Numerical computation of endomorphism rings of Jacobians

Abstract

We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.

Nonlinear algebra and applications

“…In addition, via the SageMath [235] package RiemannSurfaces [35], we compute a Riemann matrix of the curve, we can then compute the corresponding KP solution (8) by computing the corresponding Riemann theta function via the Julia [28] package Theta.jl [12]. As the Trott curve is an M -curve, i.e., it has the maximum number of ovals (see Figure 2), we can proceed our computations over the real numbers [215].…”

Theta Surfaces

“…This is the dehomogenization of (2) with respect to z. Using the partial derivatives q y (x, y) = x 3 + 3xy 2 + x and q x (x, y) = 3x 2 y + y 3 + y, we compute the differential forms in (5) and (6). We choose to evaluate the integrals in (7) over the lines y = 0 and x = 0.…”

“…The second summand is obtained by integrating over the line x = 0, with parameter y = t, so that q x (0, t) = t (t 2 + 1) in (6). The three coordinates of 2 (p 2 (t)) are found to be…”

“…Notable implementations are the library abelfunctions in SageMath [31], the package algcurves in Maple [11], and the MATLAB code presented in [16]. The software we used for our experiments is the package RiemannSurfaces in SageMath due to Bruin, Sijsling and Zotine [6].…”

Families of (3,3)-split Jacobians

“…Then the principally polarized abelian surface (E 1 × E 2 )/Γ α is isomorphic to the Jacobian of a curve of genus two whose Igusa-Clebsch invariants are as follows: When K is a number field or a finite field of characteristic char(K) > 5, it is possible to construct a genus-2 curve over K with given Igusa-Clebsch invariants (see [7] and [18]). When K is a number field, the recent work of Bruin, Sijsling, and Zotine [6] allows one to verify numerically over C that a curve obtained from Igusa-Clebsch invariants of the form above indeed has a (3, 3)-split Jacobian.…”