Elliptic modular invariants and numerical ranges

Abstract

We prove the j-invariant of an irreducible elliptic curve associated with a 3-by-3 or 4-by-4 matrix is real and greater than or equal to 1. Additionally, we correct an erroneous statement appearing in Kippenhahn's argument about the boundary generating curves of the numerical ranges of 3 × 3 matrices.

Computing the determinantal representations of hyperbolic forms

“…The second reason is more important from the viewpoint of developing the theory of numerical range. In [3], the authors of this paper proved that any irreducible curve V C (F ) associated with a weighted shift matrix has genus g 1, and in [1], they showed that the j-invariant of an irreducible elliptic curve associated with a 3 × 3 or 4 × 4 matrix is real and greater than or equal to 1. There are many tools for computing Riemann theta functions on a Riemann surface with g = 1.…”

“…where ω 1 is a positive real number and w 2 is a purely imaginary number with ℑ(ω 2 ) > 0 (cf. [1]). The τ -invariant of the curve V C (F ) is defined by τ = ω 2 /ω 1 .…”

“…In the paper [1], the so-called j-invariant of an irreducible elliptic curve associated with the following 4 × 4 matrix is explicitly formulated. The 4 × 4 cyclic weighted shift matrix is…”

On the boundary of rank-knumerical ranges

“…have discovered many interesting properties of the boundary curve of W (A). [4][5][6][7] In 2011, Wu [8] proved the following: Moreover, under any one of the three conditions, α is an eigenvalue of A with algebraic multiplicity larger than its geometric multiplicity. An elementary proof of Wu's result is given by Cheung and Li [9] by considering the zero points of another polynomial…”

Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

“…In [1], the classification of the curve F T (x, y, z) = 0 is applied to categorize the shapes of the numerical ranges of 4 × 4 matrices. In [2] and [3], we discussed rational curves and elliptic curves in the context of numerical ranges. For any pair (x 0 , y 0 ) of real numbers, the algebraic equation F T (x 0 , y 0 , z) = 0 has n real solutions in z because x 0 (T )+y 0 (T ) is a Hermitian matrix.…”