292 problems

Let $C/\mathbb{Q}$ be the specific plane quartic defined in Conjecture 1.6 of Furio and Lombardo. The conjecture asserts that $C(\mathbb{Q})$ consists of exactly four rational poin…

For every odd prime $p$ and every imaginary quadratic field $K$ whose $p$-class rank is $2$, if the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified pro-$p$ extension…

For every global function field $K$ of characteristic $p$, every integer $n\ge 1$, and every effective divisor $D$ on $K$, there are only finitely many isomorphism classes of conti…

Langlands correspondence for $\mathrm{GL}_2/K$. The representation $\rho_{\lambda}$ exists as above, is potentially semistable at every place $v$, and its associated representation…

Let $X$ be a smooth projective variety defined over a finitely generated subfield $F\subset\mathbb{C}$. Under the Artin comparison identification … let…

Serre's uniformity question. There exists a bound $C$ such that for every non-CM elliptic curve $E/\mathbb{Q}$ and every prime $\ell>C$, the representation $\rho_{E,\ell}$ is surje…

Let $K$ be a totally real field and let $p\geq3$ be prime. A continuous representation … is geometric if it satisfies the geometricity conditions of Fontaine and Mazur. The Fontain…

Let $K$ be a number field, let $p$ be a rational prime unramified in $K$, and let $G_K$ denote the absolute Galois group of $K$. Let…

Maeda's conjecture. The Galois conjugates satisfy

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, a…

Rasmussen–Tamagawa conjecture. The set $\mathcal{H}(K,g)$ is finite for any choice of $K$ and $g$. This strengthens the known finiteness of $\mathcal{H}(K,g,\ell)$ when the prime…

Let $L/\mathbb{Q}_p$ be a finite extension, let $E$ be a coefficient field, let $W_L$ be the set of Serre weights, and let $\overline{\rho}_L:G_L\to\mathrm{GL}_2(\mathbb{F})$ be a…

Let $E/F$ be a finite extension, let $\rho:\operatorname{Gal}(E/F)\rightarrow GL_n(\mathbb{C})$ be a representation, and let $L(\rho,s)$ denote its Artin $L$-function. Langlands' f…

Let $F$ be a finite extension of $\mathbb Q$, let $\mathbb A_F$ be its ring of adeles, and fix a prime $p$ together with an isomorphism $\overline{\mathbb Q}_p\cong\mathbb C$. Cons…

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let … be the exact sequence for the arithmetic fundamental…

Let $K$ be a totally real field and $L$ a number field. Let $A/L$ be an abelian variety of $\mathrm{GL}_2$-type such that … For a prime $\mathfrak{p}$ of $K$ above a rational prime…

Frey–Mazur conjecture. If $E[p]$ and $E'[p]$ are isomorphic as Galois modules for some prime $p>17$, then $E$ and $E'$ are $\mathbb{Q}$-isogenous.

Buzzard–Gee conjecture. The following are true. There exists a map

Let $E/\mathbb Q$ be an elliptic curve, let $p$ be a prime, and let $G\subseteq \operatorname{GL}(2,\mathbb Z/p\mathbb Z)$ be the image of the mod-$p$ Galois representation…

Generalized Tate conjecture. There is a closed algebraic subset $Z\subset X$ of codimension $k$ such that

Let $K$ be a finitely generated field, let $X$ be a smooth projective variety over $K$, and consider a $\mathds{Q}$-compatible system of Galois representations … Let…

Let $(R,T,\zeta)$ be a triple as in the paper's setup, where $T$ is an $R$-representation of $G_{\mathbb Q_p}$ and $\zeta$ is a $\mathbb Z_p$-basis of…

Let $F$ be a totally real field in which $p$ is unramified, let $G_F$ be its absolute Galois group, and let $\Sigma_v$ denote the embeddings associated with a place $v\mid p$. Let…

Mazur–Rubin's conjecture. If $E_1$ and $E_2$ are $p$-Selmer near-companions over $K$, then there exists a $G_K$-module isomorphism

Gouvea's dimension conjecture. In the presentation proposition for the universal deformation ring, equality always holds in the Krull-dimension bound; equivalently,