72 problems

Let $L/K$ be a Galois extension of number fields with Galois group $G=\operatorname{Gal}(L/K)$. Let $\rho:G\to\operatorname{GL}(V)$ be an irreducible representation with character…

Colmez's conjecture. The identity

Let $\rho$ be a nontrivial irreducible representation, and let $L(s,\rho)$ denote its Artin $L$-function. Artin Conjecture. The function $L(s,\rho)$ is entire. Artin's conjecture w…

Artin's holomorphicity conjecture. For every nontrivial $\chi\in\mathrm{Irr}(G^+)$, the associated Artin $L$-function $L(s;L/\mathbb Q;\chi)$ is entire.

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, and fix $s_0\in\mathbb C\setminus\{1\}$. Let $\operatorname{Irr}(G)=\{\chi_1,\ldots,\chi_r\}$, write…

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…

Let $G$ be the Galois group of a Galois extension of $\mathbb{Q}$, and let $\alpha$ be a representation of $G$ in $\mathbb{C}^n$. Let $L(s,\alpha)$ be the associated Artin $L$-func…

An Artin representation is a finite-dimensional complex representation of a Galois group, and an automorphic representation is a representation of an appropriate general linear gro…

Let $E$ be a finite extension of $\mathbb Q$ with Galois group $Gal~(E|\mathbb Q)$, and let $\sigma_{n+1}:Gal~(E|\mathbb Q)\to GL_{n+1}(\mathbb C)$ be an irreducible representation…

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 $K/F$ be an extension of number fields with Galois group $G$, and let $\rho$ be an $n$-dimensional complex representation of $G$. Write $\mathbb{A}_F$ for the adeles of $F$, an…

Let $Q$ be as above, let $\chi$ be a simple odd Artin character of $Q$, and let $N$ be a subring of $\overline{\bf Q}$. Let ${\cal M}_0$ be a motive over $Q_0$, not necessarily hom…

Let $F$ be a number field, and let $(\rho,V)$ be a continuous $n$-dimensional complex representation of $\operatorname{Gal}(\overline F/F)$, with Artin $L$-function $L(s,\rho)$. An…

Nonvanishing conjecture. If ${\rm Gal}(L/\mathbb Q)$ admits no irreducible symplectic representations, then the associated Artin $L$-functions do not vanish at $s=1/2$.

Artin's conjecture. The completed function $\Lambda(s,\psi)$ is entire. Equivalently, $L(s,\psi)$ can be analytically continued to $\mathbb{C}$ apart possibly from a pole at $s=1$…

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

Let ${\mathcal{CM}}^0$ be the space of class functions associated with CM-types, and let $ht$ and $Z(\mathord\cdot,s)$ be the linear height and logarithmic-derivative functionals d…

Let $K/{\mathbb{Q}}$ be a Galois extension with Galois group $G$, and let $\pi$ be an irreducible complex representation of $G$. Let $L(\pi,s)$ be the associated Artin $L$-function…

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, and let $\chi$ be an irreducible character of $G$. Let $L(s,\chi,K/\mathbb Q)$ denote the associated Artin $L$…

Let $F'/F$ be a $G$-Galois extension of number fields, where $G$ is finite, and let $\rho\colon G\to \mathrm{GL}_N(\mathcal{O}_E)$ be an $E$-linear Galois representation for a numb…

Artin GRH. Artin $L$-functions should satisfy the generalized Riemann hypothesis and therefore have no zeros with real part greater than $1/2$.

Artin conjecture. For any non-trivial, irreducible $\rho$, the function $L(\rho,s)$ has an analytic continuation to the whole complex plane. The source defines the Euler product in…

Artin's noncommutative-torus conjecture. Every Artin $L$-function can be written as the Euler product of the local zeta functions associated to $2n$-dimensional noncommutative tori…

Let $G$ be a Galois group and let $\mathscr{A}_{RM}^{2n}$ be a suitable even-dimensional noncommutative torus associated with $G$. Let $\sigma_n$ be an $n$-dimensional representati…

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…