Hilbert模特征簇上的平行权2点与奇偶猜想
Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture
摘要：
设F是d次全实域，p是F中全可裂的奇素数。我们定义和研究一维部分本征簇插值希尔伯特模形式在F的重量变化只在一个单一的地方v以上的p.对于这些本征簇，我们表明，刘，万和肖开发的方法适用，并推导出，在一个边界环在权空间的半径足够小，部分本征簇分解为一个不相交的联盟的组件是有限的权空间。我们应用这个结果证明的奇偶校验版本的Bloch-加藤猜想有限斜率希尔伯特模形式平凡中心字符（在某些假设下），减少到平行重量2的情况下。作为我们关于部分本征簇结果的另一个结果，我们仍然在假设p在F中完全分裂的情况下证明了，（1 + d维）尖点Hilbert模特征簇具有许多性质，（所有，如果d是偶数）不可约分量包含一个具有非临界斜率和平行权2的经典点（在p处具有其导体可以显式有界的某个特征），或任何其他代数权重。
英文摘要：
Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch--Kato conjecture for finite slope Hilbert modular forms with trivial central character (under some assumptions), by reducing to the case of parallel weight 2. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that p is totally split in F, that the full (dimension 1 + d) cuspidal Hilbert modular eigenvariety has the property that many (all, if d is even) irreducible components contain a classical point with non-critical slopes and parallel weight 2 (with some character at p whose conductor can be explicitly bounded), or any other algebraic weight.
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