17 problems

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $H/\mathbb{Q}$ be a finite Galois extension, and let $\varrho\colon G_\mathbb{Q}\to\operatorname{GL}(V_\varrho)$ be an Artin rep…

$p$-adic Artin conjecture. The element $L_{p,\Sigma_F}(M(\rho))$ belongs to $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\max}]]$.

Let $A$ be a regular local ring with maximal ideal $\mathfrak{m}$, acted on by a finite group $G$, satisfying the assumptions of the Introduction, and let $k:=A/\mathfrak{m}$ be pe…

Let $\rho=\operatorname{Ind}_{\operatorname{G}_L}^{\mathbb{Q}}\zeta$ be an odd two-dimensional dihedral Artin representation, with $L$ imaginary quadratic, and let $S(\rho)$ be the…

Minimalist conjecture for twists. For $100\%$ of elliptic curves $E/\mathbb Q$,

Parity conjecture for Artin twists.

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 $T_p$ be the fixed Galois-stable lattice attached to $\rho$, let $A_\infty'$ be the inverse limit of the $p$-split ideal class groups of the fields $H_n$, and let $G$ act on…

Let $K/F$ be a finite Galois extension of number fields, set $\Delta=\operatorname{Gal}(K/F)$, and let $E$ be an elliptic curve over $F$. For an irreducible representation $\sigma$…

Let $\rho$ be a two-dimensional Artin representation, let $\rho=\begin{pmatrix}\tau&*\\0&\tau'\tau^{-1}\bar{\tau}\bar{\tau}'^{-1}\bar{\tau}^{-1}\tau'\end{pmatrix}$ as in the paper,…

Let $\mathcal{F}/\mathcal{K}$ be a finite Galois extension, let $A$ be an abelian variety over the global field $\mathcal{K}$, and let $A(\mathcal{F})_{\mathbb{C}}$ be the…

Let $\rho:G_{\mathbb{Q}}\longrightarrow GSp_4(\mathbb{C})$ be a symplectically odd Artin representation whose image does not factor through, up to conjugacy in $GSp_4(\mathbb{C})$,…

Let $G$ be a quasi-split reductive group over $\mathbb{Q}$, let ${}^L G$ be its $L$-group, and let $G_{\mathbb{Q}}={\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. The strong Artin co…

Let $G(x)$ and $P(x)$ denote the two sums defined in the source: $G(x)$ is the sum involving $\phi(n)/\operatorname{ord}_n(a)$ over integers $n\leq x$ coprime to $a$, and $P(x)$ is…

Let $f$ be the modular form under consideration, of weight $k$, and let $\phi$ be an Artin representation of $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ of dimension…

Let $E$) be an elliptic curve and let $\rho$ be an Artin representation, both defined over $\mathbb{Q}$. Define the twisted $L$-function by … where … $H_l^1(E)$ is the dual of the…

Let $A$ be an abelian variety over a number field $K$, let $F/K$ be a Galois extension, let $p$ be a prime, and let $\tau$ be a self-dual representation of…