Perfect forms and the Vandiver conjecture

TL;DR

This paper proves that under certain conditions relating to the size of p, a specific eigenspace of the class group in p-cyclotomic extensions is trivial, using advanced number theory and quadratic form techniques.

Contribution

It establishes a new result connecting the size of p with the triviality of eigenspaces in class groups, employing K-theory and Voronoi reduction methods.

Findings

Eigenspace on C attached to (p-n)-th power of Teichmuller character is trivial when log(p) exceeds a bound.

Uses K-theory of integers and Voronoi reduction theory in the proof.

Provides new insights into the structure of class groups in cyclotomic fields.

Abstract

Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi reduction theory of quadratic forms.