发表机构
Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在虚二次域上构造高权模形式的 zeta 元素并证明显式互反律，从而在弱假设下证明了 Kato 岩泽主猜想，推广了椭圆曲线情形。
AI 中文摘要
设 $p\geq 5$ 为素数，$f$ 为偶数权 $k = 2r \geq 2$、水平 $N$、$p \nmid N$ 且平凡 nebentype 的 $p$-ordinary 尖点新形式。本文在仅假设 $f$ 的相伴 Galois 表示满足大图像假设且其剩余表示绝对不可约的条件下，证明了以 $f$ 的 $p$-adic $L$-函数表述的 Kato 岩泽主猜想。这推广了 Burungale--Castella--Skinner 关于 $\mathbb{Q}$ 上具有好普通约化的椭圆曲线的主要结果。我们的证明遵循他们的总体策略。一个关键的新成分是在 $p$ 分裂的虚二次域上构造与 $f$ 相联系的 zeta 元素，并证明其显式互反律，推广了 Burungale--Skinner--Tian--Wan 的相应工作。这些结果提供了 Burungale--Castella--Skinner 论证中关键输入的高权类比，并使得他们的方法能够对高权模形式实施。
英文摘要
Let $p\geq 5$ be a prime, and let $f$ be a $p$-ordinary cuspidal newform of even weight $k = 2r \geq 2$ and level $N$, with $p \nmid N$ and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for $f$, formulated in terms of the $p$-adic $L$-function of $f$, under the sole assumptions that the associated Galois representation of $f$ satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale--Castella--Skinner, which treats elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which $p$ splits, of a zeta element attached to $f$, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale--Skinner--Tian--Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale--Castella--Skinner and allow their method to be carried out for higher-weight modular forms.
Comments31 pages. Comments welcome!
