STNB2017, (31st edition)
CM cycles on Kuga--Sato varieties over Shimura curves and Selmer groups
Presenters
Abstract
Let $f$ be a normalized Hecke eigenform of even weight $\geq 4$ and level $N$, and let $K$ be an imaginary quadratic field. Under the Heegner hypothesis that every prime dividing $N$ splits in $K$, Nekov\'a\v{r} extended Kolyvagin's method of Euler systems to this higher weight situation by replacing CM points on modular curves by certain algebraic (CM) cycles on the Kuga--Sato variety on which the Galois representation associated to $f$ is realized. As a consequence, he was able to bound the Selmer group of $f$ under the assumption that certain cohomology class in the bottom layer of the Euler system does not vanish. In a recent joint work with Y. Elias, we have adapted Nekov\'a\v{r}'s result to the scenario where the Heegner hypothesis is relaxed in such a way that one moves to Shimura curves. The Galois representation attached to $f$ can be realized as well in the cohomology of a self-fold fiber product of the universal abelian surface over a Shimura curve, where one can still construct a family of algebraic cycles leading to an Euler system. I will sketch how to construct such cycles in this setting, and explain their main properties giving rise to the desired Euler system to which one can apply Kolyvagin's machinery.
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