Test of Vandiver's conjecture with Gauss sums -- Heuristics
Résumé
The link between Vandiver's conjecture and Gauss sums is well-known since the papers of Iwasawa (1975) and Anglès--Nuccio (2010); this context has been considered by many authors with various purposes (Iwasawa theory, Galois cohomology, Fermat curves,...).
We prove again the 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, from a lecture we gave at the Laval University (1984). Then we provide a specific use of Gauss sums of characters of order p of F_\ell^\times allowing a necessary and sufficient condition for Vandiver's conjecture (Theorem 4.6 and corollaries 4.7, 4.8, using both the sets of exponents of p-irregularity and of p-primarity of suitable products of Jacobi sums obtained as twists of Gauss sums). We propose § 5.2 new heuristics and numerical experiments to strengthen our arguments in direction of Vandiver's conjecture and we show that any counterexample leads to excessive constraints modulo p on the above twists as \ell varies. All the calculations are given with their PARI/GP programs.
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