Adjoint modular Galois representations and their Selmer groups

摘要

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 the p-adic Galois representation φ_0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ_0)) 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(φ_0)) from the proof of Wiles of the Shimura-Taniyama conjecture.