论文标题
Coleman-Mazur Eigencurve的模块化证明
A modular proof of the properness of the Coleman-Mazur eigencurve
论文作者
论文摘要
我们给出了科尔曼 - 马祖尔特征库的正常的新证明。 Coleman和Mazur在1998年首次提出了特征库来满足适当性的评估标准的问题，并使用Diao和Liu在2016年使用深，强大的Hodge和Galois-理论机制定居。我们的证明是简短而明确的，不使用Galois理论。取而代之的是，我们基于过度会员模块化形式的基本特性调整了早期的Buzzard和Calegari方法。为了促进这一点，我们将Pilloni的几何形状扩展到了超大型重量的过度转化形式到超大型基因座。在此过程中，我们表明Hecke运算符$ u_p $是在任何分析权重的大量超关联半径的空间上注入的。
We give a new proof of the properness of the Coleman-Mazur eigencurve. The question of whether the eigencurve satisfies the valuative criterion for properness was first asked by Coleman and Mazur in 1998 and settled by Diao and Liu in 2016 using deep, powerful Hodge- and Galois- theoretic machinery. Our proof is short and explicit and uses no Galois theory. Instead we adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms. To facilitate this, we extend Pilloni's geometric construction of overconvergent forms of arbitrary weight farther into the supersingular locus. Along the way, we show that the Hecke operator $U_p$ is injective on spaces of forms of large overconvergence radius of any analytic weight.
