Third Year Major Oral: On the $a$-number and $p$-rank of superelliptic curves over finite fields

Abstract: For an abelian variety $X$ over $\overline{\mathbb{F}}_p$, the $p$-rank of it is defined to be the integer $f$ such that $p^f$ is the number of $p$-torsion points on $X$. It is known that $f$ lies between 0 and the dimension of $X$, and in 1975, Neal Koblitz proved that in $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties (PPAV) of dimension $g$, the stratum of PPAVs with $p$-rank at most $f$ has pure codimension $g-f$. So, roughly speaking, there are much less PPAVs with lower $p$-rank than the ones with higher $p$-rank.

For a curve $C$ over $\overline{\mathbb{F}}_p$, we define its $p$-rank as the $p$-rank of its Jacobian. The $p$-rank of a curve $C$ is known to be equal to the stable, or semi-simple, rank of the Cartier-Manin matrix, a matrix constructed from the $p$-th power Frobenius operator on $H^1(C, \mathcal{O}_C)$, and the $a$-number of $C$ is defined to be the nullity of this matrix. One may naturally wonder if any statement about $p$-rank and $a$-number that follows Koblitz's heuristic can be made for curves. In this talk, I will discuss relevant results for superelliptic curves, that is, smooth projective curves with an affine model of the form $y^l = f(x)$. 

Advisor: Wanlin Li