11 problems

Let $a_{11},a_{22},a_{33},a_{12},a_{13},a_{23},d$ be integers, and consider the generalized Markoff equation … Let $D$ and $A_{11},A_{22},A_{33}$ denote the quantities defined earl…

Let $S(m)=\mathbb{F}_q[x_0,\dots,x_m]$, and let $S_d(m)$ be its degree-$d$ graded component. For $m,d\geq 1$ and $1\leq r\leq \binom{m+d}{d}$, let $e_r(d,m)$ be the maximum number…

Let $e_r(d,m)$ be the maximal number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous degree-$d$ polynomials in $m+1$…

Let $e_r(d,m)$ denote the maximal number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous degree-$d$ polynomials in…

Let $Y$ be a surface satisfying the projective-resolution and congruence assumptions, and let $\widetilde{Y}$ be a fixed minimal smooth model. Suppose that for some…

Let $m\geq 4$ be an even integer, and let $\mathcal{X}$ be a nonsingular hypersurface of degree $d$ in $\mathbb{P}^m$ over $\mathbb{F}_q$. Write $N_q(\mathcal{X})$ for the number o…

Even-dimensional Dwork point count conjecture. The number of points over $\mathbb F_q$ on the Dwork hypersurface can be expressed as

Let $O_6$ be the double octic defined by the equation displayed in the source, and define … For each prime $p>3$, let $a_p$ be the eigenvalue of $T_p$ for the newform of weight $4$…

Let $O_5$ be the double cover of $P^3$ branched over the octic displayed in the source. For each prime $p>2$, let $a_p$ denote the eigenvalue for $T_p$ of the newform of weight $4$…

Let $X\subset\mathbb{P}^{n}$ be a projective algebraic set defined over $\mathbb{F}_{q}$, of dimension $m\geq n/2$ and degree $d\leq q+1$, which is an i.t. complete intersection. W…

Let $X\subset\mathbb{P}^{n}(\mathbb{F}_q)$ be a projective algebraic set of degree $d$ and dimension $s$. Writing $\pi_j=|\mathbb{P}^j(\mathbb{F}_q)|$, Edoukou's point-count conjec…