On the Gross order of vanishing conjecture for large vanishing orders

TL;DR

This paper proves the Gross order of vanishing conjecture for cases with large vanishing orders, establishing new results beyond the previously known zero or one cases, by linking it to the Gross-Kuz'min conjecture.

Contribution

It provides the first proof of the conjecture for large vanishing orders, leveraging the equivalence to the Gross-Kuz'min conjecture and the Iwasawa Main Conjecture.

Findings

Proves the Gross order of vanishing conjecture in new cases with large vanishing orders.

Establishes the equivalence between the Gross conjecture and the Gross-Kuz'min conjecture.

Provides a direct proof using known Iwasawa Main Conjecture results.

Abstract

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.