Articles written in Proceedings – Mathematical Sciences

    • Invariant generalized ideal classes - structure theorems for $p$-class groups in $p$-extensions

      Georges Gras

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      We give, in sections 2 and 3, an english translation of: Classes g\acute{e}n\acute{e}ralis\acute{e}es invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: From Theory to Practice, SMM, Springer-Verlag, 2nd corrected printing 2005. We recall, in section 4, some structure theorems for finite $\mathbb{Z}_p[G]$-modules ($G\simeq\mathbb{Z}/p\mathbb{Z}$) obtained in: Sur les$\scr l$-classes d’id\acute{e}aux dans les extensions cycliques relatives de degr\acute{e} premier $\mathcal{l}$, Annales de l’Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a $p$-class group in a cyclic extension of degree $p$. In section 5, we apply this to the study of the structure of relative $p$-class groups of Abelian extensions of prime to $p$ degree, using the Thaine–Ribet–Mazur–Wiles–Kolyvagin ‘principal theorem’, and the notion of ‘admissible sets of prime numbers’ in a cyclic extension of degree $p$, from: Sur la structure des groupes de classes relatives, Annales de l’Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the $p$-ramification theory (as dual form of non-ramification theory) and which have become standard in a $p$-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.

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


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      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.

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