35 problems

Let $\mathfrak p\mid p$ be a place of $F$, with ramification index $e$, residue degree $s$, and inertia group $I_{\mathfrak p}$. Let $W_{\mathfrak p}^?(\rho)$ denote the predicted…

Let $f=\sum a_nq^n$ be an ordinary newform of type $(k,\epsilon)$ for $\Gamma_1(N)$, defined over a field $E$ of characteristic $p$, with $(p,N)=1$, and let … be the Galois represe…

Universality conjecture. Every such curve $C$ is universal.

Let $f$ be a polynomial of degree $k$ on a discrete Riemann surface, and let the ramification number of $f$ at a cycle $\gamma$ be its winding number around the origin, when define…

Let $\oX=X-\ul{x}$, let $\operatorname{LS}^{\operatorname{restr}}_{cG}(\oX)$ and $\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD_{\ul{x}})$ be the restricted moduli stacks of l…

Let $k$ be the base field, let $\mathcal{X}=(X,D)$ be an object of $\category{mSm}$, and let $q\geq 0$. Write $\mathcal{X}^{\circ}$ for the complement of the boundary,…

Let $K$ be a number field, let $p$ be a prime, and let ${\mathcal T}_{S_p}$ denote the relevant tame ramification module for the maximal extension of $K$ unramified outside the pla…

Localized function-field Boston–Markin conjectures. There exists a positive real number $\delta_q^{G,C}$ such that, as $n_1,\dots,n_{d_\lhd(G)}\to\infty$,

Möbius cancellation conjecture. As $X\to\infty$,

Function-field Boston–Markin conjecture. There exists a regular $G$-extension $K/\mathbb{F}_q(T)$, split completely at infinity, such that the number of ramified primes of…

Boston–Markin conjecture. There exists a totally real tamely ramified $G$-extension $K/\mathbb{Q}$ such that the number of finite primes of $\mathbb{Q}$ ramified in $K$ is…

Rarity conjecture. For every $r=0,1,2,\dots$, there exists an integer $r'$ such that

Let $L/K$ be a finite weakly ramified Galois extension of number fields, with $G=\operatorname{Gal}(L/K)$, ring of integers $\mathcal{O}_L$, and different $\mathcal{D}_{L/K}$. When…

Let $G$ be a nontrivial finite group. Write $d(G)$ for the minimal number of generators of $G$, $G^{\mathrm{ab}}$ for its abelianization, and $p(G^{\mathrm{ab}})$ for the subgroup…

Abhyankar's inertia conjecture. The pair $(G,I)$ is realizable if and only if

Let $p$ be a prime, let $q$ be a power of $p$, and let $G$ be a finite group. Write $p(G)$ for the subgroup generated by the $p$-Sylow subgroups of $G$; thus $G$ is cyclic-by-quasi…

Let $\mathcal S$ be a henselian trait with perfect residue field of characteristic $p>0$, with closed point $s$, generic point $\eta$, and geometric generic point $\overline\eta$.…

Deformability in towers conjecture. If $G$ is cyclic, then the natural morphism

Let $F_{\pi}/F$ be a finite Galois-algebra extension of number fields with group $G$. Write $\mathfrak{D}(\pi)$ for its different, and suppose that the square root $A(\pi)$ of…

Let $k$ be a $p$-adic field with residue field $\mathfrak{f}$, and let $\mathcal{W}=\langle \operatorname{Fr}\rangle\ltimes\mathcal{I}$ be its Weil group, with wild inertia subgrou…

Let $X$ be a smooth, irreducible, projective variety over the field of fractions $K$ of a Dedekind domain $R$, let $\varphi:X\to X$ be a PCF morphism defined over $K$, and let…

Let $X$ be a smooth, irreducible, projective variety defined over the field of fractions $K$ of a Dedekind domain $R$. Let $\varphi:X\to X$ be a morphism defined over $K$, let…

Let $S$ be the fixed field of $\ker(\pi)$, identified with $k((\overline{u}))$ for a parameter $\overline{u}$. The action of $G_F$ defines a continuous $k$-linear action $\tau$ of…

Let $p$ be a prime and let $G$ be a nontrivial finite group. Write $G^{ab}$ for the abelianization of $G$, let $p(G)$ be the normal subgroup generated by the elements of $p$-power…

Totally ramified point conjecture. There is a point $\alpha$ in the disk $V$ such that $K(\alpha)/K$ is totally ramified and