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 $f\in {\mathcal{O}}_K[x]$ of discriminant Δ 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 (2Δ), then the image of the natural 2-adic Galois representation is open in $\mathrm{GSp}\left(T_2(J)\right)$ and contains a certain congruence subgroup of $\mathrm{Sp}\left(T_2(J)\right)$ 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 $y^2=f(x)(x-\lambda )(x-{\lambda }^{\prime })$ for distinct elements $\lambda ,{\lambda }^{\prime }\in K$. 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.

中文翻译：

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与超椭圆雅可比矩阵相关联的 2-adic Galois 图像的有界结果

令K为数域，令C为K上的超椭圆曲线，其中包含雅可比J。假设C由以下形式的方程定义${是}^2=F(X)(X-\lambda )$ 对于一些不可约的单多项式 $F\in {\mathcal{哦}}_{钾}[X]$ 判别式 Δ 和某个元素 $\lambda \in {\mathcal{哦}}_{钾}$. 我们的第一个主要结果是如果有一个质数$\mathfrak{磷}$的ķ分$(F(\lambda ))$ 但不是 (2Δ)，那么自然 2-adic Galois 表示的图像在 $\mathrm{GSP}\left({吨}_2(J)\right)$ 并包含一个特定的同余子群 $\mathrm{斯普}\left({吨}_2(J)\right)$ 取决于最大功率 $\mathfrak{磷}$ 划分 $(F(\lambda ))$. 我们还提出并证明了这个结果的一个变体，当C由以下形式的方程定义时适用${是}^2=F(X)(X-\lambda )(X-{\lambda }^{\prime })$ 对于不同的元素 $\lambda ,{\lambda }^{\prime }\in 钾$. 然后我们证明前一个陈述中的假设几乎适用于所有$\lambda \in {\mathcal{哦}}_{钾}$ 并证明 Cadoret 和 Tamagawa 一致有界结果的定量形式。