Weston, Tom Algebraic cycles, modular forms and Euler systems. (English) Zbl 1045.11037 J. Reine Angew. Math. 543, 103-145 (2002). Author’s summary: Let \(\rho_f\) be the \(l\)-adic representation associated to a newform \(f\) of weight \(\geq 2\), squarefree level, and arbitrary character. In this paper, subject to some mild hypotheses on \(f\), we construct a cohesive Flach system (which is a variant of an Euler system due to Mazur) for the adjoint representation \(\text{ad}^0\rho_f\). This in turn yields an annihilator of the Selmer group of \(\text{ad}^0\rho_f\) with applications to the deformation theory of the residual representation \(\overline\rho_f\). The cohesive Flach system is constructed as the image under an Abel-Jacobi map of Hecke correspondences and modular units on Kuga-Sato varieties. Our key technical result relates the ramification of classes in the image of the Abel-Jacobi map to divisor maps in positive characteristic. Cited in 3 Documents MSC: 11F80 Galois representations 11F11 Holomorphic modular forms of integral weight 11R34 Galois cohomology 19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) Keywords:cohesive Flach system; adjoint representation; Selmer group; Abel-Jacobi map; Hecke correspondences; modular units; Kuga-Sato varieties; divisor maps PDFBibTeX XMLCite \textit{T. Weston}, J. Reine Angew. Math. 543, 103--145 (2002; Zbl 1045.11037) Full Text: DOI References: [1] Carayol, Progr Sur les repreÂsentations l - adiques associeÂes aux formes modulaires de Hilbert Ann EÂ, Math Sci pp 87– (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.