Generalized Heegner cycles on Mumford curves

Abstract

We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves, 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 $f_\infty$ and an imaginary quadratic field $K$. 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 necessarily 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 for the (one variable) anticyclotomic $p$-adic $L$-function of a modular form $f$ and an imaginary quadratic field $K$ at non-central critical integers.