31 problems

Let $k$ be an imaginary quadratic field, let $F/k$ be a finite abelian extension with Galois group $G$, let $r<0$, and let $onzerocharchiinwidehat{G}$ be a character. Write…

Fix an odd prime $p$, a field isomorphism $\iota:\mathbb{C}\simeq\mathbb{C}_p$, a character $\psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G})$, and a finite set of places $S$…

Colmez's conjecture. The identity

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 $L/K$ be a Galois extension with Galois group … For every irreducible character $\chi\colon G\to\mathbb{C}$, let $L(s,\chi)$ denote the associated Artin $L$-function. Artin's h…

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…

Let $K$ be a global field and let $\rho$ be an Artin representation with Artin $L$-function $L(s,\rho)$. Set $m=\operatorname{ord}_{s=1/2}L(s,\rho)$. Convergence conjecture. There…

Let $K$ be a global field and let $\rho\colon\operatorname{Gal}(K^{\mathrm{sep}}/K)\to\operatorname{GL}_n(\mathbb C)$ be a nontrivial irreducible representation. For a prime $v$ of…

Deep Riemann hypothesis. The limit

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 $L/K$,…

Stark's conjecture. For every automorphism $\alpha$ of $\mathbb{C}$,

Let $G$ be a finite group and let $N\unlhd G$ be a normal subgroup. For an irreducible representation $\rho$ of $G$, write $\ker\rho$ for its kernel; for a subgroup $H\subseteq G$,…

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

Let $L/k$ be a Galois extension of number fields with ${\mathrm {Gal}}(L/k) \cong G$. Let $\rho$ be a complex representation of $G$ of dimension $m$. Strong Artin conjecture. There…

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Let $W(\chi)$ be the root number of the Artin $L$-function attached to $\chi$, and let…

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Let $\chi_0$ denote the trivial character, and classify irreducible characters of…

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Write … where the union is a multiset. LI− conjecture. The multiset…

Let $k$ be a totally real number field and let $\chi:G_k\to\overline{\mathbb Q}_p$ be an even $p$-adic Artin character. Let $G_\chi(T)$ be its Iwasawa power series. p-adic Artin conj…

Let $E/{\mathbb Q}$ be an elliptic curve. For every Artin representation $\rho$ over ${\mathbb Q}$, suppose there is an invariant $\BSD(E,\rho)\in{\mathbb C}^{\times}$. If $\rho$ a…

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 a Galois extension of number fields with Galois group $G$. Let $(X,K)$ be a supercharacter theory of $G$, and let $\operatorname{Sup}(G)$ be the set of supercharacters…

Let $L/k$ be a finite Galois extension of number fields with $\operatorname{Gal}(L/k)\simeq G$, and let $\rho$ be an $m$-dimensional complex representation of $G$. Let $L(s,\rho)$…

Let $L/k$ be a finite normal extension of number fields with $\operatorname{Gal}(L/k)\simeq G$, let $\rho$ be a complex representation of $G$, and let $L(s,\rho,L/k)$ be the associ…

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$ and let $r_S(\psi)$ be the order of vanishing at $s=0$ of the corresponding complex $S$-truncated Artin $L$-function. Gros…

Stark's conjecture over the rationals. For every $\chi\in\widehat{G}$, one has $A(\chi,\mathcal{A})\in\overline{\mathbb{Q}}$ and