The $$\mathrm {al}$$ al function of a cyclic trigonal curve of genus three

Abstract

A cyclic trigonal curve of genus three is a Z 3 Galois cover of P 1 , therefore can be written as a smooth plane curve with equationFollowing Weierstrass for the hyperelliptic case, we define an "al" function for this curve and al (c) r , c = 0, 1, 2, for each one of three particular covers of the Jacobian of the curve, and r = 1, 2, 3, 4 for a finite branchpoint (b r , 0). This generalization of the Jacobi sn, cn, dn functions satisfies the relation:which generalizes sn 2 u + cn 2 u = 1. We also show that this can be viewed as a special case of the Frobenius theta identity. 4 6 9

The sigma function over a family of cyclic trigonal curves with a singular fiber

“…In this section, we review the properties of the sigma function of a non-singular cyclic trigonal curve of genus three following the papers [12,28,29].…”

“…Our strategy is the following: in [12,29], we obtained explicit properties (such as Jacobi inversion formulas that relate transcendental and meromorphic functions) of the sigma function σ Xs for the non-singular curve X s over the punctured disc s ∈ D * ε = D ε \ {0} [12,29]; in [27,22], we analyzed σ X 0 for the normalized curve X 0 of X 0 = X s=0 given by [27,22]. Now we consider the degenerating family of curves X := {(x, y, s) | (x, y) ∈ X s , s ∈ D ε }, we will exhibit the first algebraic de Rham cohomology groups, whose generators are given by first and second-kind differentials, as well as their period matrices, for X s when s ∈ D * ε , and for X 0 .…”

“…Since this degeneration can be regarded as a higher-genus version of the elliptic case, type IV in Kodaira's classifications, we applied the technique of this paper to the case of such type IV degenerate family of elliptic curves, cf. Appendix C. For example, the al function, introduced in [29], plays a crucial role in this paper, its definition is rather elaborate, whereas the al function for the elliptic case is very simple. In particular, Appendix C gives an explicit description of the behavior of sigma for the degeneration of type IV, which had not previously appeared as far as we know.…”

Triangular Schlesinger systems and superelliptic curves

“…In this way we obtain two statements. The first one is a generalization of Theorem 5 for a much larger set of parameters a, b, c than the above discrete set of the form (35) which originates from the algebro-geometric approach of Section 4.1.…”

“…These expressions for the matrix entries b kl i (a) are presented in Theorem 1 from Section 2.1. Superelliptic curves are of much interest nowadays as well as some other related classes of curves, like Z m curves or (m, N )-curves, see [1,2,13,21,30,35,38,41,42,46,47] and references therein. There are some differences and ambiguity across the literature in definitions of these classes.…”