68 problems

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

Let $G$ be a nontrivial finite group, and let $r(G)$ denote the least number of places of ${\mathbb Q}$ ramified in a finite Galois extension $K/{\mathbb Q}$ with…

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…

Gross's conjecture. For every prime $p$, there exists a non-solvable number field $K/{\mathbf Q}$ ramified only at $p$ and possibly at $\infty$.

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 $\overline\eta$.…

Let $U_n=\operatorname{Spec}\left(\mathbb{Z}\left[\frac{1}{n}\right]\right)$ for square-free $n\in\mathbb{N}$, let $\pi_A^t(U_n)$ denote the finite groups appearing as Galois group…

Freeness conjecture. The ideal ${\mathcal A}_{E/F}$ is free over the group ring $\mathbb{Z}[G]$.

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 $U\to X$ be an open immersion of schemes over $k$, with $U$ smooth of dimension $d$, and let $\mathcal F$ be a smooth $E$-sheaf on $U$. Define the naive Swan class … and the Sw…

Conjecture on $P_{\mathbb{Q},\{\infty,p,l\},p}$. One has

Generic nondegeneracy conjecture. For generic $z$, all rational functions with prescribed ramification data $z$ and $\{\mathbf w\}$ are nondegenerate.

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…

Realizability conjecture. If $C_1$, $C_2$, and $C_3$ are all 3-derangements, then the branching data $C_1,C_2,C_3$ is realizable.

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 the product of ramified primes be used as the counting invariant for extensions. Product-of-ramified-primes conjecture. This ordering is conjectured to be always suitable for o…

Let $F$ be a number field and $G$ a finite group. A finite extension $K/F$ is ramified over at most a single place when at most one place of $F$ ramifies in $K/F$. One-place ramifi…

Let $F$ be a global field and $G$ a finite group. For a finite extension $K/F$, let the ramification of $K/F$ mean the places of $F$ ramified in $K$, and consider the least possibl…

Let $S$ be the set under consideration, let $p$ be a prime, and let $\mathcal{S}_p$ denote the associated $p$-adic set. A sequence in $\mathcal{S}_p$ is pseudo-divergent when it ha…

Let $F/F_0$ be the ramified quadratic extension, let $V$ be a hermitian space over $F$ of signature $(r,s)$, and let $I$ be a nonempty subset of $\{0,\ldots,\lfloor n/2\rfloor\}$ s…

Let $X\to\mathds{P}^1$ be a finite cover from a hyperelliptic Mumford curve $X$, let $\Omega$ be the universal covering space of $X$, and consider the ramification points of the co…

Consider a five-dimensional $\mathcal{N}=1$ $\mathfrak{g}$-type quiver gauge theory on $S^1_{\mathcal{C}'}\times\mathbb{C}^{\times}_q\times\mathbb{C}_t$, with the origin of…

Infinite wild ramification conjecture. There is at least one prime $\mathfrak{p}$ of $K$ lying above $d$ for which $K_\infty/K$ is infinitely wildly ramified above $\mathfrak{p}$,…