7 problems

The paper defines the Hasse–Witt invariant of a Calabi–Yau variety in two ways: using the Cartier operator and using Calabi–Yau modular forms. Equivalence conjecture. These two def…

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…

Periodically monodromic asymptotic conjecture.

Basic higher-kernel conjecture. If $m(r,p)=1$, there is a positive integer $N_{d,r}$ such that every basic tower $\mathcal T$ with ramification invariant $d$ has a rational constan…

Basic-tower a-number conjecture. There are a positive integer $N_d$, depending only on $d$ and $p$, and a rational number $c$, depending on the tower, such that

Exact asymptotic formula conjecture. There exist an integer $m>0$, a rational number $\lambda$, and a periodic function $c\colon\mathbf Z/m\mathbf Z\to\mathbf Q$ such that

Asymptotic kernel-dimension conjecture. For each such tower and each positive integer $r$, there are rational constants $a_r,\lambda_r$, a positive integer $m_r$, and a periodic fu…