Iwasawa theory of p-adic Lie extensions
Final Report Abstract
One of the big themes in number theory is the conviction that there should be deep links between analytic data, typically special values of a complex analytic L-function, and algebraic data, namely certain localisation kernels of Galois cohomology: Selmer groups. Sofar the most successful approach to prove theorems in this spirit is Iwasawa theory: The Iwasawa Main Conjectures predict the existence of a p-adic ζ-function that on the one hand interpolates special values of the analytic L-function of a motive M and on the other hand describes part of the Galois-module structure of Selmer groups attached to this motive. The strength of the approach of Iwasawa theory is, that it investigates whole towers of Galois extensions simultaneously. In classical Iwasawa theory, this tower is a Zp-extension, in non-commutative Iwasawa theory, one allows more general p-adic Lie groups. Most recently the first general case, viz for the Tate-motive over totally real fields, has been proven by Ritter and Weiss and independently by Kakde. However, very little is known for more complicated motives, even for elliptic curves. To generalise these proofs to other motives, it is crucial to gain a better understanding of the Selmer groups involved. In classical cases one is lead to study the µ- and the λ-invariant of the dual of the Selmer group. These invariants come out of the classification theory of Iwasawa modules and describe the size of the torsion part of an Iwasawa module. In fact, most approaches to prove instances of the Main Conjecture need the vanishing of the µ-invariant as an input. The technique, that proved most successful in studying the Iwasawa invariants is deformation theory. More precisely, instead of a single motive, one looks at a family of motives, such that the Galois representations attached to the members of the family all arise as certain quotients of a universal representation associated to the family. Families of modular forms were introduced to classical Iwasawa theory by Hida. P. Barth investigated how the Iwasawa invariants behave in families. His results in this direction can be summed up as follows: If the family is parameterising by one parameter, i.e., the coefficients of the representation lie in the power series ring Zp [[t]] rather then in Zp as for single motives, then, assuming some technical conditions, he can show, that the µ-invariant is p-adically locally constant and that the λ-invariant is constant for general families and a big class of Galois extensions. Moreover, applying the work of Fukaya and Kato, he was able to show, that their most general Tamagawa Number Conjectures imply a (non-commutative) Main Conjecture for families. C. Aribam applied deformation methods to show that the µ-invariant of the fine Selmer group vanishes in certain cases if the motive is adjoint to an elliptic curve with complex multiplication. The fine Selmer group is a subgroup of the Selmer group and believed to be relatively small. In a different direction Aribam investigated Selmer groups belonging to the universal representation: a representation having coefficients in a universal deformation ring. In particular, he considered the Selmer group for the adjoint representation of the universal one and was able to show a control theorem in that case, i.e., that the difference between the invariants of the Selmer group of the whole tower and the Selmer group for intermediate steps can be controlled. As another interesting result Aribam could show that in certain special situations the deformation ring coincides with a certain Hecke algebra. Results of this kind are called R = T theorems, and they are important, as they provide insight in the structure of the universal deformation ring. In particular, it is one of the very few methods known up today to construct examples of one parameter families as used in the work of Barth.
Publications
- On the non-commutative Main Conjecture for Elliptic Curves with Complex Multiplication. Asian J. Math. 14 (2010), no. 3, 385–416
A. Bouganis and O. Venjakob
- Iwasawa Theory for One-Parameter Families of Motives. PhD-thesis, University of Heidelberg (2011)
P. Barth
