论文标题
p-rirgular模块化形式的算术：家庭和p- ad-l功能
Arithmetic of p-irregular modular forms: families and p-adic L-functions
论文作者
论文摘要
令$ f _ {\ mathrm {new}} $成为重量$ \ geq 2 $的经典新形式，prime to $ p $级别。我们研究$ f _ {\ mathrm {new}} $的算术及其唯一的$ p $ -stabilisation $ f $时，当$ f _ {\ mathrm {new}} $是$ p $ - irregular，也就是说，当它的hecke pynomial in $ p $ aft $ p $ aft $ p $ aft $ p $ aff $ p $ aft $ p $ afted单个重复的根源。特别是，我们通过$ f $研究$ p $ - 亚种的重量家庭及其基础变化到一个虚构的二次菲尔德$ f $，其中$ p $ splits，并证明各自的特征库都是gorenstein，均为$ f $ $ f $。我们使用它来构建与Coleman家族相比，通过$ f $构建了两个变量的$ P $ -ADIC $ L $ - 函数，以及该家族的基础底座上的基础变化对$ f $的基础$ F $ l $ l $ - 功能。我们通过$ p $ - 亚种artin形式主义将两种和三变量的$ p $ p $ ad-adic $ l $ - 功能联系起来。在这种情况下，这些结果用于XIN WAN的工作中，以证明Iwasawa的主要猜想。 在附录中，我们证明了模块化符号的HIDA二元性结果，在Hecke代数与过度授权模块化符号的家族之间构建了配对，并证明了它在任何cusp形式周围局部非分类。这使我们能够控制家庭中（古典和比安奇）Hecke代数的尺寸。
Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three-variable $p$-adic $L$-function over the base-change of this family to $F$. We relate the two- and three-variable $p$-adic $L$-functions via $p$-adic Artin formalism. These results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in this case. In an appendix, we prove results towards Hida duality for modular symbols, constructing a pairing between Hecke algebras and families of overconvergent modular symbols and proving that it is non-degenerate locally around any cusp form. This allows us to control the sizes of (classical and Bianchi) Hecke algebras in families.
