assume inductively that $g \geq 2$ and that the statement is true for $g - 1$. Clearly, $\{\bar{c}_{3}, ... , \bar{c}_{2g}\}$ generates a $2(g - 1)$-dimensional subspace $\mathfrak{g}'$ of $M \otimes {\mathbb{F}}_{2}$ which it generates, and the intersection pairing of any two distinct elements of this set is $1$. Therefore, the inductive assumption implies that the $(2(g - 1)^{2} + (g - 1))$-element set $\{\bar{t}_{i}\}_{3 \leq i \leq g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent; in fact, $\mathfrak{g}'$ is a copy of $\mathfrak{sp}_{2g - 2}({\mathbb{F}}_{2})$ lying inside $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$.

We first claim that the subset $S_{1} := \{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{1}, \bar{t}_{2}]\} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent. To see this, note that we have already shown that $\{\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}]\}$ is linearly independent, and from the fact that the matrices in $\mathfrak{g}'$ have all $0$’s in their first and second rows, it is clear that $\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}] \notin \mathfrak{g}'$. The elements of $S_{1}$ generating $\mathfrak{g}'$ are linearly independent by the inductive assumption, and so the claim follows.

We next claim that the subset $S_{2} := \{[\bar{t}_{1}, \bar{t}_{i}]\}_{3 \leq i \leq 2g} \cup \{[\bar{t}_{2}, \bar{t}_{i}]\}_{3 \leq i \leq 2g}$ is linearly independent. In order to show this, consider a linear combination of elements of the set $S_{2}$ written as $\sum_{i = 3}^{2g} \beta_{i} [\bar{t}_{1}, \bar{t}_{i}] + \sum_{i = 3}^{2g} \gamma_{i} [\bar{t}_{2}, \bar{t}_{i}]$ with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Using the formula in (\[eq commutator\]), we see that this is the linear operator $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $$u : v \mapsto \sum_{i = 3}^{2g} \beta_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \bar{c}_{i}) + \sum_{i = 3}^{2g} \gamma_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \bar{c}_{i})$$ $$\label{eq commutator2} = \langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} + \langle v, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i}.$$ Suppose that $u = 0$. Assume that $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \neq 0$. Then by the nondegeneracy of the symplectic pairing, we can choose $v \in M \otimes {\mathbb{F}}_{2}$ such that $\langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle = 1$. Then $u(v)$ written as a linear combination using the basis $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ has $\bar{c}_{1}$-coefficient equal to $1$, a contradiction because $u(v) = 0$. Therefore, $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} = 0$, which implies that $\beta_{3} = ... = \beta_{2g} = 0$. Now assume that $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \neq 0$. Then similarly, we can choose $w \in M \otimes {\mathbb{F}}_{2}$ such that $\langle w, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle = 1$ and get that $u(w)$ has $\bar{c}_{2}$-coefficient equal to $1$, a contradiction because $u(w) = 0$. Therefore, $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} = 0$, which implies that $\gamma_{3} = ... = \gamma_{2g} = 0$, and so $S_{2}$ is linearly independent.

We finally claim that if a linear combination of elements in $S_{1}$ is equal to a linear combination of elements in $S_{2}$, then these linear combinations must be trivial. Since the full set of $t_{i}$’s and their commutators coincides with $S_{1} \cup S_{2}$, this implies the statement of the proposition. Let $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ be a linear combination of elements in $S_{2}$, written as in (\[eq commutator2\]) with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Now it is clear from the formula there that for each of the matrices in $S_{2}$, the $(1, j)$th entries are all equal and the $(2, j)$th entries are all equal for $3 \leq j \leq 2g$. Then we see by putting $\bar{c}_{3}, ... , \bar{c}_{2g}$ into this formula that $\beta_{3} = ... = \beta_{2g}$ and $\gamma_{3} = ... = \gamma_{2g}$. But then we have $$\langle \bar{c}_{1}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{1}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = 0,$$ and we get that the upper left-hand $2 \times 2$ submatrix of $u$ is the $0$ matrix. This is also true of any matrix in $\mathfrak{g}'$, so we know from what was given in (\[eq base case\]) that $t_{1}$, $t_{2}$, and $[t_{1}, t_{2}]$ do not appear when $u$ is written as a linear combination of elements in $S_{1}$, and therefore we have $u \in \mathfrak{g}'$. As was noted above, every matrix in $\mathfrak{g}'$ has all $0$’s in its first and second rows. Thus, for any $v \in M \otimes {\mathbb{F}}_{2}$, when $u(v)$ is written as a linear combination of $\bar{c}_{i}$’s, the $\bar{c}_{1}$-coefficient and the $\bar{c}_{2}$-coefficient are both $0$. Now by the same argument that was used for the previous claim, we have $\beta_{3} = ... = \beta_{2g} = \gamma_{3} = ... = \gamma_{2g} = 0$. Therefore $u = 0$, as desired.

