The Prime-to-Adjoint Principle and Unobstructed Galois Deformations in the Borel Case

✍ Scribed by Gebhard Böckle; Ariane Mézard

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 306 KB
• Volume: 78
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2384

✦ Synopsis

For a given odd two-dimensional representation \Ä over F p of the absolute Galois group G E of a totally real field E which is unramified outside a finite set of places S, Mazur defined a universal deformation ring R G S (\Ä ). By obstruction theory, the group 2 S (E, ad \Ä ) measures to what extent R G S ( \Ä ) is determined by local relations. Using devissage on ad \Ä , we give criteria for the vanishing of 2 S (E, ad \Ä ) in terms of vanishing of S-class groups, in terms of Iwasawa invariants, and in terms of special values of p-adic L-functions. If S is the set of places above p and , the condition 2 S (E, ad \Ä )=0 implies that R G S ( \Ä ) is free of dimension 2[E : Q]+1. In this case, we obtain a reformulation of Vandiver's conjecture and asymptotic connections between Greenberg's conjecture and the freeness of R G S ( \Ä ). For larger S, we relate the freeness of the universal deformation ring for minimal deformations to the vanishing of a modified obstruction group 2 S, S p (E, ad \Ä ). Based on this, we can calculate non-free rings R G S ( \Ä ) for some explicit reducible \Ä coming from the action of G Q on p-torsion points of elliptic curves.