Darmon cycles and the Kohnen-Shintani lifting

Abstract:

Let $\mathbf {f}(q)$ be a Coleman family of cusp forms of tame level $N$. Let $k_{0}$ be the classical weight at which the specialization of $\mathbf {f}(q)$ is new. By the Kohnen-Shintani correspondence, we associate to every even classical weight $k$, a half-integral weight form (for $k \neq k_{0}$) $g_{k} = \sum \limits _{D > 0} c(D, k)q^D \in S_{\frac {k+1}{2}}(\Gamma _{0}(4N))$ and $g_{k_{0}} = \sum \limits _{D > 0} c(D, k)q^D \in S_{\frac {k+1}{2}}(\Gamma _{0}(4Np))$.

We first prove that the Fourier coefficients $c(D, k)$ for $k \in 2\mathbb {Z}_{> 0}$ can be interpolated by a $p$-adic analytic function $\tilde {c}(D, \kappa )$ with $\kappa$ varying in a neighbourhood of $k_{0}$ in the $p$-adic weight space. For discriminants $D$ such that $\tilde {c}(D, k_{0}) = 0$, which we call Type II, we show that $\frac {d}{d\kappa }[\widetilde {c}(D, \kappa )]_{k=k_{0}}$ is related to certain algebraic cycles associated to the motive $\mathcal {M}_{k_{0}}$ attached to the space of cusp forms of weight $S_{k_{0}}(\Gamma _{0}(Np))$. These algebraic cycles appear in the theory of Darmon cycles.