Title: Adjoint Selmer groups and cyclicity
Abstract: For a given elliptic cusp form f, we have a 2-dimensional p-adic Galois representation r with coefficients in a p-adic integer ring. Having r act on  SL(2)-Lie algebra by adjoint (conjugate action), we get a 3-dimensional representation Ad. We describe the formula of the order of the p-adic arithmetic cohomology group Sel(Ad) (called the adjoint Selmer group) via the L-value L(1,Ad)=L(1,Ad(f)) and explore the question when the Selmer group is cyclic (having one generator) over the coefficient ring?