21 problems

Let $Y$ be a small category and let $\mathbf{M}$ be a left Quillen presheaf on $Y$. Suppose that $\mathbf{M}$ satisfies hypothesis (o), introduced in Section 19, which in particula…

Let $Y$ be a category and let $A$ be a presheaf of Segal $n$-categories over $Y^o$. Let $horiz$ be the set of horizontal arrows in $\int_Y A$, namely the arrows from $y$ to…

For an algebraic space $X$, let $\Cmot(X; \Q(r)):=\Gamma(X;\Q(r))$ denote the rational motivic cohomology spectrum with Tate twist $r$. For every integer $r\in\Z$, consider the pre…

Descent conjecture. The descent object $\mathcal{X}/\mathcal{G}$ of $\mathcal{G}$-equivariant sheaves on $\mathcal{X}$ is also preorderlike.

Let $R$ be a discrete valuation ring (DVR) with fraction field $K$, and let $A$ and $B$ be $n\times n$ matrices over $R$. Suppose that $L/K$ is a finite extension and $S$ is the in…

Let $M$ be a closed graded manifold with a weakly involutive cover $M=\bigcup_{i=1}^N C_i$, and let $Y=\bigcup_{i=1}^N Y_i$ be a smooth rigid analytic space over $\Lambda$ with an…

Selmer's conjecture. The second descent excludes an even number of generators.

Descent conjecture. Under these assumptions, the set $X(k)$ is dense in

Let $r$ be odd, and let $\rho$ be the representation of $Sp_{2n+r-2k+1}^{(2r)}({\mathbb A})$ generated by the descent functions defined by the integral construction in the paper. D…

Work in either of the two settings considered in the subsection: the derived analytic or derived algebraic geometric setting with the associated period and Robba rings. Consider pe…

Let $E/\mathbb{Q}$ be an elliptic curve with trivial torsion subgroup, let $\Gamma\subset E(\mathbb{Q})$ be a subgroup of known points, and write $r_{2^n}=\dim_{\mathbb{F}_2}S_n$ f…

Let $X_0(\mathfrak{p}_3)$ and $X_0(\mathfrak{p}_3^2)$ be the indicated Shimura curves with models over $\mathbb{F}_{2^3}$. Descent conjecture. The canonical branched cover … and th…

Let $k$ be a number field and let $X$ be a Kummer surface over $k$. A 2-covering associated to $X$ means one of the 2-coverings of the abelian surfaces associated to $X$. Kummer-su…

Let $E/F'$ be a solvable extension of number fields, and let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ that is $\mathrm{Gal}(E/F')$-invariant.…

Let $S$ be the sphere spectrum and $MU$ the complex cobordism spectrum, with the Hopf–Galois resolution whose terms are $MU\wedge MU^{\wedge q}\simeq MU\wedge BU^q_+$. Applying alg…

Let $N\in\{1,2,3,4\}$, let $\mathcal{A}$ be a complex $N$-Lie conformal superalgebra, and let $\mathcal{L}$ be an $\widehat{\mathcal{S}}/\mathcal{R}$-form of…

Let $G$ be a commutative group variety over $\overline{\mathbb{Q}}$, and let $C$ be a connected proper real-analytic submanifold of $G(\mathbb{C})$. Suppose $C$ contains a subgroup…

Let $A$ be a simple abelian variety over $\overline{\mathbb{Q}}$, and let $\Gamma\subset A(\overline{\mathbb{Q}})$ be a subgroup of positive rank. Let $C$ be the closure of…

Let $A$ be a simple abelian variety over $\overline{\mathbb{Q}}$ of dimension $g$, and let $\Gamma\subset A(\overline{\mathbb{Q}})$ be a subgroup of positive rank. Let $C$ be the c…

Let $C$ be a curve over ${\mathbb Q}$, and let $\operatorname{Sel}^{(n)}(C)$ denote the $n$-Selmer set, consisting of the $n$-coverings of $C$ with points everywhere locally. Finit…

Let $C$ be a genus-one curve over ${\mathbb Q}$ with points everywhere locally, and let…