21 problems

Rational Stark conjecture. There exists an element $u_\chi\in(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}$ such that

Let $F$ be a quadratic extension of $\mathbb{Q}$ and let $K$ be a nontrivial finite abelian extension of $F$. If $F$ is real quadratic, assume that one infinite place of $F$ stays…

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 $H/F$ 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…

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…

Harris--Venkatesh conjecture. There are an element $u\in\mathcal{U}(\operatorname{Ad}(\rho))$ and a positive integer $m$ such that, for any primes $p,\ell\geq 5$ coprime to $N$,

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 $d=n^2+3=p$ be a prime in the specified sequence, let $K$ be the real quadratic field used in the construction, and let $e^{i\vartheta_j}$ be the phases in the almost-flat fidu…

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

Harris period conjecture. These combinations give rational bases of the corresponding coherent cohomology groups; in particular, $\omega_f^{\Sigma_\infty}/\det R_f$ is rational.

For each $n\ge 0$, let $\mathbb{Q}_n$ be the $n$th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, let $M_n=M\mathbb{Q}_n$, let…

Let $\eta,\psi\in\mathcal{W}(\mathbb{C}_p)$ be finite-order characters of orders $p^m$ and $p^n$, respectively. Let $M_m$ and $M_n$ be the fixed fields of the kernels of…

Let $L/K$ be a finite Galois extension of number fields with group $G$, let $S$ contain all archimedean places, and let $r>1$ be an integer. For a complex character $\chi$ of $G$,…

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…

Gross's $p$-adic conjecture. One has

Let $E/F$ be a finite Galois extension of totally real number fields, with $G=\operatorname{Gal}(E/F)$. Let $p$ be a prime, let $\Sigma$ be a finite set of places of $F$ containing…

A part of Stark's conjecture. For every $\sigma\in G$,