Gras conjecture

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

• 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, doi:10.1007/BF01388599, ISSN 0020-9910, MR 0742853, S2CID 122576427