• New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments

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    • Keywords

       

      Cyclotomic field of $p$-th roots of unity; Vandiver’s conjecture; Gauss sums; Jacobi sums; Kummer theory; class field theory; $p$-ramification

    • Abstract

       

      The link between Vandiver’s conjecture and Gauss sums is well known sincethe papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459,1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Anglès and Nuccio (Acta Arith 142(3):199–218, 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 $p$-th cyclotomic field (Sur la $p$-ramification abélienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order $p$ of $\mathbb{F}^{\times}_{ell}$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of $p$ irregularity and of $p$-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in $\S$5.2 new heuristics showing that any counter example 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 the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.

    • Author Affiliations

       

      GEORGES GRAS1

      1. Villa la Gardette, Chemin Château Gagnière, 38520 Le Bourg d’Oisans, France
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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