In his seminal paper in 2001, Henri Darmon proposed a systematic construction of `$p$`

-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of local (`$p$`

-adic) cohomology classes/cycles in the `$p$`

-adic Galois representation attached to a cuspidal cohomological automorphic representation of `$\mathrm{PGL}_2$`

over any number field, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato conjecture. This work generalises previous constructions of Rotger-Seveso for elliptic cusp forms and earlier joint work with Williams for Bianchi cusp forms. Time permitting, I will also talk about the plectic analogues of these objects.

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Last updated: 13 Sep 2024