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Gras conjecture

In algebraic number theory, the Gras conjecture relates the p-parts of the Galois eigenspaces of an ideal class group to the group of global units modulo cyclotomic units. It was proved by Mazur & Wiles (1984) as a corollary of their work on the main conjecture of Iwasawa theory. Kolyvagin (1990) later gave a simpler proof using Euler systems. A version of the Gras conjecture applying to ray class groups was later proven by Timothy All.

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In algebraic number theory, the Gras conjecture (Gras 1977) relates the p-parts of the Galois eigenspaces of an ideal class group to the group of global units modulo cyclotomic units. It was proved by Mazur & Wiles (1984) as a corollary of their work on the main conjecture of Iwasawa theory. Kolyvagin (1990) later gave a simpler proof using Euler systems. A version of the Gras conjecture applying to ray class groups was later proven by Timothy All.

References

References

Gras, Georges (1977), "Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés", Annales de l'Institut Fourier, 27 (1): 1–66, doi:10.5802/aif.641, ISSN 0373-0956, MR 0450238
Kolyvagin, V. A. (1990), "Euler systems", The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Boston, MA: Birkhäuser Boston, pp. 435–483, doi:10.1007/978-0-8176-4575-5_11, ISBN 978-0-8176-3428-5, MR 1106906
Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae, 76 (2): 179–330, Bibcode:1984InMat..76..179M, doi:10.1007/BF01388599, ISSN 0020-9910, MR 0742853, S2CID 122576427