Adjoint modular Galois representations and their Selmer groups

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(φ) of a two-dimensional modular Galois representation φ. We start with thep -adic Galois representation φ0 of a modular elliptic curveE and present a formula expressing in terms ofL (1, ad(φ0 )) the intersection number of the elliptic curveE 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(φ0 )) 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 universalp -ordinary Hecke algebra, and the second variable is the cyclotomic variableS . In the one-variable case, we let φ denote thep -ordinary Galois representation with values inGL 2 (Zp [[T ]]) lifting φ0 , and the characteristic power series of the Selmer group Sel(ad(φ)) is given by ap -adicL -function interpolatingL (1, ad(φk )) for weightk + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp [[T ,S ]] of Sel(ad(φ) ⊗ ν−1 ), where ν 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 usingp -adic Siegel modular forms.