Let $N = 2n$ (resp. $N = n + n'$) if $n' \leq n$ (resp. if $n' > n$); that is, $N = n + \max \{n, n'\}$. In order to show that $G$ contains the congruence subgroup $\Gamma(2^{N})$, since $G$ is closed, it suffices to show that the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for each integer $m \geq 1$. We claim that in fact we only need to show this for $m = 1$. Indeed, for any $m \geq 1$, the restriction of the logarithm map to $\Gamma(2)$ sends each element $T \in \Gamma(2^{N + m - 1})$ to an element of $\mathfrak{sp}_{2g}({\mathbb{Z}}_{2})$ which is equivalent to $T - 1$ modulo $2^{N + m}$ since $(T - 1)^{2} \equiv 0$ modulo $2^{N + m}$. In this way, one verifies that there is an isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$, given by sending an element $t \in \mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $1 + 2^{N + m - 1}\tilde{t} \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, where $\tilde{t}$ is any operator in ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}/ 2^{N + m}{\mathbb{Z}})$ whose image modulo $2$ is $t$. In particular, each $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$ is an elementary abelian group of exponent $2$ and rank $2g^{2} + g$ (this is also known from the proof of [@sato2010abelianization Corollary 2.2]). Moreover, it is easy to check that for each $m \geq 1$, the map sending each matrix in $\Gamma(2^{N})$ to its $2^{m - 1}$th power induces a group isomorphism $\Gamma(2^{N}) / \Gamma(2^{N + 1}) \stackrel{\sim}{\to} \Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$. It follows that if the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$, then the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for all $m \geq 1$. Thus, in order to prove the proposition, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$.

As above, let $t_{i}$ denote $T_{c_{i}} - 1$ for $1 \leq i \leq 2g$. Since Lemma \[lemma transvection commutator basis\] says that the image modulo $2$ of $\{t_{i}\}_{1 \leq i \leq 2g} \cup \{[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g}$ is a basis of $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains the images of the elements in $\{1 + 2^{N}t_{i}\}_{1 \leq i \leq 2g} \cup \{1 + 2^{N}[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g} \subset \Gamma(2^{2n})$. For ease of notation, we write $n'' = \max \{n, n'\}$. Clearly $G$ is generated by $\{1 + 2^{n}t_{i}\}_{1 \leq i \leq 2g - 1} \cup \{1 + 2^{n'}t_{2g}\}$. We verify using the property $t_{i}^{2} = 0$ that $(1 + 2^{n}t_{i})^{2^{n''}} \equiv 1 + 2^{n + n''}t_{i}$ (mod $2^{n + n'' + 1}$) for $1 \leq i \leq 2g - 1$; $(1 + 2^{n'}t_{2g})^{2^{n + n'' - n'}} \equiv 1 + 2^{n + n''}t_{2g}$ (mod $2^{n + n'' + 1}$); and $$(1 + 2^{n}t_{i})(1 + 2^{n''}t_{j})(1 + 2^{n}t_{i})^{-1}(1 + 2^{n''}t_{j})^{-1} \equiv 1 + 2^{n + n''}[t_{i}, t_{j}] \ (\mathrm{mod} \ 2^{n + n'' + 1})$$ for $1 \leq i < j \leq 2g$. Thus, $G \supset \Gamma(2^{n + n''})$, as desired.

