We show that the slopes of the $U_5$ operator acting on 5-adic overconvergent modular forms of weight $k$ with primitive Dirichlet character $\chi$ of conductor 25 are given by either$$\left\{\frac{1}{4}·⌊\frac{8i}{5}⌋:i\in \mathbb{N}\right\}\text{or}\left\{\frac{1}{4}·⌊\frac{8i+4}{5}⌋:i\in \mathbb{N}\right\},$$depending on $k$ and $\chi$.We also prove that the space of classical cusp forms of weight $k$ and character $\chi$ has a basis of eigenforms for the Hecke operators $T_p$ and $U_5$ which is defined over ${\mathbf{Q}}_5(\sqrt[4]{5},\sqrt{3})$.

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for ${\text{GL}}_2\times {𝒦}^{\times }$ by the method of Eisenstein congruence on $GU(3,1)$, where $𝒦$ is an imaginary quadratic field. We construct a $p$-adic family of ordinary Eisenstein series on the group of unitary similitudes $GU(3,1)$ with the optimal constant term which is basically the product of the Kubota-Leopodlt $p$-adic $L$-function and a $p$-adic $L$-function for ${\text{GL}}_2\times {𝒦}^{\times }$. This construction also provides a different point of view of $p$-adic...

We give a geometric definition of overconvergent modular forms of any $p$-adic weight. As an application, we reprove Coleman’s theory of $p$-adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.