14 problems

Estes–Guralnick conjecture. Every such polynomial $f$ is the minimal polynomial of some integer symmetric matrix.

Weak form of Greenberg's conjecture. If $X(k^\mathrm{c}_\infty)$ is nontrivial, then it has a nontrivial finite $bZ_p\llbracket\operatorname{Gal}(k^\mathrm{c}_\infty/k)\rrbracket$-…

Let $F$) be a totally real field, let $p$ be a prime, let $H_{\infty}/F$ be the relevant infinite extension with Galois group $\mathcal{G}$, and let $S$, $S_{ram}$, and $S_p$ denot…

Kala–Yatsyna's conjecture. There is no number field $K$ with a universal $\mathbb{Z}$-form over $\mathcal{O}_K$ except for

Modularity conjecture. Every elliptic curve over a totally real field $F$ is modular.

Serre's modularity conjecture. Every representation $\rho$ of Serre type should be automorphic.

Let ${\mathcal K}$ be the set of totally real number fields. For $K\in{\mathcal K}$, let $D_K$ be its discriminant, let ${\mathcal T}_K$ be the relevant $p$-torsion group, and let…

Representation-number doubling conjecture. Under these hypotheses, such a matrix $T'$ exists and the representation number satisfies the displayed doubling identity. The claim conc…

Let ${\mathcal K}_{\rm real}$ be the set of all totally real number fields, let $D_K$ be the discriminant of $K$, and fix a prime $p\geq 2$. For each $K$, let ${\mathcal T}_K$ be t…

Vanishing of the Iwasawa $\mu$-invariant. For every totally real field $F$, the Galois group over $F_{\mathrm{cyc}}$ of the maximal abelian $p$-extension of $F_{\mathrm{cyc}}$ unra…

Let $F$ be a totally real number field of degree $n$ over $\mathbb{Q}$, and let $E/F$ be an elliptic curve modular of Shimura level $1$. Let $\delta_E$ be the relevant degree of a…

Weak Serre conjecture. If $\overline{\rho}$ is continuous, irreducible, and totally odd, then it is modular.

Let $K$ be a totally real field, let $G_K$ be its absolute Galois group, and let $\ell$ be a prime. A representation … is called totally odd when it has the required oddness at eve…

Voight's conjecture. The minimal root discriminant among totally real fields of degree $10$ is approximately