We now state and prove another proposition which will be needed only for the last statements of Theorems \[thm main1\] and \[thm main2\], which pertain to the case that the hyperelliptic curve has degree $4$.

\[prop open subgroup degree 4\]

Assume the notation of Proposition \[prop open subgroup\] and that $g = 1$ and $n' = n$, and let $c_{3} \in M$ be an element which is equivalent modulo $2$ to $a_{1}$. Then the subgroup of $\Gamma(2^{n})$ generated by $G$ and the element $T_{c_{3}}^{2^{n}}$ coincides with $\Gamma(2^{n})$.

By the argument used at the beginning of the proof of Proposition \[prop open subgroup\], we only need to show that the images of $T_{c_{1}}^{2^{n}}, T_{c_{2}}^{2^{n}}, T_{c_{3}}^{2^{n}}$ modulo $2^{n + 1}$ generate $\Gamma(2^{n}) / \Gamma(2^{n + 1})$. Using the isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{n}) / \Gamma(2^{n + 1})$ which was established in that proof, we see that it suffices to show that $\{\bar{t}_{1}, \bar{t}_{2}, \bar{t}_{3}\}$ is linearly independent and thus generates the rank-$3$ elementary abelian group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, where $\bar{t}_{i}$ denotes the image modulo $2$ of $T_{c_{i}} - 1$ for $1 \leq i \leq 3$. But this is clear from noting that $c_{3} \equiv c_{1} + c_{2}$ (mod $2$) and writing out these linear operators as matrices with respect to an ordered basis of $M \otimes {\mathbb{F}}_{2}$ consisting of the images of $c_{1}$ and $c_{2}$.

Proof of main theorems and a further result {#S4}
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The main goal of this section is to prove Theorems \[thm main1\] and \[thm main2\]. Our strategy for this is to put together local results with respect to several primes of $K$ using Proposition \[prop Galois action local\] to get several elements in $G_{2}$ and then to use Proposition \[prop open subgroup\] to determine that the subgroup of $G_{2}$ generated by these elements contains a certain congruence subgroup of the symplectic group. This is realized by the following theorem, which can be used in a far more general situation than required for Theorems \[thm main1\] and \[thm main2\] and is therefore a useful result in its own right.

\[thm several primes\]

Let $J$ be the Jacobian of the hyperelliptic curve $C$ over $K$ of genus $g$ and degree $d'$ with defining equation $y^{2} = \prod_{i = 1}^{d'} (x - \alpha_{i})$ for some elements $\alpha_{i} \in \mathcal{O}_{K}$ for $1 \leq i \leq d'$, and define the $2$-adic Galois image $G_{2}$ as above. Suppose that there are distinct primes $\mathfrak{p}_{1}$, ... , $\mathfrak{p}_{d' - 1}$ of $K$ not lying over $(2)$ and such that for $1 \leq i \leq d' - 1$, the only two $\alpha_{j}$’s which are equivalent modulo $\mathfrak{p}_{i}$ are $\alpha_{i}$ and $\alpha_{d'}$; let $m_{i} \geq 1$ be the maximal integer such that $\mathfrak{p}_{i}^{m_{i}} \mid (\alpha_{d'} - \alpha_{i})$. Let $n = \max \{v_{2}(m_{i})\}_{i = 1}^{d' - 2}$. If $n' := v_{2}(m_{d' - 1}) \leq n$ or if $d' = 2g + 2$, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{2n + 2})$. Otherwise, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{n + n' + 2})$. Moreover, if $d' = 4$, then $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$.

