Newton polygon elevation (awaiting review)

The Newton polygon $P$ of an abelian variety of dimension $g$ over a finite field is a piecewise linear function satisfying the following conditions:

• Its left endpoint is $(0,0)$ and its right endpoint is $(2g, g)$.
• Its vertices are nonnegative integer lattice points.
• Its vertices are symmetric: $(i,j)$ is a vertex if and only if $(2g-i, g-j)$ is.

We call the set of polygons satisfying these conditions eligible. The elevation of such a polygon is the number of nonnegative lattice points $(x,y)$ with $0 \le x \le g$ lying strictly below $P$. The set of eligible polygons forms a partially ordered set: $P_1 \le P_2$ if every point of $P_1$ is below $P_2$. This poset is catenary: any two maximal chains between two eligible polygons have the same length, the difference in their elevations.

The Newton polygon of an ordinary abelian variety has vertices $(0,0)$, $(g,0)$, and $(2g,g)$ and is the minimal element in this poset, with elevation $0$. The Newton polygon of a supersingular abelian variety has vertices $(0,0)$ and $(2g,g)$ and is the maximal element in this poset, with elevation $\frac{(g+1)^2}{4}$ ($g$ odd) or $\frac{g(g+2)}{4}$ ($g$ even). There is a unique Newton polygon of elevation $1$, with vertices $(0,0)$, $(g-1,0)$, $(g+1,1)$, and $(2g, g)$; abelian varieties with this Newton polygon are called almost ordinary.