34 problems

Let $K/F$ be an abelian extension of number fields, let $S$ be a finite set of places containing all infinite places of $F$ and all places ramified in $K/F$, and suppose that a dis…

Let $K$ be a complex cubic field, let $K(\mathfrak{f})$ be the relevant narrow ray class field, let $e_{\mathfrak{f}}$ be the number of roots of unity in $K(\mathfrak{f})$, and fix…

Let $F$ be a real quadratic field, let Z_\\A(s) be the difference of two ray class partial zeta functions associated with a ray class, and let $\exp(-Z_\A'(0))$ be the correspond…

Let $f$) be a newform of weight $1$ for $\Gamma_1(N)$, with associated two-dimensional complex Galois representation $\rho_f$, and let $\operatorname{Ad}(\rho_f)$ be its adjoint re…

Stark's conjecture. The quotient $R_{K_0}/R_K$ is the determinant of an $(mn-n)\times(mn-n)$ matrix. This matrix should be diagonalized into $m-1$ blocks, each an $n\times n$ matri…

Stark's conjecture. The coefficient $R_K(\chi)$ has the form of a regulator: it is the determinant of a matrix whose entries are linear combinations of logarithms of units of $K_0$…

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of th…

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global…

The motivic action conjecture. The classes

Let $F$ be a real quadratic field, let $f$ be a Hilbert modular form of parallel weight one, and let…

Let $H/F$ be an abelian extension of number fields, let $W$ be the number of roots of unity in $H$, and let $S$ be a finite set of places of $F$ containing the ramified and infinit…

Stark's conjecture. Except in the trivial case $\e_\A=1$, one has

Let $f$ be a newform of weight $1$ and level $N$, with associated representation $\rho_f$. Let $\mathcal{U}(\operatorname{Ad}(\rho_f))$ be the equivariant unit module, and let…

Let $K/k$ be a CM extension, let $L/k$ be a finite abelian extension with group $G$, let $H$ be the one-dimensional torus over $k$ split by $K$, and let $I$ be the augmentation ide…

Let $L/k$ be a finite abelian extension with group $G$, let $S$ be a finite set of places containing the Archimedean and ramified places, let $T$ be an auxiliary set, and let $I$ b…

Let $K/k$ be an abelian CM extension with group $G$, let $S$ contain the Archimedean places, the places dividing $p$, and the ramified places, and let $\mathfrak p\mid p$ split com…

Let $K/k$ be a CM Galois extension with group $G$ and complex conjugation $c$. Let $U^{-1}$ be the group of minus $p$-units of $K$, let $X^{-1}$ be the minus part of the degree-zer…

Assume that $k$ is totally real, $K/k$ is a CM Galois extension with group $G$, $c\in G$ is complex conjugation, and $V$ is a totally odd representation of $G$ over $\mathbb C_p$.…

Assume that $k$ is totally real, $K/k$ is a CM abelian extension with group $G$, $p$ is a rational prime, and $\chi$ is an odd character of $G$, so that $\chi(c)=-1$. Let…

Let $K/k$ be a Galois extension with group $G$, let $X$ be the degree-zero subgroup of the free abelian group on the Archimedean places of $K$, and let $A^*$ be the unit group of…

Let $K$ be the real quadratic field in the construction, and let the Stark units be the units derived directly from Stark's predicted $L$-function values. Tate–Brumer abelianity co…

Let $K^{\mathfrak p\mathfrak j}$ be the ray class field appearing in Theorem, let $\epsilon_\sigma$ be its Stark unit, and let $\xi_\ell$ and $x_0$ be the associated quantities in…

Let $K$ be a number field, let $L/K$ be a finite Galois extension with group $G$, let $S$ be a finite set of places of $K$ containing $S_{\infty}$ and all places ramified in $L/K$,…

Let $K$ be a real quadratic field and let $\{\rho_1,\rho_2\}$ be its real embeddings. Let $\cc$ be an ideal of the maximal order $\OO_K$, let $S=\{\rho_2\}$, and let $A$ be a ray i…

Stark conjecture. If $\varrho$ does not contain the trivial representation, then