Differential operators and hyperelliptic curves over finite fields

Abstract

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over Fp in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g ≥ 2 has level 2. We provide a good number of examples and raise a conjecture.

The level of pairs of polynomials

“…The level of a polynomial has been studied in [1,6,4]. In particular, results were established relating the level of a polynomial defining a (hyper)elliptic curve to p-torsion of the Jacobian; see also Section 2.…”

“…In this case, by either [27, Theorem 2] or [28, Corollary of page 12], H is superspecial and therefore (1) is stratified for any value of a 1 , a 0 : In this case, levelðy 2 z 3 À x 5 À bz 5 Þ ! 3 by [4,Example 4.4]. In contrast, where p 1 ðmod 5Þ (e.g.…”

“…One can check that, under a M€ obius transformation of the form 4 , the hyperelliptic curve H defined by y 2 ¼ ðx À 1Þ 8 À x 8 corresponds to y 2 ¼ x 8 À 1, and therefore both have the same p-rank. As shown in [17,Section 2], H is ordinary if and only if p 1 ðmod 8Þ, and supersingular (that is, its p-rank is 0) if and only if p 7 ðmod 8Þ: In the ordinary case, we know that the level is 2, and at least three in the supersingular (not superspecial) case.…”

The TestIdeals package for Macaulay2

“…One interesting particular case is when K = F p and f is the defining homogeneous polynomial of a hyperelliptic curve of genus g; when g = 1, it was proved in [BDSV15] that the corresponding elliptic curve defined by f is ordinary if and only if its level is 1, (equivalently, if and only if (R/(f )) (x,y,z) is F -pure, ) and supersingular if and only if its level is 2. When the genus is at least 2, level 2 is a necessary (but not sufficient) condition for the curve for being ordinary [BCBFY18]; in this case, one also has that, if the curve is supersingular, then its level has to be at least 3, so the level can always distinguish these two properties in any genus. We illustrate these results by means of the following examples.…”

The level of pairs of polynomials

“…Our next goal is to show that the level of a pair is invariant under coordinate transformations. Although 3.6, 3.7 and 3.8 can also be found in [BCBFY18], we review it here for the convenience of the reader.…”