8 problems

Let $k$ be totally real, let $K/k$ be an abelian CM extension with group $G$, let $p$ be a rational prime, and let $\chi$ be a one-dimensional character of $G$. Define … Gross's hi…

In the abelian rank-one setting, let $\epsilon$ be the minus unit singled out by Stark's complex conjecture, let $\mathfrak q\mid p$ be the chosen prime, and let…

Let $F$ be a totally real field in which $p$ is inert. Let $\tauin X_p$ be a point attached to an oriented integral ideal as above, let $H$ be the narrow Hilbert class field of $F$…

Let $K$ be an imaginary quadratic field, let $p$ split as $(p)=\mathfrak q\overline{\mathfrak q}$, and let $\chi$ be the nontrivial character of $\operatorname{Gal}(K/\mathbb Q)$.…

Let $e=\dim\mathrm{H}^0(\mathbb{Q}_p,W_p^-)$, let $\rho^*$ be the dual Artin representation, let $\omega$ be the cyclotomic character, and let $R_p(W_p^*)$ be Gross's $p$-adic regu…

Let $g$ be a weight one modular form, let $g^*$ be its dual, and let $u_1$ denote the unit attached to the associated Galois representation. Let $\kappa(g_{\alpha},g_{1/\alpha}^*)$…

Let $K=\mathbb{Q}(\sqrt{d_K})$ be real quadratic, let $L=K(\mathfrak{f}\infty)^{\langle\sigma_{\wp}\rangle}$, and let $u_D(\mathfrak{a},\mathfrak{f})$ be Dasgupta's p-adic invarian…

Let $F$ be a number field, let $M/F$ be a finite abelian extension, let $S$ and $T$ satisfy the stated assumptions, and let $\mathfrak{p}\in S$ with $R=S\setminus\{\mathfrak{p}\}$.…