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Algebraic cycles, modular forms and Euler systems. (English) Zbl 1045.11037

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.

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)
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References:

[1] Carayol, Progr Sur les repreÂsentations l - adiques associeÂes aux formes modulaires de Hilbert Ann EÂ, Math Sci pp 87– (1990)
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