Selmer varieties for curves with CM Jacobians

We study the Selmer variety associated to a canonical quotient of the Qp-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over Q whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multivariable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over Q for such curves. Let X/Q be a smooth proper curve of genus g ≥ 2 and b ∈ X(Q) a rational point. We assume that X has good reduction outside a finite set S of primes and choose an odd prime p / ∈ S. In earlier papers ([14], [15], [16], [17], [18]), a p-adic Selmer variety H f (G,U) was defined and studied, with the hope of applying its structure theory to the Diophantine geometry of X . Here, G = Gal(Q̄/Q), U = π Qp,un 1 (X̄, b) is the Qp-pro-unipotent étale fundamental group of X̄ = X ×Spec(Q) Spec(Q̄), and the subscript f refers to a collection of local ‘Selmer conditions,’ carving out a moduli space of torsors for U on the étale topology of Spec(Z[1/S, 1/p]) that satisfy the condition of being crystalline at p. The Selmer variety is actually a pro-variety consisting of a projective system · · ·→H f (G,Un+1)→H f (G,Un+1)→· · ·→H f (G,U2)→H f (G,U1) of varieties over Qp associated to the descending central series filtration U = U ⊃ U ⊃ U ⊃ U = [U,U] ⊃ · · · of U and the corresponding system of quotients Un = U \U that starts out with U1 = V = TpJ ⊗Qp, the Qp-Tate module of the Jacobian J of X . As a natural extension of the map X(Q) J(Q) H f (G, V )