9 problems

Let $X$ be a projective variety defined over the finite field $\mathbb F_q$, and suppose that its effective monoid $A^+(X)$ is finitely generated. The height zeta function associat…

Manin's conjecture. As $t\to\infty$, the dominant term of $N_\infty(U(\bold A),t)$ should be asymptotically equivalent to

Let $X$ be a nonsingular projective algebraic variety over a number field $F$, let ${\cal L}=(L,(\|\cdot\|_v)_v)$ be a metrized line bundle on $X$, and let $U\subset X$ be a subvar…

Oblong-surface conjecture. The main heuristic holds with

Let $G$ be an algebraic group over a number field $K$ with a representation $\rho:G\rightarrow GL_{n+1}$, let $U=u_0G\subseteq\mathbb{P}^n(\mathbb{Q})$ for…

In the same quotient setting, let $\fie GK$ denote the set of isomorphism classes of large $G$-fields over $K$, and let $\delta_{F,m}$ be the residue invariant associated with $F$…

Let $V\subset\mathbf P^3\times\mathbf P^3$ be the diagonal cubic family above, let $\mathbf x\in\mathbf P^3$ be such that the fibre $V_{\mathbf x}$ is nonsingular, and let…

Let $V\subset\mathbf P^3\times\mathbf P^3$ be the hypersurface defined by … and let $F$ be the number field under consideration. Its anticanonical line bundle is…

Let $X$ be a Fano variety over a number field, and suppose the height zeta function associated with the anticanonical line bundle $L=\omega_X^{-1}$ has a pole of order $t$ at its l…