ncts

Let $g \in S_{k}(\Gamma_{0}(N))$ be a normalized newform and $F$ be a harmonic Maass form that is good for $g$ . The $p$ -adic properties of the holomorphic part $F^{+}$ has been studied by many authors. K. Bringmann et al. corrected $F^{+}$ to a $p$ -adic modular form by a certain $p$ -adic constant $\alpha_{g}$ . When $g$ has a complex multiplication by an imaginary quadratic field $K$ and $p$ is split in $\mathcal{O}_{K}$ and p ∤ N, it is known that $\alpha_{g}=0$ . On the other hand, we do not know much about $\alpha_{g}$ for an inert prime $p$ . In this talk, we consider a weight $2$ normalized CM form $g$ with rational integer Fourier coefficients. Then we have the map $\phi$ from $X_{0}(N)$ to a CM elliptic curve $E_{g}$ by Eichler-Shimura construction. The speaker showed that $\text{deg}(\phi)\cdot \alpha_{g}$ is a $p$ -adic unit for almost all inert primes. I will explain this result in this talk.