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Pré-Publication, Document De Travail Année : 2018

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 give again the interpretation of Vandiver's conjecture in terms of the minus part of the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field, that we had published at the Laval University (1984). Then we provide a specific use of Gauss sums allowing a numerical test of Vandiver's conjecture (cf. Theorem 4.5 using both the set of exponents of p-irregularity and the sets of exponents of p-primarity of a suitable twist of Gauss sums giving Jacobi sums of characters of order p). Then we propose new heuristics for its rightness. We show that a counterexample to Vandiver's conjecture leads to a strange phenomenon on the congruential properties modulo p of Gauss sums and to an unusual complexity of algorithms. All the techniques used are basic and classic. Many tables are given to strength our arguments; the corresponding PARI programs may be copy and past by the reader.
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Dates et versions

hal-01856083 , version 1 (09-08-2018)
hal-01856083 , version 2 (29-08-2018)
hal-01856083 , version 3 (08-12-2018)
hal-01856083 , version 4 (25-06-2019)

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Georges Gras. Test of Vandiver's conjecture with Gauss sums -- Heuristics. 2018. ⟨hal-01856083v2⟩
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