We prove the Gross order of vanishing conjecture in special cases where the vanishing order of the character in question can be arbitrarily large. In almost all previously known cases the vanishing order is zero or one. One major ingredient of our proofs is the equivalence of this conjecture to the Gross–Kuz’min conjecture. We present here a direct proof of this equivalence, using only the known validity of the Iwasawa Main Conjecture over totally real fields.

中文翻译：

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关于大消失订单的Gross的两个猜想

我们在特殊情况下证明了消失猜想的Gross阶，其中所讨论角色的消失序可以任意大。在几乎所有先前已知的情况下，消失顺序为零或一。我们的证明的主要组成部分是该猜想与Gross–Kuz'min猜想的等价性。在这里，我们仅使用岩泽主猜想在完全真实域上的已知有效性来提供这种等效性的直接证明。