Using the embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ fixed at the beginning of §\[S2\], we identify $T_{\ell}(J)$ with $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ as we did in §\[S2\]. Let $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ be a symplectic basis of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ as in Definition \[dfn symplectic basis\](b) with $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g + 1}, \infty)$ if $d'$ is odd and $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g}, \alpha_{2g + 2}, \alpha_{2g + 1})$ if $d'$ is even. Then Proposition \[prop Galois action local\] says that for $1 \leq i \leq 2g$, the image of $G_{K, \mathfrak{p}_{i}} \subset G_{K}$ in ${\mathrm{GSp}}(T_{2}(J))$ contains $T_{c_{i}}^{2m_{i}}$, where $T_{c_{i}} \in {\mathrm{Sp}}(T_{2}(J))$ is the transvection with respect to an element $c_{i} \in T_{2}(J)$ which is equivalent modulo $2$ to the formula given in (\[eq c\_i\]). Meanwhile, if $d'$ is even, the image of $G_{K, \mathfrak{p}_{2g + 1}}$ contains $T_{c_{d'}}^{2m_{2g + 1}}$ for some other element $c_{d'} \in T_{2}(J)$ corresponding to the lift of a loop separating $\{\alpha_{2g + 1}, \alpha_{2g + 2}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{2g + 2}$. In any case, we have $T_{c_{1}}^{2m_{1}}, ... , T_{c_{2g}}^{2m_{2g}} \in G_{2}$; note that $m_{i} / 2^{v_{2}(m_{i})} \in {\mathbb{Z}}_{2}^{\times}$ and so by taking suitable powers we get that $T_{c_{1}}^{2^{v_{2}(m_{1}) + 1}}, ... , T_{c_{2g}}^{2^{v_{2}(m_{2g}) + 1}} \in G_{2}$. It then follows from Proposition \[prop open subgroup\] applied to $M := H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ that $G_{2} \supset \Gamma(2^{2n + 2})$ if $n' \leq n$ or if $d' = 2g + 2$ and $G_{2} \supset \Gamma(2^{n + n' + 2})$ otherwise.

If $d' = 4$, then in particular, we have $T_{c_{1}}^{2^{\max\{n, n'\} + 1}}, T_{c_{2}}^{2^{\max\{n, n'\} + 1}}, T_{c_{3}}^{2^{\max\{n, n'\} + 1}} \in G_{2}$ with $c_{1}, c_{2}, c_{3}$ as described above. Clearly we have $c_{1} \equiv a_{1} + b_{1}$, $c_{2} \equiv b_{1}$, and $c_{3} \equiv a_{1}$ (mod $2$), and so Proposition \[prop open subgroup degree 4\] implies that $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$.

\[rmk index bound\]

We note that it is generally not difficult to compute the order of $G / \Gamma(2^{2n + 2})$ or $G / \Gamma(2^{n + n' + 2})$, where $G \subseteq G_{2}$ is the subgroup generated by the powers of transvections given in the proof above. Therefore, one may improve the upper bound for $[\Gamma(2) : G_{2}]$ which directly follows from the statement of Theorem \[thm several primes\]. For example, we have $[\Gamma(2) : G_{2}] \leq 2^{(2n + 1)(2g^{2} + g) - (n + 1)(d' - 1)}$ in the case that $n = n'$.

[(Legendre curve)]{} \[ex Legendre\]

For any $\lambda \in \mathcal{O}_{K} \smallsetminus \{0, 1\}$, let $E_{\lambda}$ be the elliptic curve over $K$ given by $y^{2} = x(x - 1)(x - \lambda)$. Suppose that there exist (necessarily distinct) primes $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ of $K$ not lying over $(2)$ and integers $m_{1}, m_{2}, \geq 1$ such that $\mathfrak{p}_{1}^{m_{1}}$ exactly divides $(\lambda)$ and $\mathfrak{p}_{2}^{m_{2}}$ exactly divides $(\lambda - 1)$. Then Theorem \[thm several primes\] tells us that the $2$-adic Galois image $G_{2}$ (strictly) contains $\Gamma(2^{v_{2}(m_{1}) + v_{2}(m_{2}) + 2})$. In the case that $m_{1} = m_{2} = 1$ (e.g. $K = {\mathbb{Q}}$, $\lambda = 6$, $\mathfrak{p}_{1} = (3)$, $\mathfrak{p}_{2} = (5)$), we get $\Gamma(2) \supset G_{2} \supsetneq \Gamma(4)$ and can therefore directly compute the precise subgroup $G_{2} \cap {\mathrm{Sp}}(T_{2}(E_{\lambda})) \subset \Gamma(2)$ using the well-known fact that the $4$-division field $K(E_{\lambda}[4])$ is generated over $K$ by $\{\sqrt{-1}, \sqrt{\lambda}, \sqrt{\lambda - 1}\}$.

