A control theorem for the torsion Selmer pointed set
Let \(F\) be a number field, let \(p\) be an odd prime, and let \(F_n\) be the \(n\)th level of the cyclotomic \(\mathbb Z_p\)-extension of \(F\), with \(\text{Gal}(F_{\infty}/F_n)=\Gamma_n\). \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl. 0245.14015)] showed that if \(E\) is an elliptic curve over \(F\) that is ordinary at all primes above \(p\), then the maps \[ \text{ Sel}_p(E, F_n) \to \text{Sel}_p(E, F_{\infty})^{\Gamma_n} \] have finite kernel and cokernel whose orders are bounded independently of \(n\). The main result of the present paper gives a non-abelian generalization of this theorem by defining a torsion analogue of the Selmer variety of \textit{M. Kim} [Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl. 1165.14020)]. Let \(R\) be a finite flat commutative \(\mathbb Z_p\)-algebra and let \(R^{\text{mon}}\) be the monoid associated to the multiplicative structure of \(R\). The author considers a subcategory \((R^{\text{mon}}\)-\(P\)-\(\text{Sets})\) of pointed sets and defines the notion of \(p\)-exponent of cokernels in this subcategory. Also, for a smooth curve \(X\) over \(F\), a torsion analogue \(H_f^1(F, \mathfrak{g}^{\le m}(X)_{R/(p^r),a})\) of the Selmer variety is defined in this subcategory, where \(\mathfrak{g}^{\le m}(X)_{R/(p^r),a}\) is the set of \(R/(p^r)\)-valued points of a certain algebraic group. Assume \(F\) is a finite abelian extension of \(\mathbb Q\) with Galois group \(\Delta\) of order prime to \(p\). Also, assume that the completion of \(F\) at each prime above \(p\) is linearly disjoint from \(\mathbb Q_p(\mu_p)\) over \(\mathbb Q_p\). Let \(X\) be a smooth curve over \(\mathbb Q\) that is isomorphic over \(\mathbb Q\) to the projective line minus finitely many \(\mathbb Q\)-rational points, or an elliptic curve with good ordinary reduction at \(p\), minus the origin, or a proper smooth curve whose Jacobian is isogenous to a product of elliptic curves with good ordinary reduction at \(p\) and satisfying a certain distinctness condition for their Galois representations. The author proves that for any character \(\chi\) of \(\Delta\) such that the restrictions of \(\chi\) and \(\chi^2\) to the decomposition subgroup \(\Delta_p\) of \(\Delta\) are non-trivial, the \(p\)-exponents of the kernel and cokernel of the map \[ H_f^1(F_n, \mathfrak{g}^{\le m}(X)_{\mathbb Z_p[\chi]/(p^r), a})^{\langle \chi\rangle} \to H_f^1(F_{\infty},\mathfrak{g}^{\le m}(X)_{\mathbb Z_p[\chi]/(p^r), a})^{\langle \chi\rangle,\Gamma_n} \] are finite and bounded independently of \(r\) and \(n\).
scientific article; zbMATH DE number 6929333
scientific article