We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves in [BDP13], in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family f∞ and an imaginary quadratic field K, constructed in [BD07] and [Sev14]. While in [BD07] and [Sev14] only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necesserely central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel-Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu [Mas12] for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.
Let π be a cuspidal, cohomological automorphic representation of an inner form G of PGL 2 over a number field F of arbitrary signature. Further, let p be a prime of F such that G is split at p and the local component πp of π at p is the Steinberg representation. Assuming that the representation is non-critical at p we construct automorphic L-invariants for the representation π. If the number field F is totally real, we show that these automorphic Linvariants agree with the Fontaine-Mazur L-invariant of the associated p-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight 2 to arbitrary cohomological weights.
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