It is also possible to prove the statement of Proposition \[prop Galois action local\], and hence the subgroup $G_{2}$ for this example for the particular cases of $C = E_{\lambda}$ over $\mathcal{K}_{\mathfrak{p}_{1}}$ and over $\mathcal{K}_{\mathfrak{p}_{2}}$ using formulas for generators of $2$-power division fields of $E_{\lambda}$ over $K$ found in [@yelton2015dyadic], as the author has done in [@yelton2015hyperelliptic §3.4].

We now prove Theorems \[thm main1\] and \[thm main2\] together.

First assume the notation and hypotheses of Theorem \[thm main1\]. Let $\alpha_{1}, ... , \alpha_{d}$ denote the roots of $f$, and write $L = K(\alpha_{1}, ... , \alpha_{d})$ for the splitting field of $f$ over $K$. Note that ${\mathrm{Gal}}(L / K)$ acts transitively on the $\alpha_{i}$’s since $f$ is irreducible. It then follows from the well-known description of the $2$-division field of a hyperelliptic Jacobian (see for instance [@mumford1984tata Corollary 2.11]) that $G_{K}$ does not fix the $2$-torsion points of $J$ and so $G_{2}$ is not contained in $\Gamma(2)$, while the image of ${\mathrm{Gal}}(\bar{K} / L)$ under $\rho_{2}$ coincides with $G_{2} \cap \Gamma(2)$.

The fact that $\mathfrak{p} \nmid (2\Delta)$ implies that the extension $L / K$ is not ramified at $\mathfrak{p}$, and so $\mathfrak{p}$ splits into a product $\mathfrak{p}_{1} ... \mathfrak{p}_{r}$ of distinct primes in $L$, for some integer $r$ dividing $[L : K]$. Then since $\mathfrak{p}^{m}$ exactly divides $(f(\lambda)) = \prod_{i = 1}^{d}(\lambda - \alpha_{i})$, we have $\mathfrak{p}_{i} \mid (\lambda - \alpha_{i})$ for some $i$; we assume without loss of generality that $\mathfrak{p}_{1} \mid (\lambda - \alpha_{1})$. Then $\mathfrak{p}_{1}$ cannot divide $(\lambda - \alpha_{i})$ for any $i \in \{2, ... , d\}$, because otherwise for such an $i$ we would have $\mathfrak{p}_{1} \mid (\alpha_{i} - \alpha_{1}) \mid (\Delta)$, which contradicts the hypothesis that $\mathfrak{p} \nmid (2\Delta)$. It follows that $\mathfrak{p}_{1}^{m}$ exactly divides $(\lambda - \alpha_{1})$. Then by applying elements of ${\mathrm{Gal}}(L / K)$ that take $\alpha_{1}$ to each $\alpha_{i}$, we get other primes lying over $\mathfrak{p}$ whose $m$th powers exactly divide the ideals $(\lambda - \alpha_{i})$; we assume without loss of generality that $\mathfrak{p}_{i}^{m}$ exactly divides $(\lambda - \alpha_{i})$ for $1 \leq i \leq d$. Now since the $\mathfrak{p} \nmid (2\Delta)$ hypothesis implies that none of the $\mathfrak{p}_{i}$’s lie over $(2)$, we can apply Theorem \[thm several primes\] with $K$ replaced by $L$, $d' = d + 1$, $\alpha_{d'} = \lambda$, and $m_{1} = ... = m_{d' - 1} = m$ to get the statement of Theorem \[thm main1\].

Now assume the notation and hypotheses of Theorem \[thm main2\]. Then the argument for proving Theorem \[thm main2\] is the same except that when applying Theorem \[thm several primes\] we choose $\mathfrak{p}_{d' - 1}$ to be a prime of $L$ lying over $\mathfrak{p}'$, and we put $d' = d + 2$, $\alpha_{d'} = \lambda$, $\alpha_{d' - 1} = \lambda'$, $m = m_{1} = ... = m_{d' - 2}$, and $m' = m_{d' - 1}$.

