Gras conjecture

{{short description|Result on the p-parts of the Galois eigenspaces of an ideal class group}}

In algebraic number theory, the Gras conjecture {{harv|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 {{harvtxt|Mazur|Wiles|1984}} as a corollary of their work on the main conjecture of Iwasawa theory. {{harvtxt|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

{{Citation | last1=Gras | first1=Georges | title=Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés | url=http://aif.cedram.org/item?id=AIF_1977__27_1_1_0 |mr=0450238 | year=1977 | journal=Annales de l'Institut Fourier | issn=0373-0956 | volume=27 | issue=1 | pages=1–66| doi=10.5802/aif.641 | doi-access=free }}
{{Citation | last1=Kolyvagin | first1=V. A. | title=The Grothendieck Festschrift, Vol. II | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-3428-5 | doi=10.1007/978-0-8176-4575-5_11 |mr=1106906 | year=1990 | volume=87 | chapter=Euler systems | pages=435–483}}
{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Wiles | first2=Andrew | author2-link=Andrew Wiles | title=Class fields of abelian extensions of Q | doi=10.1007/BF01388599 |mr=742853 | year=1984 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=76 | issue=2 | pages=179–330| bibcode=1984InMat..76..179M | s2cid=122576427 }}

Category:Theorems in algebraic number theory

Category:Conjectures that have been proved