Conductors and newforms for U(1,1)
Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL2(F). This theory is analogous to the results of Casselman forGL2(F) and Jacquet, Piatetski-Shapiro, and Shalika forGLn(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.
Iwasawa main conjecture for Rankin–Selberg
p-adic L-functions
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersingular elliptic curves.
Kirillov theory for compact $p$-adic groups.
The purpose here is to describe a method by which one may obtain a reasonably explicit and "global" picture of the unitary representation theory of compact p-adic groups, and to indicate some applications. (By p-aάic, we refer to Q v , or local fields of characteristic zero.) The basic inspira-tion for such a description goes back to Kirillov's work on nilpotent lie groups. The main ingredients are the exponential map and the co-adjoint action. The Campbell-Hausdorff formula is used heavily as a tool.