Realizing uniform boundedness along one-parameter families {#S5}
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Fix an irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$. Cadoret and Tamagawa have shown in [@cadoret2012uniform Theorems 1.1 and 5.1] that for all but finitely many $\lambda \in K$, the $\ell$-adic Galois image $G_{\ell, \lambda}$ associated to the Jacobian of the curve given by $y^{2} = f(x)(x - \lambda)$ is open in the $\ell$-adic Galois image $G_{\ell, \eta}$ associated to the generic fiber of the family parametrized by $\lambda$. These theorems also assert that there is some integer $B \geq 1$ depending only on $f$ and $\ell$ such that the index of $G_{\ell, \lambda}$ in $G_{\ell, \eta}$ is bounded by $B$ for all but finitely many $\lambda \in K$. The following theorem recovers the openness result for $\ell = 2$ when $d \geq 4$ and explicitly provides the aforementioned uniform bound when $d$ is even, under the assumption that $K$ has class number $1$. (Note that for the elliptic curve case, where $d \in \{ 2, 3\}$, such openness results are already known from the celebrated Open Image Theorem of Serre given by [@serre1989abelian IV-11], while uniform bounds are given by [@arai2008uniform Theorem 1.3].) It is interesting to note that Faltings’ Theorem is used both in the proof of [@cadoret2012uniform Theorem 1.1] and in our proof of the theorem below.

\[thm uniform bounds\]

Assume that $\mathcal{O}_{K}$ is a PID. Let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 3$ with discriminant $\Delta$. For each $\lambda \in K$, let $J_{\lambda}$ denote the Jacobian of the hyperelliptic curve $C_{\lambda}$ with defining equation $y^{2} = f(x)(x - \lambda)$, and write $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ for the image of the associated $2$-adic Galois representation.

a\) If $d \geq 4$, then the Lie subgroup $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ is open for all but finitely many $\lambda \in K$.

b\) If $d = 3$ (resp. if $d \geq 5$), then $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{4}$ (resp. all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{2}$).

c\) If $d = 4$, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(16)$ for all but finitely many $\lambda \in K$. If $d \geq 6$ is even, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in K$.

Let $\Sigma \subset K^{\times}$ denote the multiplicative subgroup generated by the elements $\xi \in \mathcal{O}_{K}$ such that $\xi$ is divisible only by primes which divide $2\Delta$. We claim that $\Sigma$ is finitely generated. Indeed, there is an obvious map from $\Sigma$ to the free ${\mathbb{Z}}$-module formally generated by the (finite) set of prime ideals of $\mathcal{O}_{K}$ which divide $2\Delta$, and its kernel is the unit group $\mathcal{O}_{K}^{\times}$, which is also well known to be finitely generated.

