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 \(g \ge 2\), 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.

我们研究了闭模子组\（G \ subseteq \ mathrm {Sp} _ {2g}（\ mathbb {Z} _2）\），其模2图像与图像\（\ mathfrak {S} _ {2g + 1} \ subseteq \ mathrm {SP} _ {2克}（\ mathbb {F} _2）\）的\（S_ {2克+ 1} \）或图像\（\ mathfrak {S} _ {2克+ 2} \ subseteq \ mathrm {SP} _ {2克}（\ mathbb {F} _2）\）的\（S_ {2克+ 2} \）标准表示下。我们证明，当\（g \ ge 2 \）时，唯一封闭的子组\（G \ subseteq \ mathrm {Sp} _ {2g}（\ mathbb {Z} _2）\）投射到\（\ mathfrak {S} _ {2克+ 2} \）处于其全反图像\（\ mathrm {SP} _ {2克}（\ mathbb {Z} _2）\） ，而所有的亚组\（G \ subseteq \ mathrm {Sp} _ {2g}（\ mathbb {Z} _2）\）冲向\（\ mathfrak {S} _ {2g + 1} \）已打开，并包含8级本金\（\ mathrm {Sp} _ {2g}（\ mathbb {Z} _2）\）的全等子组。作为立即应用，我们能够增强Zarhin在与超椭圆曲线相关的2-adic Galois表示上的结果。我们还证明了具有完整Galois群的偶数多项式的基本推论。