30 problems

Universality conjecture. The formal group ${\frak F}(x\otimes 1,1\otimes x)$ is the universal object in the category of formal groups over a (topological) Hopf algebra.

Lubin's conjecture. Under these hypotheses,

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}^{\mathrm{ord}}$ be a map from a smooth connected variety, let $x\in X(\mathbb{F})$, and let…

Cartier–Witt divisor conjecture. An $R$-point of $\widehat{\bbF}_1^\mathrm{Syn}$ is given by a formal group over $R$, a Cartier–Witt divisor on this pullback, and an isomorphism ov…

Let $\bbS^{\widehat{\mathbbl{\Delta}}}$ be the complement of the zero section in the universal formal group $\widehat{\bbG}_\mathrm{univ}=\bbS^{\widehat{\calN}}$, regarded as an an…

Let $\bbS^\mathrm{Syn}$ be the syntomic stack, and let $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn}$ be the associated monoid stack. Consider the map from…

Let $\bbS^\mathrm{Syn}$ be the syntomic stack and let $\bbM_{\mathrm{m}\ifthenelse{\equal{\bbS}{}}{}{,\bbS}}^\mathrm{Syn}$ be its associated monoid stack. Let…

Let $\bbS^{\widehat{\calN}}$ be the stack of one-dimensional commutative formal groups with a section, and let…

Let $C:\operatorname{Spec} R\to \mathcal{M}_{cub}$ be a cubic curve, and let $\widehat{C}$ denote its formal completion, a formal group over $R$. Write…

Let $K$ be a finite extension of $\mathbf{Q}_p$, with ring of integers $\mathcal{O}_K$ and maximal ideal $\mathfrak{m}_K$. Let $P,U\in T\cdot\mathcal{O}_K[[T]]$ satisfy … with…

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. Let $\operatorname{Preper}(u)$ denote the preperiodic points of $u$, and let $\Lambda(f)$ denote the zero set of…

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. For a power series $h\in\mathbb{Z}_p[[X]]^d$, let $\Lambda(h)$ denote its zero set. Commuting noninvertible forma…

Let $H_\Sigma$ be the formal group over the stack $\Sigma$ introduced above. An algebraization of $H_\Sigma$ is an isomorphism class of a smooth affine group scheme over $\Sigma$ w…

Formal-group isogeny conjecture. There exists a formal group $F$ with coefficients in $\mathcal{O}_K$, two endomorphisms $f_F$ and $u_F$ of $F$, and a nonzero power series $h$ such…

Let $\formalgroup$ be a formal group over a perfect field $k$. For a simplicial commutative $k$-algebra $A$, let $\theta(A)$ be its underlying $E_\infty$-algebra, and write…

Let $G$ be a semi-abelian variety. Define a special subvariety to be a translate of a semi-abelian subvariety of $G$ by a torsion point, and call the zero-dimensional special subva…

Let $\Gamma$ be a formal group of height $n=(p-1)k$, and let $\mathbb{C}$ denote the complex line bundle appearing in the definition of the orientability order $\Theta$. Orientabil…

Universal strict formal-group conjecture. There exists a universal strict ${\mathcal O}_K^\triangleright$-formal group over some ring $S$.

Let $K/\mathbb{Q}_p$ be a finite field extension with ring of integers $A$, uniformizer $\pi$, and residue field $\mathbb{F}_q$. Ravenel's Local Conjecture. … where $I_n$ is the id…

Let $K/\mathbb{Q}$ be a finite Galois extension with ring of integers $A$. Let $J_n$ denote the minimal $n$-congruing ideal of $A$. Ravenel's Global Conjecture, stronger precise fo…

Let $K/\mathbb{Q}$ be a finite Galois extension with ring of integers $A$. An ideal $I$ of $A$ is $n$-congruing if, for every $a\in A$, there exists $N\in\mathbb{N}$ such that…

Let $K/\mathbb{Q}$ be a finite field extension with ring of integers $A$. For a prime ideal $\mathfrak{p}$ of $A$, write $A_{\mathfrak{p}}$ for the localization of $A$ at…

Lubin's conjecture. Then $g$ is an endomorphism of some Lubin–Tate formal group $F(x,y)$ over ${\mathcal O}_K$, and

Let $A$ be a commutative $L^{\natural}$-algebra, and let the injection in Proposition 6.8m be the functorial map from $\mathcal{CF}_{L/k}(A)$ to the formal group $\hat{G}_3(A)$. Bi…

Let $q$ be a parameter and let the inclusion in Proposition 5.24i be the inclusion of the corresponding formal Galois functors. Equality conjecture. If $q$ is not a root of unity,…