85 problems

Let $p$) be a prime, let $\bar G$ be a pro-$p$ group, let $n$ be a positive integer, and let $\bar\omega\in H^n(\bar G,\mathbb{F}_p)$. For a field $F$ containing a root of unity of…

Let $p$ be a prime with $p\geqslant 3$, let $F^{cyc}$ be the cyclotomic extension of $F$, and let $F^S$ be the maximal extension of $F$ unramified outside the specified set $S$. Fo…

Let $K$ be the imaginary quadratic field and let $M_{\theta\mathbb{Q}_p}(n)$ be the $p$-adic realization of the Hecke-character motive defined in the source, with weight $w$. Let…

Let $F$ be a number field, let $\mathfrak{p}$ be a finite place, and let $F_{\mathfrak{p}}$ be the completion of $F$ at $\mathfrak{p}$ with algebraic closure…

Let $F$ be a number field with algebraic closure $\overline{F}$, let $X$ be a smooth projective variety over $F$, let $p$ be a prime, and let $S$ contain the places above $\infty$…

For each prime $p$, let $V_p:=V_{f,\mathfrak{p}}$ be the $p$-adic Galois representation attached to a newform $f$ of weight $2k-2$, let $T_p:=T_{f,\mathfrak{p}}$ be a stable lattic…

Vanishing conjecture for high tensor powers. One has $Sel^0_T(V_n)=0$ for $n\gg0$. This conjecture concerns the disappearance of globally unramified Selmer classes in sufficiently…

Hesselholt's descent conjecture. For all positive integers $q$, this canonical map is an isomorphism of pro-abelian groups, and the higher continuous cohomology groups of the pro-…

Let $F$ be a number field, let $F_{\infty}$ be its cyclotomic $\mathbf{Z}_{p}$-extension, let $V$ be a critical nearly ordinary Galois representation, and let $A=V/T$ be a torsion…

Let $K$ be a field with valuation ring $V$, residue field $k$, and let $\bar{K}$ be an algebraic closure of $K$ with Galois group $G_K$. Let $\bar{V}$ be the integral closure of…

La conjecture $C_{EP,F}(V)$. Pour tout $\omega\in\Delta_F(V)$, on a

Let $F$ be a field and let $\Theta\subset H^1(G_F,\mathbb Z/2)$ be a two-dimensional subspace. Let $I_\Theta$ denote the ideal generated by $\Theta$ in the cohomology algebra…

Let $\ell$ be a prime number and let $F$ be a field containing a primitive $\ell$th root of unity. For $u\in H^1(G_F,\mathbb Z/\ell)$, let … be its annihilator, and for a cyclic ex…

Let $\mathfrak g$ be the graded ${\mathbb Z}_p$-Lie algebra associated with the weight filtration on the pro-$p$ Galois action, let $\mathcal D$ be Ihara's stable derivation algebr…

Localisation injectivity and vanishing conjecture. For all $0\leq i\leq n$, this map is injective, and

Serre's Conjecture I. Does

Let $k$ be a number field, let $G$ be a transitive permutation group of degree $n$, and let $N\normal G$. Write $q:G\to G/N$ for the quotient map and … for the induced pushforward.…

Let $p$ be a prime, $K$ a number field, $S$ a finite set of primes of $K$, and $\overline{\rho}:G_{K,S}\to \operatorname{GL}_{n}(\mathbb{F})$ a continuous absolutely irreducible re…

Let $\mathcal{K}$ be a finite extension of $\mathbb{Q}_p$, let $\mathcal{O}$ be its valuation ring, and let $\mathbf{A}(\rho)=\mathbf{V}_\rho/\mathbf{T}_\rho$ be the $p$-divisible mod…

Let $k$ be a number field, $G$ a transitive permutation group of degree $n$, and $T\normal G$ a proper normal subgroup. Let $q:G\to G/T$ be the canonical quotient map, let $G_k$ be…

Let $(\operatorname{G},\mathrm{H})$ be a symmetric pair over $k$ such that $\operatorname{G}$ is quasi-split over $k$ and $\operatorname{G}_{\mathrm{der}}$ is simply connected. Set…

Let $F$ be a field, let $d$ be an integer, and let $\ell$ be the prime appearing in the cohomology group. The diophantine dimension $\operatorname{dd}(F)$ is the smallest integer…

Let $q$ be a non-negative integer and let $K$ be a field. Say that $K$ has the $C^q_{\mathrm{sc}}$ property if, for every finite extension $L/K$ and every principal homogeneous spa…

Let $K$ be a field of cohomological dimension at most $2$, and let $Z$ be a principal homogeneous space under a semisimple simply connected $K$-group. Serre's conjecture II. The va…

Weight $k-2$ p-newness conjecture. The class $c^{”}$ is $p$-new for every $p\in\mathcal{S}$. This is motivated by the expectation that the general newform Eisenstein congruence con…