20 problems

Positivity conjecture. The invariants

Let $N_q(3)$ denote the maximum number of $F_q$-rational points on a smooth projective curve of genus $3$ over $F_q$, and let $[2\sqrt{q}]$ denote the greatest integer less than or…

Nonexistence conjecture. No genus-$2$ curve over ${\mathbf F}_q$ has characteristic polynomial $g_q$.

Let $\mathcal X$ be a curve defined over $\mathbf F_q$ of genus $g$, and write its zeta function as … If $h(t)=t^{2g}P(t^{-1})$, let $\alpha_1,\ldots,\alpha_{2g}$ be the roots of…

Let $C$ and $D$ be smooth projective curves of equal genus at least $2$ over a finite field ${\mathbb F}$. Embed $C$ into its Jacobian and define its Hilbert class field cover…

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field ${\mathbb F}$, and let $K:= {\mathbb F}(D)$. Write $C$ for the base change…

${\bf n}_m$-derived Riemann hypothesis. The function $\widehat\zeta^{\,({\bf n}_m)}_{X,\Bbb F_q}(s)$ satisfies the Riemann hypothesis: all its zeros lie on

Let $C$ be a curve of genus $g$ over a field of characteristic $p>0$. Write $a(C)$ for the a-number of its Jacobian, and let $r$ be the rank of the Cartier operator on…

Let $q$ be a prime power, let $\gamma$ be a positive integer, and let $g$ be a positive integer. Let $N_q(g,\gamma)$ denote the maximal number of $\mathbb F_q$-rational points on a…

Let $(C,[M])$ and $(C',[M'])$ be smooth, projective, irreducible pointed curves over ${\mathbb F}_q$, with moduli ${\mathfrak m},{\mathfrak m}'$ on $C,C'$ respectively. Suppose the…

Let $(C,[M])$ and $(C',[M'])$ be smooth irreducible projective pointed curves over a finite field ${\mathbb F}_q$. Let ${\mathfrak m},{\mathfrak m}'$ be moduli on $C,C'$ disjoint f…

Let $p$ be a prime and let $C$ and $C'$ be smooth proper irreducible curves of genera $g$ and $g'$, with Newton polygons $P$ and $P'$, respectively. A Newton polygon is determined…

Genus stability conjecture. There is a polynomial $G(x)\in\mathbb{Q}[x]$ of degree at most ${\rm dim}(G)+1$, depending on the tower, such that for all sufficiently large $n$,

Let $X$ and $Y$ be smooth projective curves of genus at least two over a finite field ${\mathbb F}_q$. Let $H_0(X):=X$, let $H_1(X):=H(X)$, and recursively let…

Let $q$ be fixed, let $g$ be the genus of a curve over the finite field $\mathbb{F}_q$, and let $n$ be a possible number of rational points. Consider the fraction of curves of genu…

Rational-point formulation. When $\gcd(k,m)=1$, the curves $D_1$ and $D_k$ have the same number of rational points over ${\mathbb F}_{2^m}$:

Let $C$ be a maximal curve over ${\mathbb F}_{q^2}$, and let $H$ be the Hermitian curve defined by … Here, saying that $C$ is dominated by $H$ means that there is a dominant morphi…

Let $C$ be a maximal curve over ${\mathbb F}_q$, meaning that it reaches the Hasse–Weil upper bound, and let $g$ be its genus. Stichtenoth and Xing's conjecture. Either … or … The…

Let $k$ be a finite field of characteristic $p$, let $U$ be a smooth geometrically connected curve over $k$, and let $\mathscr{F}$ be an irreducible smooth…

Let $Y$ and $Y'$ be curves over the finite field $\mathbb F_q$. For each curve, let $\mathcal T_{Y,\mathbb F_q}^{\operatorname{unr}}$ denote the subtower of exceptional covers unra…