论文标题
Eisenstein系列$ G_2 $和对称的Cube Bloch - 喀托猜想
Eisenstein series for $G_2$ and the symmetric cube Bloch--Kato conjecture
论文作者
论文摘要
让$ f $是重量和琐碎的nebentypus的cuspidal特征，让$ p $是$ f $的水平，让$ρ_f$是$ p $ - ad的galois代表，附加到$ f $。假设$ρ_f$ $ρ_f$的对称立方所附加的$ l $功能在其中心点消失了。然后在一些温和的假设下，并以亚瑟猜想的某些后果为条件，我们根据bloch-kato猜想，在$ρ_f$的对称立方体的适当扭曲的bloch-kato selmer群中构建了一个非平凡的元素。 我们的技术基于Skinner和Urban的方法。我们在适当的Selmer组中通过$ p $ - 原发性变形的Eisenstein系列为特殊的cuspidal家族中的$ G_2 $构建，然后在相应的$ G_2 $ -GALOIS表示的相应家族中学习一个格子。我们还对所使用的特定猜想进行了详细的研究，并解释了人们如何尝试证明它们。
Let $F$ be a cuspidal eigenform of even weight and trivial nebentypus, let $p$ be a prime not dividing the level of $F$, and let $ρ_F$ be the $p$-adic Galois representation attached to $F$. Assume that the $L$-function attached to the symmetric cube of $ρ_F$ vanishes to odd order at its central point. Then under some mild hypotheses, and conditional on certain consequences of Arthur's conjectures, we construct a nontrivial element in the Bloch--Kato Selmer group of an appropriate twist of the symmetric cube of $ρ_F$, in accordance with the Bloch--Kato conjectures. Our technique is based on the method of Skinner and Urban. We construct a class in the appropriate Selmer group by $p$-adically deforming Eisenstein series for the exceptional group $G_2$ in a generically cuspidal family and then studying a lattice in the corresponding family of $G_2$-Galois representations. We also make a detailed study of the specific conjectures used and explain how one might try to prove them.
