Control Theorem for Greenberg's Selmer Groups of Galois Deformations

✍ Scribed by Tadashi Ochiai

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 234 KB
• Volume: 88
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2611

✦ Synopsis

We give sufficient conditions for the Selmer group of a p-adic deformation of a motive over a number field to be controlled. Then we apply this result to the Selmer groups of various Galois representations. For example, we treat the cyclotomic deformations and the Hida deformations of the representations associated to modular forms.

2001 Academic Press

Mazur's control theorem [M]. Let E be an elliptic curve over a number field F. Assume that E has good ordinary reduction at all places of F dividing p. Let F ÂF be the cyclotomic Z p -extension and let 1 n be Gal(F ÂF n ). Then the kernel and the cokernel of the restriction map:

are finite groups whose orders are bounded independent of n.