title: Adjoint modular Galois representations and their Selmer groups.
abstract: In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(phi) of a two-dimensional modular Galois representation phi. We start with the p-adic Galois representation phi0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(phi0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(phi0)) from the proof of Wiles of the Shimura-Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let phi denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting phi0, and the characteristic power series of the Selmer group Sel(ad(phi)) is given by a p-adic L-function interpolating L(1, ad(phik)) for weight k + 2 specialization phik of phi. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(phi) [symbol, see text] nu-1), where nu is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.
authors.affiliation: Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA.
authors.forename: H
authors.initials: H
authors.lastname: Hida
authors.forename: J
authors.initials: J
authors.lastname: Tilouine
authors.forename: E
authors.initials: E
authors.lastname: Urban