9 problems

Let $k$ be a totally real number field, let ${\mathfrak{f}}\neq\mathcal{O}_k$ be a proper integral ideal, let $K=k({\mathfrak{f}})$, let $G=\operatorname{Gal}(K/k)$, and let $S$ be…

Let $k$ be a totally real number field, let ${\mathfrak{f}}\neq\mathcal{O}_k$ be a proper integral ideal, let $K=k({\mathfrak{f}})$ be the associated ray-class field, and write…

A SIC in dimension $d$ is a set of $d^2$ equiangular unit vectors obtained from a fiducial vector by a Weyl–Heisenberg orbit. In the setting above, $d$ is also the modulus defining…

Fix $d\geq4$, write $\Delta=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and let $f'\mid f$. Let $\mathcal M$ satisfy the extended order-to-multiplet conject…

Let $K$) be an imaginary quadratic field with discriminant $d_K$, let $(2)=\mathfrak p_1\mathfrak p_2$, let $h(d_K)$ denote the class number, and let $\textsf{K}_1$ denote the rele…

Let $K$ be a CM-field with reflex field $K^*$, let $\mathfrak{f}$ be a modulus of $K$, and let $\mathfrak{f}^*$ be the corresponding modulus of $K^*$. Write … for the natural homom…

Let $K$ be a totally real field containing a unit of norm $-1$ and satisfying condition $\mathbf{(G1)}$, let $n=|G|$, and let $L_p$, $H$, $\zeta_p^+$, and $P_l$ be as defined in th…

Let $K$ be an imaginary quadratic field and let $\mathfrak{f}$ be a nontrivial integral ideal of $K$. Let $K_\mathfrak{f}$ be the ray class field modulo $\mathfrak{f}$, let…

Let $K$ be a real quadratic field with $p$ inert, let $f$ and $n$ be prime to $pNd_K$, and let $u_C(r,\tau)\in K_p^{\times}$ be the invariant attached to a modular-unit family, wit…