3 problems

Let $\mathcal{R}$ be a primitive branch of a Hida family, let $\tilde{\mathcal{X}}(\mathcal{R})$ be the metaplectic covering of its arithmetic weight space, and let…

Assume that $E(F)$ has rank $1$, choose a generator $P_0$ of $E(F)$ modulo torsion, and for each admissible totally positive $t\in O_F$ let $P_t\in E(F)$ be the trace of the Darmon…

Let $f = \sum_{n\geq 1} a_n q^n \in S_{2k}(N)$ be a normalized newform with rational coefficients. Assume that the sign of the functional equation of $L(f,s)$ is $\varepsilon=-1$ a…