Choose any $\lambda \in K$, which we may write as $\mu / \nu$ for some coprime $\mu, \nu \in \mathcal{O}_{K}$, because $\mathcal{O}_{K}$ is a PID. Let $h(x) = \nu^{d}f(\nu^{-1} x)$, which is a monic polynomial in $\mathcal{O}_{K}[x]$; note that the discriminant of $h$ is equal to $\nu^{2d^{2} - d}\Delta$. Then there is a $K(\sqrt{\nu})$-isomorphism from $C_{\lambda}$ to the hyperelliptic curve $C_{\lambda}'$ whose defining equation is $y^{2} = \nu^{d}f(\nu^{-1}x)(x - \mu) \in \mathcal{O}_{K}[x]$, given by $(x, y) \mapsto (\nu x, \nu^{(d + 1) / 2}y)$. Thus, letting $J_{\lambda}'$ denote the Jacobian of $C_{\lambda}'$, the $2$-adic Tate modules $T_{2}(J_{\lambda})$ and $T_{2}(J_{\lambda}')$ are isomorphic as ${\mathrm{Gal}}(\bar{K} / K(\sqrt{\nu}))$-modules. In light of this, we replace $K$ with $K(\sqrt{\nu})$ and consider the $2$-adic Galois image $G_{2, \lambda}$ associated to $J_{\lambda}'$. Now Theorem \[thm main1\] says that if there is a prime element $\mathfrak{p}$ dividing $\nu^{d}f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$, then $G_{2, \lambda} \subset {\mathrm{GSp}}(T_{2}(J_{\lambda})) \cong {\mathrm{GSp}}(T_{2}(J_{\lambda}'))$ is open. It follows from the fact that $\mu$ and $\nu$ are coprime that $\nu^{d}f(\lambda)$ is not divisible by any prime element dividing $\nu$, so a prime $\mathfrak{p}$ satisfying the above condition does not divide $2\Delta$. The existence of such a prime is equivalent to the condition that $\nu^{d} f(\lambda) \notin \Sigma$, so to prove part (a) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{d}$ for only finitely many $\lambda \in K$. Note that any such $\lambda$ yields a solution $(x = \lambda, y) \in K \times K$ to an equation of the form $\xi y^{d} = f(x)$ with $\xi \in \Sigma'$, where $\Sigma' \subset \Sigma$ is a set of representatives of elements in $\Sigma / \Sigma^{d}$. If $d \geq 4$, then an application of the Riemann-Hurwitz formula shows that such an equation defines a smooth curve of genus $\geq 2$, and then Faltings’ Theorem implies that there are only finitely many solutions defined over $K$ to each such equation. Therefore, to prove (a) it suffices to show that there are only finitely many choices of $\xi$. But this follows from the fact that $\Sigma / \Sigma^{d}$ is finite because $\Sigma$ is finitely generated.

Now assume that $d = 3$ or $d \geq 5$ and that $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{s}$, with $s = 4$ if $d = 3$ and $s = 2$ otherwise. Then if we write $\lambda = \mu / \nu$ as above, we have $\nu \in \Sigma \cdot (K^{\times})^{s}$. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ both when $d = 3$ and when $d \geq 5$. Therefore, to prove (b) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$. This follows from the same argument as above, once we observe by Riemann-Hurwitz that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$.

Finally, assume that $d \geq 4$ is even and choose any $\lambda = \mu / \nu \in K$ as before. Let $s = 4$ if $d = 4$ and let $s = 2$ otherwise. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which in both cases is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda}))$ strictly contains $\Gamma(16)$ (resp. $\Gamma(4)$). Therefore, to prove (c) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$, which likewise follows from checking that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$.

\[rmk not PID\]

a\) If we drop the assumption that $\mathcal{O}_{K}$ is a PID in the statement of Theorem \[thm uniform bounds\], then we observe from the proof above that we still get the statement of (a) when “all but finitely many $\lambda \in K$" is replaced by “all but finitely many $\lambda \in \mathcal{O}_{K}$" (and this statement holds for $d \in \{2, 3\}$ as well). In particular, this shows that the elements $\lambda \in \mathcal{O}_{K}$ which satisfy the hypothesis in Theorem \[thm main1\] account for all but finitely many of the elements $\lambda \in \mathcal{O}_{K}$ such that $G_{2, \lambda}$ is open.

b\) In any case, the hypothesis “$\mathcal{O}_{K}$ is a PID" may be weakened to “the Hilbert class field tower of $K$ terminates". This follows from the fact that under this hypothesis, there is a finite extension of $K$ with class number $1$, and it clearly suffices to prove the assertions of Theorem \[thm uniform bounds\] when $K$ is replaced with a finite extension of $K$.

\[rmk 4-torsion\]

Parts (b) and (c) of Theorem \[thm uniform bounds\] say that for a given polynomial $f$ of degree $d \neq 4$, there are many elements $\lambda \in K$ such that $G_{2, \lambda} \supsetneq \Gamma(4)$. In these cases, it is always possible to compute the full structure of $G_{2}$ and determine its index in ${\mathrm{GSp}}(T_{2}(J_{\lambda}))$ by considering the Galois action on the $4$-torsion subgroup of $J$ and using formulas for the generators of the $4$-division field $K(J[4])$ over $K$. Such formulas are provided by [@yelton2015images Proposition 3.1] in the case that $d$ is odd and are found in [@yelton2015hyperelliptic §2.4] in the case that $d$ is even.
