论文标题
希尔伯特模块化形式的光谱光环
Spectral halo for Hilbert modular forms
论文作者
论文摘要
让$ f $是一个完全真实的领域，$ p $是一个奇怪的素数，完全分为$ f $。我们证明，与$ f $ $ f $的确定四个代数相关的特征变化满足了以下属性：在重量空间的边界环上，特征变量是无限无限的许多连接组件的不连接，这些组件在重量空间中是有限的；在每个固定连接的组件上，$ u_ \ mathfrak {p} $ - 点的斜率与$ \ mathfrak {p} $ - 参数的$ p $ - ad-adic估值是由显式数字界的，对于所有PRIMES $ \ mathfrak {p} $ f $ p $ po $ p $ to $ p $。应用汉森的$ p $ - 辅助插值定理，我们能够将结果转移到希尔伯特模块化特征室。特别是，我们证明，在希尔伯特模块化特征值的每个不可还原部分上，当点向边界移动时，其$ u_p $ slope占零。就特征库而言，这完成了科尔曼 - 马祖尔的“光晕”猜想的证明。
Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$ over $p$. Applying Hansen's $p$-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its $U_p$ slope goes to zero. In the case of eigencurves, this completes the proof of Coleman-Mazur's `halo' conjecture.
