Differential forms on the curves associated to Appell-Lauricella hypergeometric series and the Cartier operator on them

Abstract

Archinard studied the curve C over C associated to an Appell-Lauricella hypergeometric series and differential forms on its desingularization. In this paper, firstly as a generalization of Archinard's results, we describe a partial desingularization of C over a perfect field K under a mild condition on its characteristic and the space of global sections of its dualizing sheaf, especially we give an explicit basis of it. Secondly, when the characteristic is positive, we show that the Cartier operator on the space can be defined and describe it in terms of Appell-Lauricella hypergeometric series.

Arithmetic inflection of superelliptic curves

“…Comparing ( 16) against the first item of Theorem 3.2 and substituting ones for instances of D n y x| (0,0) , we now obtain (17) det…”

“…In this section, we further develop this conjectural picture to include atomic inflectionary curves associated to superelliptic Legendre and Weierstrass pencils with affine presentations y n = x a (x − 1) b (x − λ) c and y n = x 3 + λx + 2, respectively. The geometry of superelliptic Legendre pencils is closely linked to algebraic differential equations and hypergeometric series; see, e.g., [12,17]. The conjectural number of singularities (3) of Weierstrass inflectionary curves appears in blue as it is not stated explicitly in our earlier papers [3,6,7,5].…”

Computing the space of differential forms of a plane curve and its Cartier-Manin matrix

“…A matrix representing V with respect to a suitable basis for H 0 ( C, Ω C ) is called a Cartier-Manin matrix for C. Computing a basis of H 0 ( C, Ω C ) and the Cartier-Manin matrix is a very important task both in theory and computation, since they are used to compute various invariants such as a-number, p-rank, and so on for the classification of curves. Indeed, there are many works on this task, e.g., [14], [21], [7], [1], [3], [11], [19], [17]. As in [7], some works were on the first cohomology H 1 ( C, O C ) of the structure sheaf O C , which is the dual notion of the space of regular differential forms and on which the natural action of the Frobenius with respect to a basis is called the Hasse-Witt matrix.…”

“…As for the computation of a basis of H 0 ( C, Ω C ), when C ′ is already nonsingular, it is known that under the assumptions (A1) and (A2) in Section 4 below, (x i y j /(∂F/∂y))dx with 0 ≤ i + j ≤ deg(F ) − 3 form a basis of H 0 ( C, Ω C ) (this fact can be viewed as a particular case of Gorenstein's result described below). On the other hand, when C ′ is singular, such explicit bases are found only for particular cases, e.g., [14], [21] for hyperelliptic curves, [7] for Fermat curves, [19] for superelliptic curves, and [1], [17] for curves associated to Appell-Lauricella hypergeometic series. For example, Sutherland recently constructed an efficient algorithm to compute Cartier-Manin matrices for superelliptic curves in [19], where he found a basis of H 0 ( C, Ω C ) to apply Stöhr-Voloch's formula.…”