论文标题
与模块化galois表示相关的理想班级数字组
Ideal class groups of number fields associated to modular Galois representations
论文作者
论文摘要
令$ p $为模块化表格的奇数数字，$ f $。我们考虑$ \ mathbb {f} _p $ - valued galois表示$ \barρ_f$附加到$ f $及其twist $ \barρ_{f，d} $ by Quadratic Chartic $χ_d$对应于Quadratic Indientant $ d $ d $。我们将$ k_ {f，d} $定义为与$ \barρ_{f，d} $的内核相对应的字段。在本文中，我们调查了数字字段$ k_ {f，d} $的理想类$ \ mathrm {cl}（k_ {f，d}）$作为$ \ mathrm {gal}（k_ {f，d}/d}/\ mathbb {q}）$ - 模块。我们给出了一种条件，该条件意味着存在$ \ mathrm {gal}（k_ {f，d}/\ mathbb {q}）$ - 来自$ \ mathrm {cl}（k_ {f，d}）\ otimes \ otimimes \ ot \ mathbb {$ d $ d $ d $ d $ d p yp的$ \ mathrm {Clrm {cl}的equivariant earivariant formormorthism $ \barρ_{f，d} $，使用Bloch和Kato的Selmer Group $ \barρ_{f，d} $。我们还提供了一些数值示例，其中通过计算Bloch和Kato的猜想下的$ f $ twist $ f $ twist的$ l $ function的核心价值来进行这种过渡。本文我们的主要结果是对Prasad和Shekhar先前在椭圆曲线上的结果部分概括为更高的重量模块化形式。
Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\barρ_f$ attached to $f$ and its twist $\barρ_{f, D}$ by the quadratic character $χ_D$ corresponding to a quadratic discriminant $D$. We define $K_{f, D}$ to be the field corresponding to the kernel of $\barρ_{f, D}$. In this article, we investigate the ideal class group $\mathrm{Cl}(K_{f, D})$ of the number field $K_{f, D}$ as a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-module. We give a condition which implies the existence of a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-equivariant surjective homomorphism from $\mathrm{Cl}(K_{f, D})\otimes \mathbb{F}_p$ to the representation space $M_{f, D}$ of $\barρ_{f, D}$, using Bloch and Kato's Selmer group of $\barρ_{f, D}$. We also give some numerical examples where we have such surjections by calculating the central value of the $L$-function of $f$ twisted by $χ_D$ under Bloch and Kato's conjecture. Our main result in this paper is a partial generalization of the previous result of Prasad and Shekhar on elliptic curves to higher weight modular forms.
