13 problems

Let $n$ be an integer, let $d$ divide $n$, and suppose that $4n+1$ is prime. The expression $daa+bb$ denotes $d a^2+b^2$, with $a,b$ allowed to be integers or rational numbers. Eul…

Let $f(X)=X^3-X+9$, and for $n\in\mathbb{Z}$ define the integral quadratic forms … The form $q(n)$ has discriminant $4n^3-4n+36$, and $q_0(n)$ is the principal form of the same dis…

Let $\ell_1$ and $\ell_2$ be binary $\mathbb{Z}$-lattices with equal discriminants … Assume that the pair $(\ell_1,\ell_2)$ is buried in a genus of rank $3$. Burying conjecture. Th…

Quadratic successive-maxima conjecture. As $D$ increases from its minimum, the successive maxima of $C_{\varepsilon}(D)$ occur only for prime conductors $f_K=\ell$ with…

Non-genus $\varepsilon$-conjecture. Except for subfamilies of density zero,

Successive-maxima conjecture. (i) As $f$ increases from its minimum, the successive maxima of $C_{\varepsilon}({\mathcal F}_{G,f}^s)$, $C'_{\varepsilon}({\mathcal F}'{}_{G,f}^s)$,…

Let $K$ be a number field of degree $d$, let $H_K$ be its Hilbert class field, and let $H_{K/\mathbb Q}$ be its genus field. Define the exceptional class group by … Let $p$ be a pr…

The $p$-rank $\varepsilon$-conjecture. For $d=p$, this assertion holds for the subfamily of cyclic extensions $F/\mathbb Q$ of degree $p$.

Genus-range classification conjecture. For any $m\in\mathbb N$, there is an integer $\Psi_m\geq\psi_m$ such that

Maximum-genus-range conjecture. The maximum genus range of assembly graphs with $2n$ vertices is

Genus-range characterization. Any DOW whose corresponding graph has genus range $\{0,1\}$ is obtained from the word

Uniform sphere-retraction conjecture. For every integer $d$ there is an integer $m>0$ such that, for every polyhedron $X$ of dimension $d$, there are maps

Bounded genus conjecture. For every positive integer $n$ there is an integer $c_n$ such that