10 problems

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…

${\bf n}_m$-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 $(C,[M])$ 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…

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 $k$ 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…