Test of Vandiver's conjecture with Gauss sums -- Heuristics

TL;DR

This paper explores the connection between Vandiver's conjecture and Gauss sums, offering new criteria and heuristics, supported by numerical experiments, to assess the conjecture's validity in cyclotomic fields.

Contribution

It introduces novel criteria based on Gauss sums for Vandiver's conjecture and proposes heuristics suggesting the conjecture's likely truth, supported by computational evidence.

Findings

New criteria for Vandiver's conjecture involving Gauss sums.

Heuristics indicating counterexamples impose excessive constraints.

Numerical experiments supporting the conjecture's validity.

Abstract

The link between Vandiver's conjecture and Gauss sums is well known since the papers of Iwasawa (1975), Thaine (1995-1999) and Angl{\`e}s-Nuccio (2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver's conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field (1984). Then we provide a specific use of Gauss sums of characters of order p of F\_ell^x and prove new criteria for Vandiver's conjecture to hold (Theorem 1.2(a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 1.2(b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in \S5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as ell varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in direction of the very probable truth of Vandiver's conjecture. All the calculations are given with their PARI/GP programs.